Properties

Degree 48
Conductor $ 3^{24} \cdot 19^{24} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 9·3-s − 9·4-s − 6·7-s + 42·9-s + 81·12-s + 6·13-s + 39·16-s − 12·19-s + 54·21-s + 6·25-s − 144·27-s + 54·28-s − 18·31-s − 378·36-s − 54·39-s + 54·43-s − 351·48-s + 63·49-s − 54·52-s + 108·57-s − 6·61-s − 252·63-s − 107·64-s + 90·73-s − 54·75-s + 108·76-s + 30·79-s + ⋯
L(s)  = 1  − 5.19·3-s − 9/2·4-s − 2.26·7-s + 14·9-s + 23.3·12-s + 1.66·13-s + 39/4·16-s − 2.75·19-s + 11.7·21-s + 6/5·25-s − 27.7·27-s + 10.2·28-s − 3.23·31-s − 63·36-s − 8.64·39-s + 8.23·43-s − 50.6·48-s + 9·49-s − 7.48·52-s + 14.3·57-s − 0.768·61-s − 31.7·63-s − 13.3·64-s + 10.5·73-s − 6.23·75-s + 12.3·76-s + 3.37·79-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut &\left(3^{24} \cdot 19^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{24} \, L(s)\cr =\mathstrut & \,\Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut &\left(3^{24} \cdot 19^{24}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{24} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(48\)
\( N \)  =  \(3^{24} \cdot 19^{24}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{57} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(48,\ 3^{24} \cdot 19^{24} ,\ ( \ : [1/2]^{24} ),\ 1 )$
$L(1)$  $\approx$  $0.00212561$
$L(\frac12)$  $\approx$  $0.00212561$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;19\}$, \(F_p\) is a polynomial of degree 48. If $p \in \{3,\;19\}$, then $F_p$ is a polynomial of degree at most 47.
$p$$F_p$
bad3 \( 1 + p^{2} T + 13 p T^{2} + 13 p^{2} T^{3} + 98 p T^{4} + 76 p^{2} T^{5} + 1538 T^{6} + 125 p^{3} T^{7} + 2377 p T^{8} + 1586 p^{2} T^{9} + 9029 p T^{10} + 203 p^{5} T^{11} + 86851 T^{12} + 203 p^{6} T^{13} + 9029 p^{3} T^{14} + 1586 p^{5} T^{15} + 2377 p^{5} T^{16} + 125 p^{8} T^{17} + 1538 p^{6} T^{18} + 76 p^{9} T^{19} + 98 p^{9} T^{20} + 13 p^{11} T^{21} + 13 p^{11} T^{22} + p^{13} T^{23} + p^{12} T^{24} \)
19 \( ( 1 + 6 T + 36 T^{2} + 132 T^{3} + 738 T^{4} + 4650 T^{5} + 21761 T^{6} + 4650 p T^{7} + 738 p^{2} T^{8} + 132 p^{3} T^{9} + 36 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
good2 \( 1 + 9 T^{2} + 21 p T^{4} + 67 p T^{6} + 321 T^{8} + 315 p T^{10} + 1093 T^{12} + 1665 T^{14} + 243 p^{3} T^{16} + 1247 T^{18} + 27 p^{2} T^{20} + 951 p T^{22} + 7945 T^{24} + 951 p^{3} T^{26} + 27 p^{6} T^{28} + 1247 p^{6} T^{30} + 243 p^{11} T^{32} + 1665 p^{10} T^{34} + 1093 p^{12} T^{36} + 315 p^{15} T^{38} + 321 p^{16} T^{40} + 67 p^{19} T^{42} + 21 p^{21} T^{44} + 9 p^{22} T^{46} + p^{24} T^{48} \)
