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Properties

Label 12-2e48-1.1-c11e6-0-3
Degree $12$
Conductor $2.815\times 10^{14}$
Sign $1$
Analytic cond. $5.79123\times 10^{13}$
Root an. cond. $14.0248$
Motivic weight $11$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $6$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.25e3·5-s − 4.52e5·9-s − 1.97e6·13-s + 5.65e6·17-s − 1.16e8·25-s − 7.85e7·29-s − 1.87e8·37-s − 7.29e8·41-s − 1.47e9·45-s − 6.53e9·49-s − 1.34e9·53-s + 1.36e10·61-s − 6.42e9·65-s + 2.08e10·73-s + 6.86e10·81-s + 1.83e10·85-s + 1.65e11·89-s − 1.98e11·97-s − 4.40e11·101-s + 6.29e11·109-s − 9.43e11·113-s + 8.92e11·117-s − 2.21e11·121-s + 7.50e10·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  + 0.465·5-s − 2.55·9-s − 1.47·13-s + 0.965·17-s − 2.37·25-s − 0.710·29-s − 0.444·37-s − 0.983·41-s − 1.18·45-s − 3.30·49-s − 0.442·53-s + 2.07·61-s − 0.687·65-s + 1.17·73-s + 2.18·81-s + 0.449·85-s + 3.14·89-s − 2.34·97-s − 4.17·101-s + 3.91·109-s − 4.81·113-s + 3.76·117-s − 0.774·121-s + 0.219·125-s − 0.331·145-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(12-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48}\right)^{s/2} \, \Gamma_{\C}(s+11/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{48}\)
Sign: $1$
Analytic conductor: \(5.79123\times 10^{13}\)
Root analytic conductor: \(14.0248\)
Motivic weight: \(11\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(6\)
Selberg data: \((12,\ 2^{48} ,\ ( \ : [11/2]^{6} ),\ 1 )\)

