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Properties

Label 8-75e4-1.1-c11e4-0-2
Degree $8$
Conductor $31640625$
Sign $1$
Analytic cond. $1.10272\times 10^{7}$
Root an. cond. $7.59116$
Motivic weight $11$
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  + 7.20e3·4-s − 1.18e5·9-s + 5.91e5·11-s + 3.06e7·16-s − 3.52e7·19-s + 4.03e8·29-s − 1.42e8·31-s − 8.51e8·36-s − 6.55e8·41-s + 4.26e9·44-s − 3.68e9·49-s + 2.95e9·59-s − 1.65e10·61-s + 9.75e10·64-s − 4.04e10·71-s − 2.54e11·76-s − 4.46e10·79-s + 1.04e10·81-s − 1.58e11·89-s − 6.98e10·99-s − 4.17e11·101-s − 7.84e10·109-s + 2.90e12·116-s − 5.88e10·121-s − 1.02e12·124-s + 127-s + 131-s + ⋯
L(s)  = 1  + 3.51·4-s − 2/3·9-s + 1.10·11-s + 7.30·16-s − 3.26·19-s + 3.65·29-s − 0.891·31-s − 2.34·36-s − 0.883·41-s + 3.89·44-s − 1.86·49-s + 0.538·59-s − 2.50·61-s + 11.3·64-s − 2.65·71-s − 11.4·76-s − 1.63·79-s + 1/3·81-s − 3.00·89-s − 0.737·99-s − 3.95·101-s − 0.488·109-s + 12.8·116-s − 0.206·121-s − 3.13·124-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s+11/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(31640625\)    =    \(3^{4} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(1.10272\times 10^{7}\)
Root analytic conductor: \(7.59116\)
Motivic weight: \(11\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 31640625,\ (\ :11/2, 11/2, 11/2, 11/2),\ 1)\)