5 \( 1 - 6 T^{2} + 6 T^{4} + 136 T^{6} - 1074 T^{8} - 1068 T^{10} + 5213 p T^{12} - 153882 T^{14} - 196983 T^{16} + 4270354 T^{18} - 10566177 T^{20} - 22411704 T^{22} + 533547889 T^{24} - 22411704 p^{2} T^{26} - 10566177 p^{4} T^{28} + 4270354 p^{6} T^{30} - 196983 p^{8} T^{32} - 153882 p^{10} T^{34} + 5213 p^{13} T^{36} - 1068 p^{14} T^{38} - 1074 p^{16} T^{40} + 136 p^{18} T^{42} + 6 p^{20} T^{44} - 6 p^{22} T^{46} + p^{24} T^{48} \)
7 \( ( 1 + 3 T - 18 T^{2} - 31 T^{3} + 207 T^{4} + 30 T^{5} - 2102 T^{6} - 981 T^{7} + 14661 T^{8} + 12452 T^{9} - 18687 p T^{10} - 25335 T^{11} + 1212499 T^{12} - 25335 p T^{13} - 18687 p^{3} T^{14} + 12452 p^{3} T^{15} + 14661 p^{4} T^{16} - 981 p^{5} T^{17} - 2102 p^{6} T^{18} + 30 p^{7} T^{19} + 207 p^{8} T^{20} - 31 p^{9} T^{21} - 18 p^{10} T^{22} + 3 p^{11} T^{23} + p^{12} T^{24} )^{2} \)
11 \( 1 + 75 T^{2} + 2694 T^{4} + 65653 T^{6} + 1298559 T^{8} + 22471266 T^{10} + 345751618 T^{12} + 4859483697 T^{14} + 64669896495 T^{16} + 6799769704 p^{2} T^{18} + 10006559574339 T^{20} + 117281480137203 T^{22} + 1320841962229513 T^{24} + 117281480137203 p^{2} T^{26} + 10006559574339 p^{4} T^{28} + 6799769704 p^{8} T^{30} + 64669896495 p^{8} T^{32} + 4859483697 p^{10} T^{34} + 345751618 p^{12} T^{36} + 22471266 p^{14} T^{38} + 1298559 p^{16} T^{40} + 65653 p^{18} T^{42} + 2694 p^{20} T^{44} + 75 p^{22} T^{46} + p^{24} T^{48} \)
13 \( ( 1 - 3 T + 24 T^{2} - 24 T^{3} + 153 T^{4} + 159 T^{5} + 2212 T^{6} + 1980 T^{7} + 44301 T^{8} + 47100 T^{9} + 522177 T^{10} + 500616 T^{11} + 5728113 T^{12} + 500616 p T^{13} + 522177 p^{2} T^{14} + 47100 p^{3} T^{15} + 44301 p^{4} T^{16} + 1980 p^{5} T^{17} + 2212 p^{6} T^{18} + 159 p^{7} T^{19} + 153 p^{8} T^{20} - 24 p^{9} T^{21} + 24 p^{10} T^{22} - 3 p^{11} T^{23} + p^{12} T^{24} )^{2} \)
17 \( 1 - 69 T^{2} + 3378 T^{4} - 123487 T^{6} + 3911979 T^{8} - 107605653 T^{10} + 2696625433 T^{12} - 61660751742 T^{14} + 1320300336015 T^{16} - 26463229527589 T^{18} + 29691124830003 p T^{20} - 9144820731031068 T^{22} + 159116471027390479 T^{24} - 9144820731031068 p^{2} T^{26} + 29691124830003 p^{5} T^{28} - 26463229527589 p^{6} T^{30} + 1320300336015 p^{8} T^{32} - 61660751742 p^{10} T^{34} + 2696625433 p^{12} T^{36} - 107605653 p^{14} T^{38} + 3911979 p^{16} T^{40} - 123487 p^{18} T^{42} + 3378 p^{20} T^{44} - 69 p^{22} T^{46} + p^{24} T^{48} \)
23 \( 1 + 33 T^{2} - 72 p T^{4} - 43934 T^{6} + 2205477 T^{8} + 37186647 T^{10} - 2055901568 T^{12} - 19262923632 T^{14} + 1522824630273 T^{16} + 7117383776656 T^{18} - 916193620186959 T^{20} - 1149641221063212 T^{22} + 503902625739574717 T^{24} - 1149641221063212 p^{2} T^{26} - 916193620186959 p^{4} T^{28} + 7117383776656 p^{6} T^{30} + 1522824630273 p^{8} T^{32} - 19262923632 p^{10} T^{34} - 2055901568 p^{12} T^{36} + 37186647 p^{14} T^{38} + 2205477 p^{16} T^{40} - 43934 p^{18} T^{42} - 72 p^{21} T^{44} + 33 p^{22} T^{46} + p^{24} T^{48} \)
