Properties

Label 32-21e32-1.1-c3e16-0-0
Degree $32$
Conductor $2.047\times 10^{42}$
Sign $1$
Analytic cond. $4.41442\times 10^{22}$
Root an. cond. $5.10096$
Motivic weight $3$
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  − 16·4-s + 145·16-s + 612·19-s + 490·25-s − 1.12e3·31-s − 1.19e3·37-s + 328·43-s + 1.63e3·61-s + 452·64-s + 308·67-s − 4.06e3·73-s − 9.79e3·76-s − 2.17e3·79-s − 7.84e3·100-s − 1.38e3·103-s − 1.07e3·109-s − 3.20e3·121-s + 1.80e4·124-s + 127-s + 131-s + 137-s + 139-s + 1.91e4·148-s + 149-s + 151-s + 157-s + 163-s + ⋯
L(s)  = 1  − 2·4-s + 2.26·16-s + 7.38·19-s + 3.91·25-s − 6.53·31-s − 5.31·37-s + 1.16·43-s + 3.42·61-s + 0.882·64-s + 0.561·67-s − 6.52·73-s − 14.7·76-s − 3.09·79-s − 7.83·100-s − 1.32·103-s − 0.945·109-s − 2.40·121-s + 13.0·124-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 10.6·148-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 7^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 7^{32}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(3^{32} \cdot 7^{32}\)
Sign: $1$
Analytic conductor: \(4.41442\times 10^{22}\)
Root analytic conductor: \(5.10096\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{441} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 3^{32} \cdot 7^{32} ,\ ( \ : [3/2]^{16} ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(6.615789355\)
\(L(\frac12)\) \(\approx\) \(6.615789355\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + p^{4} T^{2} + 111 T^{4} - 249 p^{2} T^{6} - 22563 T^{8} - 99699 p T^{10} - 3667 p^{6} T^{12} + 1401253 p^{3} T^{14} + 9152001 p^{4} T^{16} + 1401253 p^{9} T^{18} - 3667 p^{18} T^{20} - 99699 p^{19} T^{22} - 22563 p^{24} T^{24} - 249 p^{32} T^{26} + 111 p^{36} T^{28} + p^{46} T^{30} + p^{48} T^{32} \)
5 \( 1 - 98 p T^{2} + 4047 p^{2} T^{4} - 13467734 T^{6} + 2004738389 T^{8} - 76183349844 p T^{10} + 60059957637514 T^{12} - 6902983281820528 T^{14} + 756156244566152106 T^{16} - 6902983281820528 p^{6} T^{18} + 60059957637514 p^{12} T^{20} - 76183349844 p^{19} T^{22} + 2004738389 p^{24} T^{24} - 13467734 p^{30} T^{26} + 4047 p^{38} T^{28} - 98 p^{43} T^{30} + p^{48} T^{32} \)
11 \( 1 + 3202 T^{2} + 1797447 T^{4} - 5608813218 T^{6} - 4531249774539 T^{8} + 15353827063546164 T^{10} + 27034229520225124106 T^{12} + \)\(48\!\cdots\!88\)\( T^{14} - \)\(31\!\cdots\!02\)\( T^{16} + \)\(48\!\cdots\!88\)\( p^{6} T^{18} + 27034229520225124106 p^{12} T^{20} + 15353827063546164 p^{18} T^{22} - 4531249774539 p^{24} T^{24} - 5608813218 p^{30} T^{26} + 1797447 p^{36} T^{28} + 3202 p^{42} T^{30} + p^{48} T^{32} \)
13 \( ( 1 - 4502 T^{2} + 10097461 T^{4} - 24107307194 T^{6} + 60089639013832 T^{8} - 24107307194 p^{6} T^{10} + 10097461 p^{12} T^{12} - 4502 p^{18} T^{14} + p^{24} T^{16} )^{2} \)