Particular Values

\(L(6)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
good3 \( 1 + 452258 T^{2} + 5034013909 p^{3} T^{4} + 13250589170612 p^{7} T^{6} + 5034013909 p^{25} T^{8} + 452258 p^{44} T^{10} + p^{66} T^{12} \)
5 \( ( 1 - 1628 T + 12403419 p T^{2} - 13271359448 p^{2} T^{3} + 12403419 p^{12} T^{4} - 1628 p^{22} T^{5} + p^{33} T^{6} )^{2} \)
7 \( 1 + 6538237834 T^{2} + 521970279540139823 p^{2} T^{4} + \)\(25\!\cdots\!04\)\( p^{4} T^{6} + 521970279540139823 p^{24} T^{8} + 6538237834 p^{44} T^{10} + p^{66} T^{12} \)
11 \( 1 + 221068875442 T^{2} + \)\(12\!\cdots\!59\)\( T^{4} + \)\(41\!\cdots\!96\)\( T^{6} + \)\(12\!\cdots\!59\)\( p^{22} T^{8} + 221068875442 p^{44} T^{10} + p^{66} T^{12} \)
13 \( ( 1 + 987076 T + 1852730507855 T^{2} - 275678720550858776 T^{3} + 1852730507855 p^{11} T^{4} + 987076 p^{22} T^{5} + p^{33} T^{6} )^{2} \)
17 \( ( 1 - 166202 p T + 34967042051999 T^{2} - \)\(28\!\cdots\!36\)\( T^{3} + 34967042051999 p^{11} T^{4} - 166202 p^{23} T^{5} + p^{33} T^{6} )^{2} \)
19 \( 1 + 88882443036610 T^{2} + \)\(26\!\cdots\!31\)\( T^{4} + \)\(19\!\cdots\!76\)\( T^{6} + \)\(26\!\cdots\!31\)\( p^{22} T^{8} + 88882443036610 p^{44} T^{10} + p^{66} T^{12} \)
23 \( 1 + 2035040686055338 T^{2} + \)\(44\!\cdots\!63\)\( T^{4} - \)\(11\!\cdots\!36\)\( T^{6} + \)\(44\!\cdots\!63\)\( p^{22} T^{8} + 2035040686055338 p^{44} T^{10} + p^{66} T^{12} \)
29 \( ( 1 + 39255300 T + 19628193501315039 T^{2} + \)\(16\!\cdots\!72\)\( T^{3} + 19628193501315039 p^{11} T^{4} + 39255300 p^{22} T^{5} + p^{33} T^{6} )^{2} \)
31 \( 1 + 89673279014756410 T^{2} + \)\(45\!\cdots\!71\)\( T^{4} + \)\(13\!\cdots\!64\)\( T^{6} + \)\(45\!\cdots\!71\)\( p^{22} T^{8} + 89673279014756410 p^{44} T^{10} + p^{66} T^{12} \)
37 \( ( 1 + 93760660 T + 473997403453396711 T^{2} + \)\(37\!\cdots\!52\)\( T^{3} + 473997403453396711 p^{11} T^{4} + 93760660 p^{22} T^{5} + p^{33} T^{6} )^{2} \)
41 \( ( 1 + 364696718 T + 695303423366735703 T^{2} + \)\(43\!\cdots\!76\)\( T^{3} + 695303423366735703 p^{11} T^{4} + 364696718 p^{22} T^{5} + p^{33} T^{6} )^{2} \)
43 \( 1 + 278346256675495666 T^{2} + \)\(35\!\cdots\!87\)\( T^{4} + \)\(14\!\cdots\!36\)\( T^{6} + \)\(35\!\cdots\!87\)\( p^{22} T^{8} + 278346256675495666 p^{44} T^{10} + p^{66} T^{12} \)
47 \( 1 + 6326063597385630298 T^{2} + \)\(25\!\cdots\!95\)\( T^{4} + \)\(76\!\cdots\!60\)\( T^{6} + \)\(25\!\cdots\!95\)\( p^{22} T^{8} + 6326063597385630298 p^{44} T^{10} + p^{66} T^{12} \)
53 \( ( 1 + 672903012 T + 19772902165094161911 T^{2} + \)\(57\!\cdots\!32\)\( T^{3} + 19772902165094161911 p^{11} T^{4} + 672903012 p^{22} T^{5} + p^{33} T^{6} )^{2} \)
59 \( 1 + \)\(12\!\cdots\!50\)\( T^{2} + \)\(79\!\cdots\!91\)\( T^{4} + \)\(30\!\cdots\!96\)\( T^{6} + \)\(79\!\cdots\!91\)\( p^{22} T^{8} + \)\(12\!\cdots\!50\)\( p^{44} T^{10} + p^{66} T^{12} \)
61 \( ( 1 - 6841350540 T + 80478976424907574911 T^{2} - \)\(28\!\cdots\!56\)\( T^{3} + 80478976424907574911 p^{11} T^{4} - 6841350540 p^{22} T^{5} + p^{33} T^{6} )^{2} \)
67 \( 1 + \)\(17\!\cdots\!34\)\( T^{2} + \)\(39\!\cdots\!67\)\( T^{4} + \)\(51\!\cdots\!44\)\( T^{6} + \)\(39\!\cdots\!67\)\( p^{22} T^{8} + \)\(17\!\cdots\!34\)\( p^{44} T^{10} + p^{66} T^{12} \)
71 \( 1 + 2876479608389123750 p T^{2} + \)\(39\!\cdots\!11\)\( T^{4} + \)\(46\!\cdots\!84\)\( T^{6} + \)\(39\!\cdots\!11\)\( p^{22} T^{8} + 2876479608389123750 p^{45} T^{10} + p^{66} T^{12} \)
73 \( ( 1 - 10444877022 T + \)\(79\!\cdots\!47\)\( T^{2} - \)\(52\!\cdots\!40\)\( T^{3} + \)\(79\!\cdots\!47\)\( p^{11} T^{4} - 10444877022 p^{22} T^{5} + p^{33} T^{6} )^{2} \)
79 \( 1 - \)\(12\!\cdots\!42\)\( T^{2} + \)\(37\!\cdots\!79\)\( T^{4} + \)\(17\!\cdots\!04\)\( T^{6} + \)\(37\!\cdots\!79\)\( p^{22} T^{8} - \)\(12\!\cdots\!42\)\( p^{44} T^{10} + p^{66} T^{12} \)
83 \( 1 + \)\(43\!\cdots\!78\)\( T^{2} + \)\(10\!\cdots\!23\)\( T^{4} + \)\(16\!\cdots\!24\)\( T^{6} + \)\(10\!\cdots\!23\)\( p^{22} T^{8} + \)\(43\!\cdots\!78\)\( p^{44} T^{10} + p^{66} T^{12} \)
89 \( ( 1 - 82909938158 T + \)\(10\!\cdots\!47\)\( T^{2} - \)\(46\!\cdots\!24\)\( T^{3} + \)\(10\!\cdots\!47\)\( p^{11} T^{4} - 82909938158 p^{22} T^{5} + p^{33} T^{6} )^{2} \)
97 \( ( 1 + 99193897654 T + \)\(54\!\cdots\!19\)\( T^{2} + \)\(22\!\cdots\!56\)\( T^{3} + \)\(54\!\cdots\!19\)\( p^{11} T^{4} + 99193897654 p^{22} T^{5} + p^{33} T^{6} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−5.33802553776454894351648899396, −4.98445090193227482054675880227, −4.91767420251117003484363966805, −4.74830833290769059810233883262, −4.69203414654114747017094020120, −4.23386842198677122138746908062, −4.18296161876610664163254576558, −3.77419436834114419782678687472, −3.77340051236182758889157724402, −3.50868368104496498224456254440, −3.26571789932817555632874018523, −3.25826847370375443008291360573, −3.10631116826494753054544252488, −2.82097990418468614960719942016, −2.66543586339957802707225488561, −2.20169534392811377701574810828, −2.19470556072620107823373605271, −2.18444665842121470899135888254, −2.11865578105725389039369959857, −1.81877752207899591856128520908, −1.36934952520986085374793217383, −1.24465522926010169484312721802, −1.14765736787594757145361281718, −0.873146782705064245244667114470, −0.820218915217850485777000819775, 0, 0, 0, 0, 0, 0, 0.820218915217850485777000819775, 0.873146782705064245244667114470, 1.14765736787594757145361281718, 1.24465522926010169484312721802, 1.36934952520986085374793217383, 1.81877752207899591856128520908, 2.11865578105725389039369959857, 2.18444665842121470899135888254, 2.19470556072620107823373605271, 2.20169534392811377701574810828, 2.66543586339957802707225488561, 2.82097990418468614960719942016, 3.10631116826494753054544252488, 3.25826847370375443008291360573, 3.26571789932817555632874018523, 3.50868368104496498224456254440, 3.77340051236182758889157724402, 3.77419436834114419782678687472, 4.18296161876610664163254576558, 4.23386842198677122138746908062, 4.69203414654114747017094020120, 4.74830833290769059810233883262, 4.91767420251117003484363966805, 4.98445090193227482054675880227, 5.33802553776454894351648899396

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

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