29 \( 1 - 99 T^{2} + 4992 T^{4} - 195868 T^{6} + 7379160 T^{8} - 248063697 T^{10} + 7160526256 T^{12} - 201232695384 T^{14} + 6286176323784 T^{16} - 206487578099362 T^{18} + 6661892311584336 T^{20} - 212891682345566460 T^{22} + 6469497331402979491 T^{24} - 212891682345566460 p^{2} T^{26} + 6661892311584336 p^{4} T^{28} - 206487578099362 p^{6} T^{30} + 6286176323784 p^{8} T^{32} - 201232695384 p^{10} T^{34} + 7160526256 p^{12} T^{36} - 248063697 p^{14} T^{38} + 7379160 p^{16} T^{40} - 195868 p^{18} T^{42} + 4992 p^{20} T^{44} - 99 p^{22} T^{46} + p^{24} T^{48} \)
31 \( ( 1 + 9 T + 6 p T^{2} + 1431 T^{3} + 17559 T^{4} + 116496 T^{5} + 1115116 T^{6} + 6590691 T^{7} + 1746849 p T^{8} + 292205718 T^{9} + 2157977877 T^{10} + 10730167623 T^{11} + 72603978219 T^{12} + 10730167623 p T^{13} + 2157977877 p^{2} T^{14} + 292205718 p^{3} T^{15} + 1746849 p^{5} T^{16} + 6590691 p^{5} T^{17} + 1115116 p^{6} T^{18} + 116496 p^{7} T^{19} + 17559 p^{8} T^{20} + 1431 p^{9} T^{21} + 6 p^{11} T^{22} + 9 p^{11} T^{23} + p^{12} T^{24} )^{2} \)
37 \( ( 1 - 288 T^{2} + 40500 T^{4} - 3694102 T^{6} + 245452068 T^{8} - 341374140 p T^{10} + 519878846091 T^{12} - 341374140 p^{3} T^{14} + 245452068 p^{4} T^{16} - 3694102 p^{6} T^{18} + 40500 p^{8} T^{20} - 288 p^{10} T^{22} + p^{12} T^{24} )^{2} \)
41 \( 1 - 93 T^{2} + 147 p T^{4} - 300610 T^{6} + 15353097 T^{8} - 583462713 T^{10} + 20159081857 T^{12} - 724569793578 T^{14} + 37029223955898 T^{16} - 1786117305603274 T^{18} + 95160087491005398 T^{20} - 4863589436492931732 T^{22} + \)\(22\!\cdots\!80\)\( T^{24} - 4863589436492931732 p^{2} T^{26} + 95160087491005398 p^{4} T^{28} - 1786117305603274 p^{6} T^{30} + 37029223955898 p^{8} T^{32} - 724569793578 p^{10} T^{34} + 20159081857 p^{12} T^{36} - 583462713 p^{14} T^{38} + 15353097 p^{16} T^{40} - 300610 p^{18} T^{42} + 147 p^{21} T^{44} - 93 p^{22} T^{46} + p^{24} T^{48} \)
43 \( ( 1 - 27 T + 330 T^{2} - 1828 T^{3} - 1695 T^{4} + 106821 T^{5} - 794246 T^{6} + 2474712 T^{7} + 7698843 T^{8} - 142539160 T^{9} + 905672439 T^{10} - 2526983520 T^{11} + 4261600267 T^{12} - 2526983520 p T^{13} + 905672439 p^{2} T^{14} - 142539160 p^{3} T^{15} + 7698843 p^{4} T^{16} + 2474712 p^{5} T^{17} - 794246 p^{6} T^{18} + 106821 p^{7} T^{19} - 1695 p^{8} T^{20} - 1828 p^{9} T^{21} + 330 p^{10} T^{22} - 27 p^{11} T^{23} + p^{12} T^{24} )^{2} \)