Particular Values

\(L(6)\) \(\approx\) \(3.809438895\)
\(L(\frac12)\) \(\approx\) \(3.809438895\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad3$C_2$ \( ( 1 + p^{10} T^{2} )^{2} \)
5 \( 1 \)
good2$D_4\times C_2$ \( 1 - 7207 T^{2} + 332777 p^{6} T^{4} - 7207 p^{22} T^{6} + p^{44} T^{8} \)
7$D_4\times C_2$ \( 1 + 75264516 p^{2} T^{2} + 4527064334200678 p^{4} T^{4} + 75264516 p^{24} T^{6} + p^{44} T^{8} \)
11$D_{4}$ \( ( 1 - 295568 T + 160442914534 T^{2} - 295568 p^{11} T^{3} + p^{22} T^{4} )^{2} \)
13$D_4\times C_2$ \( 1 - 2292597030668 T^{2} + \)\(67\!\cdots\!58\)\( T^{4} - 2292597030668 p^{22} T^{6} + p^{44} T^{8} \)
17$D_4\times C_2$ \( 1 - 94789945385980 T^{2} + \)\(43\!\cdots\!78\)\( T^{4} - 94789945385980 p^{22} T^{6} + p^{44} T^{8} \)
19$D_{4}$ \( ( 1 + 17627976 T + 302871116598118 T^{2} + 17627976 p^{11} T^{3} + p^{22} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 - 1142595191533724 T^{2} + \)\(11\!\cdots\!38\)\( T^{4} - 1142595191533724 p^{22} T^{6} + p^{44} T^{8} \)
29$D_{4}$ \( ( 1 - 201881948 T + 26564448133170958 T^{2} - 201881948 p^{11} T^{3} + p^{22} T^{4} )^{2} \)
31$D_{4}$ \( ( 1 + 71057008 T + 33404626522449662 T^{2} + 71057008 p^{11} T^{3} + p^{22} T^{4} )^{2} \)
37$D_4\times C_2$ \( 1 - 314299519826577836 T^{2} + \)\(37\!\cdots\!42\)\( p^{2} T^{4} - 314299519826577836 p^{22} T^{6} + p^{44} T^{8} \)
41$D_{4}$ \( ( 1 + 327655148 T + 975285625454018902 T^{2} + 327655148 p^{11} T^{3} + p^{22} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 + 1390333460700664660 T^{2} + \)\(21\!\cdots\!98\)\( T^{4} + 1390333460700664660 p^{22} T^{6} + p^{44} T^{8} \)
47$D_4\times C_2$ \( 1 - 7724634446695861180 T^{2} + \)\(27\!\cdots\!18\)\( T^{4} - 7724634446695861180 p^{22} T^{6} + p^{44} T^{8} \)
53$D_4\times C_2$ \( 1 - 23136628073904172204 T^{2} + \)\(27\!\cdots\!58\)\( T^{4} - 23136628073904172204 p^{22} T^{6} + p^{44} T^{8} \)
59$D_{4}$ \( ( 1 - 1478770576 T + 34879178115908872198 T^{2} - 1478770576 p^{11} T^{3} + p^{22} T^{4} )^{2} \)
61$D_{4}$ \( ( 1 + 8264891460 T + 97961878141180941838 T^{2} + 8264891460 p^{11} T^{3} + p^{22} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 - \)\(15\!\cdots\!40\)\( T^{2} + \)\(23\!\cdots\!78\)\( T^{4} - \)\(15\!\cdots\!40\)\( p^{22} T^{6} + p^{44} T^{8} \)
71$D_{4}$ \( ( 1 + 20218888256 T + \)\(25\!\cdots\!26\)\( T^{2} + 20218888256 p^{11} T^{3} + p^{22} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 - \)\(17\!\cdots\!64\)\( T^{2} - \)\(44\!\cdots\!02\)\( T^{4} - \)\(17\!\cdots\!64\)\( p^{22} T^{6} + p^{44} T^{8} \)
79$D_{4}$ \( ( 1 + 22324995440 T + \)\(14\!\cdots\!58\)\( T^{2} + 22324995440 p^{11} T^{3} + p^{22} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - \)\(37\!\cdots\!88\)\( T^{2} + \)\(64\!\cdots\!58\)\( T^{4} - \)\(37\!\cdots\!88\)\( p^{22} T^{6} + p^{44} T^{8} \)
89$D_{4}$ \( ( 1 + 79209683076 T + \)\(58\!\cdots\!98\)\( T^{2} + 79209683076 p^{11} T^{3} + p^{22} T^{4} )^{2} \)
97$D_4\times C_2$ \( 1 - \)\(27\!\cdots\!60\)\( T^{2} + \)\(29\!\cdots\!18\)\( T^{4} - \)\(27\!\cdots\!60\)\( p^{22} T^{6} + p^{44} T^{8} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.241929340180591801774147832486, −8.090156323438697338575153740832, −8.087035350159907845264294793394, −7.10602869238327033259703459156, −7.01409652882461435616368042707, −7.01138350820244042453617161598, −6.57994418812998748581155659262, −6.35360366903336033817774698864, −6.07992754225463552015366862378, −5.98415105949353363813956820952, −5.55793132076411557049227913736, −5.03036382819089020418351893887, −4.44194641141565809357522013916, −4.25094889481221188561230933559, −3.96221341898186269708131758379, −3.09569407100046179688279306752, −3.06182276315773274114416984147, −2.79236759944644210807983095685, −2.62564721354921405290832439307, −2.01993023831638046132753960468, −1.72991967101191072134478993511, −1.46282877435857133469932798388, −1.43033110611313781459876979066, −0.71366662398528229604792584127, −0.15624714658301114180173171977, 0.15624714658301114180173171977, 0.71366662398528229604792584127, 1.43033110611313781459876979066, 1.46282877435857133469932798388, 1.72991967101191072134478993511, 2.01993023831638046132753960468, 2.62564721354921405290832439307, 2.79236759944644210807983095685, 3.06182276315773274114416984147, 3.09569407100046179688279306752, 3.96221341898186269708131758379, 4.25094889481221188561230933559, 4.44194641141565809357522013916, 5.03036382819089020418351893887, 5.55793132076411557049227913736, 5.98415105949353363813956820952, 6.07992754225463552015366862378, 6.35360366903336033817774698864, 6.57994418812998748581155659262, 7.01138350820244042453617161598, 7.01409652882461435616368042707, 7.10602869238327033259703459156, 8.087035350159907845264294793394, 8.090156323438697338575153740832, 8.241929340180591801774147832486

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