Properties

Label 24-684e12-1.1-c2e12-0-0
Degree $24$
Conductor $1.049\times 10^{34}$
Sign $1$
Analytic cond. $1.75668\times 10^{15}$
Root an. cond. $4.31713$
Motivic weight $2$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 16·7-s − 16·13-s + 144·25-s − 40·31-s − 32·37-s + 92·43-s − 166·49-s − 48·61-s − 88·67-s + 148·73-s − 56·79-s − 256·91-s + 72·97-s − 208·103-s + 384·109-s + 568·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 908·169-s + 173-s + ⋯
L(s)  = 1  + 16/7·7-s − 1.23·13-s + 5.75·25-s − 1.29·31-s − 0.864·37-s + 2.13·43-s − 3.38·49-s − 0.786·61-s − 1.31·67-s + 2.02·73-s − 0.708·79-s − 2.81·91-s + 0.742·97-s − 2.01·103-s + 3.52·109-s + 4.69·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 5.37·169-s + 0.00578·173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{24} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{24} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s+1)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(2^{24} \cdot 3^{24} \cdot 19^{12}\)
Sign: $1$
Analytic conductor: \(1.75668\times 10^{15}\)
Root analytic conductor: \(4.31713\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{684} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 2^{24} \cdot 3^{24} \cdot 19^{12} ,\ ( \ : [1]^{12} ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(5.600093314\)
\(L(\frac12)\) \(\approx\) \(5.600093314\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
19 \( ( 1 - p T^{2} )^{6} \)
good5 \( 1 - 144 T^{2} + 10971 T^{4} - 585416 T^{6} + 4836807 p T^{8} - 806383944 T^{10} + 22143061586 T^{12} - 806383944 p^{4} T^{14} + 4836807 p^{9} T^{16} - 585416 p^{12} T^{18} + 10971 p^{16} T^{20} - 144 p^{20} T^{22} + p^{24} T^{24} \)
7 \( ( 1 - 8 T + 179 T^{2} - 176 p T^{3} + 15947 T^{4} - 94664 T^{5} + 914066 T^{6} - 94664 p^{2} T^{7} + 15947 p^{4} T^{8} - 176 p^{7} T^{9} + 179 p^{8} T^{10} - 8 p^{10} T^{11} + p^{12} T^{12} )^{2} \)
11 \( 1 - 568 T^{2} + 145243 T^{4} - 22836920 T^{6} + 2687288227 T^{8} - 293350971280 T^{10} + 34140265046930 T^{12} - 293350971280 p^{4} T^{14} + 2687288227 p^{8} T^{16} - 22836920 p^{12} T^{18} + 145243 p^{16} T^{20} - 568 p^{20} T^{22} + p^{24} T^{24} \)
13 \( ( 1 + 8 T + 550 T^{2} + 5288 T^{3} + 159399 T^{4} + 1366608 T^{5} + 32302788 T^{6} + 1366608 p^{2} T^{7} + 159399 p^{4} T^{8} + 5288 p^{6} T^{9} + 550 p^{8} T^{10} + 8 p^{10} T^{11} + p^{12} T^{12} )^{2} \)
17 \( 1 - 1432 T^{2} + 1143643 T^{4} - 655305688 T^{6} + 295209942243 T^{8} - 109703420939664 T^{10} + 34406929925155602 T^{12} - 109703420939664 p^{4} T^{14} + 295209942243 p^{8} T^{16} - 655305688 p^{12} T^{18} + 1143643 p^{16} T^{20} - 1432 p^{20} T^{22} + p^{24} T^{24} \)
23 \( 1 - 4552 T^{2} + 10193446 T^{4} - 14740519208 T^{6} + 15265202376847 T^{8} - 11896313415879568 T^{10} + 7143697798842655316 T^{12} - 11896313415879568 p^{4} T^{14} + 15265202376847 p^{8} T^{16} - 14740519208 p^{12} T^{18} + 10193446 p^{16} T^{20} - 4552 p^{20} T^{22} + p^{24} T^{24} \)