47 \( 1 + 138 T^{2} + 7572 T^{4} - 38306 T^{6} - 22914762 T^{8} - 783505734 T^{10} + 34453373350 T^{12} + 3500681070681 T^{14} + 81786603926376 T^{16} - 2428850433482510 T^{18} - 169721165196903795 T^{20} + 983943932988432855 T^{22} + \)\(30\!\cdots\!69\)\( T^{24} + 983943932988432855 p^{2} T^{26} - 169721165196903795 p^{4} T^{28} - 2428850433482510 p^{6} T^{30} + 81786603926376 p^{8} T^{32} + 3500681070681 p^{10} T^{34} + 34453373350 p^{12} T^{36} - 783505734 p^{14} T^{38} - 22914762 p^{16} T^{40} - 38306 p^{18} T^{42} + 7572 p^{20} T^{44} + 138 p^{22} T^{46} + p^{24} T^{48} \)
53 \( 1 + 249 T^{2} + 39441 T^{4} + 4815556 T^{6} + 497858907 T^{8} + 45463220709 T^{10} + 3748987222471 T^{12} + 283322157570882 T^{14} + 19845468367702974 T^{16} + 1298079639719427616 T^{18} + 79710021912349022082 T^{20} + \)\(46\!\cdots\!36\)\( T^{22} + \)\(25\!\cdots\!40\)\( T^{24} + \)\(46\!\cdots\!36\)\( p^{2} T^{26} + 79710021912349022082 p^{4} T^{28} + 1298079639719427616 p^{6} T^{30} + 19845468367702974 p^{8} T^{32} + 283322157570882 p^{10} T^{34} + 3748987222471 p^{12} T^{36} + 45463220709 p^{14} T^{38} + 497858907 p^{16} T^{40} + 4815556 p^{18} T^{42} + 39441 p^{20} T^{44} + 249 p^{22} T^{46} + p^{24} T^{48} \)
59 \( 1 - 84 T^{2} + 11136 T^{4} - 659473 T^{6} + 52026123 T^{8} - 2015195853 T^{10} + 119175559654 T^{12} - 5662799821827 T^{14} + 362426766850593 T^{16} - 607655636974952 p T^{18} + 2096182407219853863 T^{20} - \)\(19\!\cdots\!45\)\( T^{22} + \)\(90\!\cdots\!81\)\( T^{24} - \)\(19\!\cdots\!45\)\( p^{2} T^{26} + 2096182407219853863 p^{4} T^{28} - 607655636974952 p^{7} T^{30} + 362426766850593 p^{8} T^{32} - 5662799821827 p^{10} T^{34} + 119175559654 p^{12} T^{36} - 2015195853 p^{14} T^{38} + 52026123 p^{16} T^{40} - 659473 p^{18} T^{42} + 11136 p^{20} T^{44} - 84 p^{22} T^{46} + p^{24} T^{48} \)
61 \( ( 1 + 3 T + 84 T^{2} - 218 T^{3} + 3972 T^{4} - 61737 T^{5} + 153490 T^{6} - 6838092 T^{7} + 12127752 T^{8} - 283195976 T^{9} + 3001052298 T^{10} - 9112676748 T^{11} + 239421388351 T^{12} - 9112676748 p T^{13} + 3001052298 p^{2} T^{14} - 283195976 p^{3} T^{15} + 12127752 p^{4} T^{16} - 6838092 p^{5} T^{17} + 153490 p^{6} T^{18} - 61737 p^{7} T^{19} + 3972 p^{8} T^{20} - 218 p^{9} T^{21} + 84 p^{10} T^{22} + 3 p^{11} T^{23} + p^{12} T^{24} )^{2} \)
67 \( ( 1 - 48 T^{2} + 414 T^{3} + 12156 T^{4} - 2574 T^{5} - 428894 T^{6} + 3968190 T^{7} + 76435956 T^{8} + 20296044 T^{9} - 1751196492 T^{10} + 170917146 p T^{11} + 375429646395 T^{12} + 170917146 p^{2} T^{13} - 1751196492 p^{2} T^{14} + 20296044 p^{3} T^{15} + 76435956 p^{4} T^{16} + 3968190 p^{5} T^{17} - 428894 p^{6} T^{18} - 2574 p^{7} T^{19} + 12156 p^{8} T^{20} + 414 p^{9} T^{21} - 48 p^{10} T^{22} + p^{12} T^{24} )^{2} \)