17 \( 1 - 16948 T^{2} + 84203208 T^{4} - 399083701352 T^{6} + 6508712666429186 T^{8} - 42615253832024962764 T^{10} + \)\(11\!\cdots\!04\)\( T^{12} - \)\(92\!\cdots\!84\)\( T^{14} + \)\(77\!\cdots\!75\)\( T^{16} - \)\(92\!\cdots\!84\)\( p^{6} T^{18} + \)\(11\!\cdots\!04\)\( p^{12} T^{20} - 42615253832024962764 p^{18} T^{22} + 6508712666429186 p^{24} T^{24} - 399083701352 p^{30} T^{26} + 84203208 p^{36} T^{28} - 16948 p^{42} T^{30} + p^{48} T^{32} \)
19 \( ( 1 - 306 T + 57757 T^{2} - 8122770 T^{3} + 990235135 T^{4} - 107235980892 T^{5} + 10750308055972 T^{6} - 990048157019604 T^{7} + 85251569497003126 T^{8} - 990048157019604 p^{3} T^{9} + 10750308055972 p^{6} T^{10} - 107235980892 p^{9} T^{11} + 990235135 p^{12} T^{12} - 8122770 p^{15} T^{13} + 57757 p^{18} T^{14} - 306 p^{21} T^{15} + p^{24} T^{16} )^{2} \)
23 \( 1 + 54964 T^{2} + 1326151704 T^{4} + 25415665738344 T^{6} + 524305919802157986 T^{8} + \)\(89\!\cdots\!72\)\( T^{10} + \)\(12\!\cdots\!96\)\( T^{12} + \)\(17\!\cdots\!32\)\( T^{14} + \)\(23\!\cdots\!87\)\( T^{16} + \)\(17\!\cdots\!32\)\( p^{6} T^{18} + \)\(12\!\cdots\!96\)\( p^{12} T^{20} + \)\(89\!\cdots\!72\)\( p^{18} T^{22} + 524305919802157986 p^{24} T^{24} + 25415665738344 p^{30} T^{26} + 1326151704 p^{36} T^{28} + 54964 p^{42} T^{30} + p^{48} T^{32} \)
29 \( ( 1 - 102034 T^{2} + 6213617737 T^{4} - 245008733294114 T^{6} + 7064972052959541764 T^{8} - 245008733294114 p^{6} T^{10} + 6213617737 p^{12} T^{12} - 102034 p^{18} T^{14} + p^{24} T^{16} )^{2} \)
31 \( ( 1 + 564 T + 243388 T^{2} + 77468784 T^{3} + 21469133071 T^{4} + 5073861272220 T^{5} + 1094289736794676 T^{6} + 211880592093523344 T^{7} + 38213365513366414528 T^{8} + 211880592093523344 p^{3} T^{9} + 1094289736794676 p^{6} T^{10} + 5073861272220 p^{9} T^{11} + 21469133071 p^{12} T^{12} + 77468784 p^{15} T^{13} + 243388 p^{18} T^{14} + 564 p^{21} T^{15} + p^{24} T^{16} )^{2} \)
37 \( ( 1 + 598 T + 47601 T^{2} - 11729418 T^{3} + 10699876539 T^{4} + 3891656263464 T^{5} + 114219217143080 T^{6} + 66547288562775176 T^{7} + 48227467333244423670 T^{8} + 66547288562775176 p^{3} T^{9} + 114219217143080 p^{6} T^{10} + 3891656263464 p^{9} T^{11} + 10699876539 p^{12} T^{12} - 11729418 p^{15} T^{13} + 47601 p^{18} T^{14} + 598 p^{21} T^{15} + p^{24} T^{16} )^{2} \)
41 \( ( 1 + 148396 T^{2} + 19955386648 T^{4} + 1828971842691940 T^{6} + \)\(14\!\cdots\!50\)\( T^{8} + 1828971842691940 p^{6} T^{10} + 19955386648 p^{12} T^{12} + 148396 p^{18} T^{14} + p^{24} T^{16} )^{2} \)
43 \( ( 1 - 82 T + 122239 T^{2} - 47448038 T^{3} + 6939760430 T^{4} - 47448038 p^{3} T^{5} + 122239 p^{6} T^{6} - 82 p^{9} T^{7} + p^{12} T^{8} )^{4} \)
47 \( 1 - 358312 T^{2} + 90535322508 T^{4} - 16795335458693168 T^{6} + \)\(25\!\cdots\!46\)\( T^{8} - \)\(71\!\cdots\!68\)\( p T^{10} + \)\(39\!\cdots\!64\)\( T^{12} - \)\(42\!\cdots\!76\)\( T^{14} + \)\(45\!\cdots\!75\)\( T^{16} - \)\(42\!\cdots\!76\)\( p^{6} T^{18} + \)\(39\!\cdots\!64\)\( p^{12} T^{20} - \)\(71\!\cdots\!68\)\( p^{19} T^{22} + \)\(25\!\cdots\!46\)\( p^{24} T^{24} - 16795335458693168 p^{30} T^{26} + 90535322508 p^{36} T^{28} - 358312 p^{42} T^{30} + p^{48} T^{32} \)
