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Properties

Label 8-50e4-1.1-c19e4-0-0
Degree $8$
Conductor $6250000$
Sign $1$
Analytic cond. $1.71328\times 10^{8}$
Root an. cond. $10.6961$
Motivic weight $19$
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  − 5.24e5·4-s + 1.73e9·9-s + 7.09e9·11-s + 2.06e11·16-s − 7.66e11·19-s − 3.33e14·29-s + 8.36e14·31-s − 9.12e14·36-s − 6.25e15·41-s − 3.72e15·44-s + 4.17e16·49-s + 5.09e16·59-s − 1.55e17·61-s − 7.20e16·64-s − 5.88e16·71-s + 4.01e17·76-s + 2.47e18·79-s + 8.99e17·81-s − 8.34e18·89-s + 1.23e19·99-s − 3.73e19·101-s − 1.15e20·109-s + 1.74e20·116-s − 1.11e20·121-s − 4.38e20·124-s + 127-s + 131-s + ⋯
L(s)  = 1  − 4-s + 1.49·9-s + 0.907·11-s + 3/4·16-s − 0.544·19-s − 4.26·29-s + 5.68·31-s − 1.49·36-s − 2.98·41-s − 0.907·44-s + 3.66·49-s + 0.765·59-s − 1.70·61-s − 1/2·64-s − 0.152·71-s + 0.544·76-s + 2.32·79-s + 0.665·81-s − 2.52·89-s + 1.35·99-s − 3.39·101-s − 5.10·109-s + 4.26·116-s − 1.82·121-s − 5.68·124-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6250000 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6250000 ^{s/2} \, \Gamma_{\C}(s+19/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(6250000\)    =    \(2^{4} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(1.71328\times 10^{8}\)
Root analytic conductor: \(10.6961\)
Motivic weight: \(19\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 6250000,\ (\ :19/2, 19/2, 19/2, 19/2),\ 1)\)