71 \( 1 - 282 T^{2} + 32736 T^{4} - 2370872 T^{6} + 220790298 T^{8} - 26938952094 T^{10} + 2442496542598 T^{12} - 177925947913872 T^{14} + 14785877597531274 T^{16} - 1309308608276156072 T^{18} + 99675617883788734818 T^{20} - \)\(68\!\cdots\!60\)\( T^{22} + \)\(48\!\cdots\!95\)\( T^{24} - \)\(68\!\cdots\!60\)\( p^{2} T^{26} + 99675617883788734818 p^{4} T^{28} - 1309308608276156072 p^{6} T^{30} + 14785877597531274 p^{8} T^{32} - 177925947913872 p^{10} T^{34} + 2442496542598 p^{12} T^{36} - 26938952094 p^{14} T^{38} + 220790298 p^{16} T^{40} - 2370872 p^{18} T^{42} + 32736 p^{20} T^{44} - 282 p^{22} T^{46} + p^{24} T^{48} \)
73 \( ( 1 - 45 T + 954 T^{2} - 12420 T^{3} + 112869 T^{4} - 885717 T^{5} + 8520496 T^{6} - 99987552 T^{7} + 1075321953 T^{8} - 9943866180 T^{9} + 87957258507 T^{10} - 804217963374 T^{11} + 7161088701801 T^{12} - 804217963374 p T^{13} + 87957258507 p^{2} T^{14} - 9943866180 p^{3} T^{15} + 1075321953 p^{4} T^{16} - 99987552 p^{5} T^{17} + 8520496 p^{6} T^{18} - 885717 p^{7} T^{19} + 112869 p^{8} T^{20} - 12420 p^{9} T^{21} + 954 p^{10} T^{22} - 45 p^{11} T^{23} + p^{12} T^{24} )^{2} \)
79 \( ( 1 - 15 T + 162 T^{2} - 1272 T^{3} + 20784 T^{4} - 124377 T^{5} + 646294 T^{6} + 1584900 T^{7} + 27228702 T^{8} + 757475484 T^{9} - 7939700580 T^{10} + 97343125308 T^{11} - 326207276637 T^{12} + 97343125308 p T^{13} - 7939700580 p^{2} T^{14} + 757475484 p^{3} T^{15} + 27228702 p^{4} T^{16} + 1584900 p^{5} T^{17} + 646294 p^{6} T^{18} - 124377 p^{7} T^{19} + 20784 p^{8} T^{20} - 1272 p^{9} T^{21} + 162 p^{10} T^{22} - 15 p^{11} T^{23} + p^{12} T^{24} )^{2} \)
83 \( 1 + 735 T^{2} + 278802 T^{4} + 73092469 T^{6} + 14921133183 T^{8} + 2529284726598 T^{10} + 370369221557998 T^{12} + 48094693311854745 T^{14} + 5639951469532219875 T^{16} + \)\(60\!\cdots\!52\)\( T^{18} + \)\(59\!\cdots\!95\)\( T^{20} + \)\(55\!\cdots\!91\)\( T^{22} + \)\(47\!\cdots\!01\)\( T^{24} + \)\(55\!\cdots\!91\)\( p^{2} T^{26} + \)\(59\!\cdots\!95\)\( p^{4} T^{28} + \)\(60\!\cdots\!52\)\( p^{6} T^{30} + 5639951469532219875 p^{8} T^{32} + 48094693311854745 p^{10} T^{34} + 370369221557998 p^{12} T^{36} + 2529284726598 p^{14} T^{38} + 14921133183 p^{16} T^{40} + 73092469 p^{18} T^{42} + 278802 p^{20} T^{44} + 735 p^{22} T^{46} + p^{24} T^{48} \)
89 \( 1 + 63 T^{2} - 6030 T^{4} + 796261 T^{6} + 66304323 T^{8} - 2461774077 T^{10} + 935450513407 T^{12} - 25781224826466 T^{14} - 7124524708356927 T^{16} + 1098444669099071821 T^{18} + 2954940430585918905 T^{20} - 45479572090020644220 p T^{22} + 65355776187152044807 p^{2} T^{24} - 45479572090020644220 p^{3} T^{26} + 2954940430585918905 p^{4} T^{28} + 1098444669099071821 p^{6} T^{30} - 7124524708356927 p^{8} T^{32} - 25781224826466 p^{10} T^{34} + 935450513407 p^{12} T^{36} - 2461774077 p^{14} T^{38} + 66304323 p^{16} T^{40} + 796261 p^{18} T^{42} - 6030 p^{20} T^{44} + 63 p^{22} T^{46} + p^{24} T^{48} \)