29 \( 1 - 4236 T^{2} + 10599858 T^{4} - 18905173916 T^{6} + 26207505332751 T^{8} - 29363209475654040 T^{10} + 27126581834481200828 T^{12} - 29363209475654040 p^{4} T^{14} + 26207505332751 p^{8} T^{16} - 18905173916 p^{12} T^{18} + 10599858 p^{16} T^{20} - 4236 p^{20} T^{22} + p^{24} T^{24} \)
31 \( ( 1 + 20 T + 4558 T^{2} + 75780 T^{3} + 9398519 T^{4} + 128151304 T^{5} + 11406077236 T^{6} + 128151304 p^{2} T^{7} + 9398519 p^{4} T^{8} + 75780 p^{6} T^{9} + 4558 p^{8} T^{10} + 20 p^{10} T^{11} + p^{12} T^{12} )^{2} \)
37 \( ( 1 + 16 T + 2834 T^{2} + 132816 T^{3} + 5608135 T^{4} + 259266080 T^{5} + 10505049484 T^{6} + 259266080 p^{2} T^{7} + 5608135 p^{4} T^{8} + 132816 p^{6} T^{9} + 2834 p^{8} T^{10} + 16 p^{10} T^{11} + p^{12} T^{12} )^{2} \)
41 \( 1 - 12940 T^{2} + 84411010 T^{4} - 363167319260 T^{6} + 1141055976209839 T^{8} - 2749019527045293592 T^{10} + \)\(51\!\cdots\!60\)\( T^{12} - 2749019527045293592 p^{4} T^{14} + 1141055976209839 p^{8} T^{16} - 363167319260 p^{12} T^{18} + 84411010 p^{16} T^{20} - 12940 p^{20} T^{22} + p^{24} T^{24} \)
43 \( ( 1 - 46 T + 4063 T^{2} - 75222 T^{3} + 33665 p T^{4} + 210829108 T^{5} - 9698344598 T^{6} + 210829108 p^{2} T^{7} + 33665 p^{5} T^{8} - 75222 p^{6} T^{9} + 4063 p^{8} T^{10} - 46 p^{10} T^{11} + p^{12} T^{12} )^{2} \)
47 \( 1 - 16864 T^{2} + 141197371 T^{4} - 779609502968 T^{6} + 3161413727801347 T^{8} - 9899882846918052040 T^{10} + \)\(24\!\cdots\!34\)\( T^{12} - 9899882846918052040 p^{4} T^{14} + 3161413727801347 p^{8} T^{16} - 779609502968 p^{12} T^{18} + 141197371 p^{16} T^{20} - 16864 p^{20} T^{22} + p^{24} T^{24} \)
53 \( 1 - 7804 T^{2} + 33518674 T^{4} - 130176848204 T^{6} + 500999063284687 T^{8} - 1626062668517107576 T^{10} + \)\(46\!\cdots\!88\)\( T^{12} - 1626062668517107576 p^{4} T^{14} + 500999063284687 p^{8} T^{16} - 130176848204 p^{12} T^{18} + 33518674 p^{16} T^{20} - 7804 p^{20} T^{22} + p^{24} T^{24} \)
59 \( 1 - 26380 T^{2} + 346619938 T^{4} - 2982250314268 T^{6} + 18759240001228815 T^{8} - 91319178554095057944 T^{10} + \)\(35\!\cdots\!76\)\( T^{12} - 91319178554095057944 p^{4} T^{14} + 18759240001228815 p^{8} T^{16} - 2982250314268 p^{12} T^{18} + 346619938 p^{16} T^{20} - 26380 p^{20} T^{22} + p^{24} T^{24} \)
61 \( ( 1 + 24 T + 9939 T^{2} - 97960 T^{3} + 41328315 T^{4} - 2132875440 T^{5} + 123276967602 T^{6} - 2132875440 p^{2} T^{7} + 41328315 p^{4} T^{8} - 97960 p^{6} T^{9} + 9939 p^{8} T^{10} + 24 p^{10} T^{11} + p^{12} T^{12} )^{2} \)
67 \( ( 1 + 44 T + 19630 T^{2} + 666172 T^{3} + 170659231 T^{4} + 4545003704 T^{5} + 13760742380 p T^{6} + 4545003704 p^{2} T^{7} + 170659231 p^{4} T^{8} + 666172 p^{6} T^{9} + 19630 p^{8} T^{10} + 44 p^{10} T^{11} + p^{12} T^{12} )^{2} \)
71 \( 1 - 31852 T^{2} + 501742210 T^{4} - 5304357659708 T^{6} + 42596325964990255 T^{8} - \)\(27\!\cdots\!40\)\( T^{10} + \)\(15\!\cdots\!32\)\( T^{12} - \)\(27\!\cdots\!40\)\( p^{4} T^{14} + 42596325964990255 p^{8} T^{16} - 5304357659708 p^{12} T^{18} + 501742210 p^{16} T^{20} - 31852 p^{20} T^{22} + p^{24} T^{24} \)
73 \( ( 1 - 74 T + 23975 T^{2} - 1469042 T^{3} + 277629131 T^{4} - 13834699364 T^{5} + 1877303415770 T^{6} - 13834699364 p^{2} T^{7} + 277629131 p^{4} T^{8} - 1469042 p^{6} T^{9} + 23975 p^{8} T^{10} - 74 p^{10} T^{11} + p^{12} T^{12} )^{2} \)