53 \( 1 + 506674 T^{2} + 85866141819 T^{4} + 10208516436240318 T^{6} + \)\(32\!\cdots\!49\)\( T^{8} + \)\(63\!\cdots\!00\)\( T^{10} + \)\(61\!\cdots\!74\)\( T^{12} + \)\(12\!\cdots\!24\)\( T^{14} + \)\(28\!\cdots\!38\)\( T^{16} + \)\(12\!\cdots\!24\)\( p^{6} T^{18} + \)\(61\!\cdots\!74\)\( p^{12} T^{20} + \)\(63\!\cdots\!00\)\( p^{18} T^{22} + \)\(32\!\cdots\!49\)\( p^{24} T^{24} + 10208516436240318 p^{30} T^{26} + 85866141819 p^{36} T^{28} + 506674 p^{42} T^{30} + p^{48} T^{32} \)
59 \( 1 - 399442 T^{2} + 89994325083 T^{4} - 32591407040941742 T^{6} + \)\(78\!\cdots\!45\)\( T^{8} - \)\(13\!\cdots\!56\)\( T^{10} + \)\(43\!\cdots\!26\)\( T^{12} - \)\(98\!\cdots\!64\)\( T^{14} + \)\(16\!\cdots\!22\)\( T^{16} - \)\(98\!\cdots\!64\)\( p^{6} T^{18} + \)\(43\!\cdots\!26\)\( p^{12} T^{20} - \)\(13\!\cdots\!56\)\( p^{18} T^{22} + \)\(78\!\cdots\!45\)\( p^{24} T^{24} - 32591407040941742 p^{30} T^{26} + 89994325083 p^{36} T^{28} - 399442 p^{42} T^{30} + p^{48} T^{32} \)
61 \( ( 1 - 816 T + 837478 T^{2} - 502269216 T^{3} + 333755818642 T^{4} - 188353178559720 T^{5} + 93910648157586304 T^{6} - 50370689927417233344 T^{7} + \)\(21\!\cdots\!87\)\( T^{8} - 50370689927417233344 p^{3} T^{9} + 93910648157586304 p^{6} T^{10} - 188353178559720 p^{9} T^{11} + 333755818642 p^{12} T^{12} - 502269216 p^{15} T^{13} + 837478 p^{18} T^{14} - 816 p^{21} T^{15} + p^{24} T^{16} )^{2} \)
67 \( ( 1 - 154 T - 1009305 T^{2} + 101871994 T^{3} + 602011276205 T^{4} - 36961847361936 T^{5} - 261850611517253414 T^{6} + 66134569555683020 p T^{7} + \)\(89\!\cdots\!66\)\( T^{8} + 66134569555683020 p^{4} T^{9} - 261850611517253414 p^{6} T^{10} - 36961847361936 p^{9} T^{11} + 602011276205 p^{12} T^{12} + 101871994 p^{15} T^{13} - 1009305 p^{18} T^{14} - 154 p^{21} T^{15} + p^{24} T^{16} )^{2} \)
71 \( ( 1 - 2546056 T^{2} + 2933736697204 T^{4} - 1997708473027518488 T^{6} + \)\(87\!\cdots\!74\)\( T^{8} - 1997708473027518488 p^{6} T^{10} + 2933736697204 p^{12} T^{12} - 2546056 p^{18} T^{14} + p^{24} T^{16} )^{2} \)
73 \( ( 1 + 2034 T + 2450047 T^{2} + 2178403830 T^{3} + 1347031106821 T^{4} + 430919574332148 T^{5} - 113107388808572534 T^{6} - \)\(30\!\cdots\!80\)\( T^{7} - \)\(25\!\cdots\!10\)\( T^{8} - \)\(30\!\cdots\!80\)\( p^{3} T^{9} - 113107388808572534 p^{6} T^{10} + 430919574332148 p^{9} T^{11} + 1347031106821 p^{12} T^{12} + 2178403830 p^{15} T^{13} + 2450047 p^{18} T^{14} + 2034 p^{21} T^{15} + p^{24} T^{16} )^{2} \)
79 \( ( 1 + 1088 T - 269028 T^{2} - 1109682680 T^{3} - 391267378201 T^{4} + 358084200932532 T^{5} + 285753619209423700 T^{6} - 15705523238748286852 T^{7} - \)\(10\!\cdots\!68\)\( T^{8} - 15705523238748286852 p^{3} T^{9} + 285753619209423700 p^{6} T^{10} + 358084200932532 p^{9} T^{11} - 391267378201 p^{12} T^{12} - 1109682680 p^{15} T^{13} - 269028 p^{18} T^{14} + 1088 p^{21} T^{15} + p^{24} T^{16} )^{2} \)