97 \( ( 1 + 39 T + 612 T^{2} + 5748 T^{3} + 48261 T^{4} + 206265 T^{5} - 4231184 T^{6} - 85255992 T^{7} - 771090135 T^{8} - 4826106264 T^{9} - 4752301719 T^{10} + 484519514904 T^{11} + 7286643045849 T^{12} + 484519514904 p T^{13} - 4752301719 p^{2} T^{14} - 4826106264 p^{3} T^{15} - 771090135 p^{4} T^{16} - 85255992 p^{5} T^{17} - 4231184 p^{6} T^{18} + 206265 p^{7} T^{19} + 48261 p^{8} T^{20} + 5748 p^{9} T^{21} + 612 p^{10} T^{22} + 39 p^{11} T^{23} + p^{12} T^{24} )^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{48} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−4.03065015756233248617212531846, −3.99407877930334346704347257383, −3.98278496552045537555988837595, −3.97175359149696364371862271458, −3.92111462231285298760360254686, −3.85511470514252493513529143072, −3.84256181240069650223152206107, −3.82645612123430162176176594497, −3.76310103476034690224379768737, −3.51545549577397121983584028476, −3.43969106937715938386105244166, −3.38830114165394806824490771880, −3.05437170889461401330356583545, −2.70849373730960705652318018339, −2.63875223321940275010252570548, −2.61813033520753190504151594410, −2.56207921107022638495579854103, −2.54586146430352091830856281159, −2.48419763154511860551357595394, −2.38270525398588913453961998960, −2.06958223671315624321030731545, −1.75617596536792581522094399510, −1.24989845106979644859933641810, −1.21369979428468023501115898218, −0.978855310011888840618543987108, 0.978855310011888840618543987108, 1.21369979428468023501115898218, 1.24989845106979644859933641810, 1.75617596536792581522094399510, 2.06958223671315624321030731545, 2.38270525398588913453961998960, 2.48419763154511860551357595394, 2.54586146430352091830856281159, 2.56207921107022638495579854103, 2.61813033520753190504151594410, 2.63875223321940275010252570548, 2.70849373730960705652318018339, 3.05437170889461401330356583545, 3.38830114165394806824490771880, 3.43969106937715938386105244166, 3.51545549577397121983584028476, 3.76310103476034690224379768737, 3.82645612123430162176176594497, 3.84256181240069650223152206107, 3.85511470514252493513529143072, 3.92111462231285298760360254686, 3.97175359149696364371862271458, 3.98278496552045537555988837595, 3.99407877930334346704347257383, 4.03065015756233248617212531846

Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.