79 \( ( 1 + 28 T + 11522 T^{2} + 419276 T^{3} + 47869143 T^{4} + 4520058168 T^{5} + 123047457132 T^{6} + 4520058168 p^{2} T^{7} + 47869143 p^{4} T^{8} + 419276 p^{6} T^{9} + 11522 p^{8} T^{10} + 28 p^{10} T^{11} + p^{12} T^{12} )^{2} \)
83 \( 1 - 52192 T^{2} + 1290888646 T^{4} - 20657183459680 T^{6} + 245496643899605871 T^{8} - \)\(23\!\cdots\!44\)\( T^{10} + \)\(17\!\cdots\!12\)\( T^{12} - \)\(23\!\cdots\!44\)\( p^{4} T^{14} + 245496643899605871 p^{8} T^{16} - 20657183459680 p^{12} T^{18} + 1290888646 p^{16} T^{20} - 52192 p^{20} T^{22} + p^{24} T^{24} \)
89 \( 1 - 43372 T^{2} + 969200722 T^{4} - 15596837215420 T^{6} + 199693920360164079 T^{8} - \)\(20\!\cdots\!04\)\( T^{10} + \)\(17\!\cdots\!68\)\( T^{12} - \)\(20\!\cdots\!04\)\( p^{4} T^{14} + 199693920360164079 p^{8} T^{16} - 15596837215420 p^{12} T^{18} + 969200722 p^{16} T^{20} - 43372 p^{20} T^{22} + p^{24} T^{24} \)
97 \( ( 1 - 36 T + 47106 T^{2} - 1757076 T^{3} + 982957935 T^{4} - 33646593864 T^{5} + 11821752170524 T^{6} - 33646593864 p^{2} T^{7} + 982957935 p^{4} T^{8} - 1757076 p^{6} T^{9} + 47106 p^{8} T^{10} - 36 p^{10} T^{11} + p^{12} T^{12} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−3.11934220337638689486526870576, −3.08144792923977352527011556521, −2.84502116866579817883467014236, −2.81376600500207418615330310219, −2.78469577584006922933283510414, −2.77110265359096898211499835035, −2.52723009891828486767925363507, −2.36514199342682211766949383064, −2.35269864540260126078627991704, −2.27786050417900066606390341237, −1.97963095329113809115040010902, −1.95829754866115195545942750463, −1.80957691891340475328426439226, −1.68450523489865902457231668626, −1.56571743584623185652863554662, −1.44808807953220604856563984371, −1.44752628328879239752694722921, −1.30707181656737745187155756851, −1.11665968961109240194198091715, −0.900649245790446282441797890995, −0.76477416333816228785994789530, −0.60854912148259470129057392168, −0.53613223488668531145845719864, −0.36238574840059989202123130944, −0.088470561905602449401428179451, 0.088470561905602449401428179451, 0.36238574840059989202123130944, 0.53613223488668531145845719864, 0.60854912148259470129057392168, 0.76477416333816228785994789530, 0.900649245790446282441797890995, 1.11665968961109240194198091715, 1.30707181656737745187155756851, 1.44752628328879239752694722921, 1.44808807953220604856563984371, 1.56571743584623185652863554662, 1.68450523489865902457231668626, 1.80957691891340475328426439226, 1.95829754866115195545942750463, 1.97963095329113809115040010902, 2.27786050417900066606390341237, 2.35269864540260126078627991704, 2.36514199342682211766949383064, 2.52723009891828486767925363507, 2.77110265359096898211499835035, 2.78469577584006922933283510414, 2.81376600500207418615330310219, 2.84502116866579817883467014236, 3.08144792923977352527011556521, 3.11934220337638689486526870576

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

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