83 \( ( 1 + 2632486 T^{2} + 3636170382349 T^{4} + 3366834686358200962 T^{6} + \)\(22\!\cdots\!84\)\( T^{8} + 3366834686358200962 p^{6} T^{10} + 3636170382349 p^{12} T^{12} + 2632486 p^{18} T^{14} + p^{24} T^{16} )^{2} \)
89 \( 1 - 2548864 T^{2} + 4462766350596 T^{4} - 5306850100326352640 T^{6} + \)\(51\!\cdots\!26\)\( T^{8} - \)\(40\!\cdots\!68\)\( T^{10} + \)\(27\!\cdots\!80\)\( T^{12} - \)\(17\!\cdots\!04\)\( T^{14} + \)\(11\!\cdots\!27\)\( T^{16} - \)\(17\!\cdots\!04\)\( p^{6} T^{18} + \)\(27\!\cdots\!80\)\( p^{12} T^{20} - \)\(40\!\cdots\!68\)\( p^{18} T^{22} + \)\(51\!\cdots\!26\)\( p^{24} T^{24} - 5306850100326352640 p^{30} T^{26} + 4462766350596 p^{36} T^{28} - 2548864 p^{42} T^{30} + p^{48} T^{32} \)
97 \( ( 1 - 2978570 T^{2} + 5898756336049 T^{4} - 7901916867899068322 T^{6} + \)\(83\!\cdots\!24\)\( T^{8} - 7901916867899068322 p^{6} T^{10} + 5898756336049 p^{12} T^{12} - 2978570 p^{18} T^{14} + p^{24} T^{16} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−2.57446855419167866906624838429, −2.51792118825530157534471730859, −2.46307608294943041055149095904, −2.21719447960024873746664939592, −2.16072464017595473136054007757, −2.02601964695031155260099189148, −1.88931978382610265040616871670, −1.87629509431534096101415649260, −1.76638808693315432262294070571, −1.53879571524846070291179282857, −1.49094912627312242960532741941, −1.48584108937301688305168016180, −1.33079457449511523854287290800, −1.32239510987665189875527550431, −1.15526092867928384227814334270, −1.10949179434415875684113430141, −1.02080192005771003167659199757, −0.974010489005056213576194949680, −0.77958743585801421946128310638, −0.64531099126977892958229515523, −0.42148548398819927003681707297, −0.39926928109252492405490738615, −0.30340652502928514009697281045, −0.22707523569578432692393564734, −0.12287582739852679291358856690, 0.12287582739852679291358856690, 0.22707523569578432692393564734, 0.30340652502928514009697281045, 0.39926928109252492405490738615, 0.42148548398819927003681707297, 0.64531099126977892958229515523, 0.77958743585801421946128310638, 0.974010489005056213576194949680, 1.02080192005771003167659199757, 1.10949179434415875684113430141, 1.15526092867928384227814334270, 1.32239510987665189875527550431, 1.33079457449511523854287290800, 1.48584108937301688305168016180, 1.49094912627312242960532741941, 1.53879571524846070291179282857, 1.76638808693315432262294070571, 1.87629509431534096101415649260, 1.88931978382610265040616871670, 2.02601964695031155260099189148, 2.16072464017595473136054007757, 2.21719447960024873746664939592, 2.46307608294943041055149095904, 2.51792118825530157534471730859, 2.57446855419167866906624838429

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

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