Properties

Degree $32$
Conductor $6.158\times 10^{28}$
Sign $1$
Motivic weight $3$
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 + 56·7-s + 145·16-s − 612·19-s + 490·25-s − 896·28-s + 1.12e3·31-s − 1.19e3·37-s + 328·43-s + 1.96e3·49-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 + 8.12e3·112-s − 3.20e3·121-s − 1.80e4·124-s + 127-s + 131-s − 3.42e4·133-s + 137-s + 139-s + ⋯
L(s)  = 1  − 2·4-s + 3.02·7-s + 2.26·16-s − 7.38·19-s + 3.91·25-s − 6.04·28-s + 6.53·31-s − 5.31·37-s + 1.16·43-s + 40/7·49-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 + 6.85·112-s − 2.40·121-s − 13.0·124-s + 0.000698·127-s + 0.000666·131-s − 22.3·133-s + 0.000623·137-s + 0.000610·139-s + ⋯

Functional equation

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

Invariants

Degree: \(32\)
Conductor: \(3^{32} \cdot 7^{16}\)
Sign: $1$
Motivic weight: \(3\)
Character: induced by $\chi_{63} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 3^{32} \cdot 7^{16} ,\ ( \ : [3/2]^{16} ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(0.0395433\)
\(L(\frac12)\) \(\approx\) \(0.0395433\)
\(L(\frac{5}{2})\) 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 \( ( 1 - 4 p T + 4 p^{2} T^{2} + 4 p^{4} T^{3} - 5359 p^{2} T^{4} + 4 p^{7} T^{5} + 4 p^{8} T^{6} - 4 p^{10} T^{7} + p^{12} T^{8} )^{2} \)
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

−4.19985560806196076935180483389, −4.05034657919307656217657668826, −3.90569740179097622457595372378, −3.88228458493218805690499534006, −3.87646189838834203563707491956, −3.87433447861295001991420833508, −3.41706807510489796740302718193, −3.30845585859995166577590503523, −3.12619248864572356698416935020, −2.93193098160778545824400468752, −2.89687491927329116345255401691, −2.75381209992742197203586368941, −2.47647433328230877142271468005, −2.32335623688397647169209057820, −2.30128925902671239552605327053, −2.11424882182833281682433378112, −2.02458764618207795642870856539, −1.88409262890720601702671405750, −1.51051420637737272417373234747, −1.27174938885264029213036465338, −1.21399393135536948119625264470, −0.987657281397556252075140643053, −0.61864738482685430683049847803, −0.55163673812741434068905960815, −0.02081761808774684410761913369, 0.02081761808774684410761913369, 0.55163673812741434068905960815, 0.61864738482685430683049847803, 0.987657281397556252075140643053, 1.21399393135536948119625264470, 1.27174938885264029213036465338, 1.51051420637737272417373234747, 1.88409262890720601702671405750, 2.02458764618207795642870856539, 2.11424882182833281682433378112, 2.30128925902671239552605327053, 2.32335623688397647169209057820, 2.47647433328230877142271468005, 2.75381209992742197203586368941, 2.89687491927329116345255401691, 2.93193098160778545824400468752, 3.12619248864572356698416935020, 3.30845585859995166577590503523, 3.41706807510489796740302718193, 3.87433447861295001991420833508, 3.87646189838834203563707491956, 3.88228458493218805690499534006, 3.90569740179097622457595372378, 4.05034657919307656217657668826, 4.19985560806196076935180483389

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

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