Particular Values

\(L(10)\) \(\approx\) \(5.008620131\times10^{-6}\)
\(L(\frac12)\) \(\approx\) \(5.008620131\times10^{-6}\)
\(L(\frac{21}{2})\) 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)$
bad2$C_2$ \( ( 1 + p^{18} T^{2} )^{2} \)
5 \( 1 \)
good3$D_4\times C_2$ \( 1 - 1739709980 T^{2} + 2918124642930982 p^{6} T^{4} - 1739709980 p^{38} T^{6} + p^{76} T^{8} \)
7$D_4\times C_2$ \( 1 - 41738972809617740 T^{2} + \)\(28\!\cdots\!98\)\( p^{4} T^{4} - 41738972809617740 p^{38} T^{6} + p^{76} T^{8} \)
11$D_{4}$ \( ( 1 - 3549480144 T + 6780691163172962306 p T^{2} - 3549480144 p^{19} T^{3} + p^{38} T^{4} )^{2} \)
13$D_4\times C_2$ \( 1 + 2645742594230176660 p^{2} T^{2} + \)\(57\!\cdots\!78\)\( p^{4} T^{4} + 2645742594230176660 p^{40} T^{6} + p^{76} T^{8} \)
17$D_4\times C_2$ \( 1 - \)\(13\!\cdots\!60\)\( p^{2} T^{2} + \)\(81\!\cdots\!58\)\( p^{4} T^{4} - \)\(13\!\cdots\!60\)\( p^{40} T^{6} + p^{76} T^{8} \)
19$D_{4}$ \( ( 1 + 383076809800 T + \)\(37\!\cdots\!58\)\( T^{2} + 383076809800 p^{19} T^{3} + p^{38} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 - \)\(19\!\cdots\!20\)\( T^{2} + \)\(20\!\cdots\!38\)\( T^{4} - \)\(19\!\cdots\!20\)\( p^{38} T^{6} + p^{76} T^{8} \)
29$D_{4}$ \( ( 1 + 166587684602220 T + \)\(15\!\cdots\!38\)\( T^{2} + 166587684602220 p^{19} T^{3} + p^{38} T^{4} )^{2} \)
31$D_{4}$ \( ( 1 - 418151119038664 T + \)\(87\!\cdots\!66\)\( T^{2} - 418151119038664 p^{19} T^{3} + p^{38} T^{4} )^{2} \)
37$D_4\times C_2$ \( 1 - \)\(14\!\cdots\!00\)\( T^{2} + \)\(10\!\cdots\!58\)\( T^{4} - \)\(14\!\cdots\!00\)\( p^{38} T^{6} + p^{76} T^{8} \)
41$D_{4}$ \( ( 1 + 3127192714527996 T + \)\(48\!\cdots\!26\)\( T^{2} + 3127192714527996 p^{19} T^{3} + p^{38} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - \)\(80\!\cdots\!80\)\( T^{2} + \)\(19\!\cdots\!98\)\( T^{4} - \)\(80\!\cdots\!80\)\( p^{38} T^{6} + p^{76} T^{8} \)
47$D_4\times C_2$ \( 1 - \)\(19\!\cdots\!60\)\( T^{2} + \)\(15\!\cdots\!78\)\( T^{4} - \)\(19\!\cdots\!60\)\( p^{38} T^{6} + p^{76} T^{8} \)
53$D_4\times C_2$ \( 1 - \)\(21\!\cdots\!40\)\( T^{2} + \)\(18\!\cdots\!78\)\( T^{4} - \)\(21\!\cdots\!40\)\( p^{38} T^{6} + p^{76} T^{8} \)
59$D_{4}$ \( ( 1 - 25484316660294360 T + \)\(89\!\cdots\!78\)\( T^{2} - 25484316660294360 p^{19} T^{3} + p^{38} T^{4} )^{2} \)
61$D_{4}$ \( ( 1 + 77840503559636276 T + \)\(11\!\cdots\!26\)\( T^{2} + 77840503559636276 p^{19} T^{3} + p^{38} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 - \)\(10\!\cdots\!00\)\( T^{2} + \)\(73\!\cdots\!18\)\( T^{4} - \)\(10\!\cdots\!00\)\( p^{38} T^{6} + p^{76} T^{8} \)
71$D_{4}$ \( ( 1 + 29427662567526696 T + \)\(15\!\cdots\!66\)\( T^{2} + 29427662567526696 p^{19} T^{3} + p^{38} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 + \)\(12\!\cdots\!20\)\( T^{2} + \)\(63\!\cdots\!38\)\( T^{4} + \)\(12\!\cdots\!20\)\( p^{38} T^{6} + p^{76} T^{8} \)
79$D_{4}$ \( ( 1 - 1235969155825112720 T + \)\(13\!\cdots\!38\)\( T^{2} - 1235969155825112720 p^{19} T^{3} + p^{38} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - \)\(32\!\cdots\!80\)\( T^{2} + \)\(12\!\cdots\!18\)\( T^{4} - \)\(32\!\cdots\!80\)\( p^{38} T^{6} + p^{76} T^{8} \)
89$D_{4}$ \( ( 1 + 4172929077000032820 T + \)\(20\!\cdots\!18\)\( T^{2} + 4172929077000032820 p^{19} T^{3} + p^{38} T^{4} )^{2} \)
97$D_4\times C_2$ \( 1 - \)\(73\!\cdots\!20\)\( T^{2} + \)\(21\!\cdots\!78\)\( T^{4} - \)\(73\!\cdots\!20\)\( p^{38} T^{6} + p^{76} T^{8} \)
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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

−7.71151566435928984158117542335, −7.61833166595229432950179686904, −7.20000621384617264478028679674, −6.77086773895658191399135554089, −6.70632109226115886085407918240, −6.20242912785246667214964559843, −6.18375373971857637843275841181, −5.53979235609469217217540497385, −5.23836378639303074355942443853, −4.99645635141766035807604281441, −4.82158112378960269952407416452, −4.07133480292001590559103890102, −3.97684252973236866686426127814, −3.95129645516118252116849557836, −3.93569651948522497922963900315, −3.10604014386202536621977364290, −2.73242016353571775501212404654, −2.51719797748099171836364028877, −2.19666260302353954073040025187, −1.48044444954281655570949487487, −1.45955206519320708873931429349, −1.21312530731795357928093852069, −0.996540123565983341334007289321, −0.43118998061997429984035201457, −0.00028001691630226852488572902, 0.00028001691630226852488572902, 0.43118998061997429984035201457, 0.996540123565983341334007289321, 1.21312530731795357928093852069, 1.45955206519320708873931429349, 1.48044444954281655570949487487, 2.19666260302353954073040025187, 2.51719797748099171836364028877, 2.73242016353571775501212404654, 3.10604014386202536621977364290, 3.93569651948522497922963900315, 3.95129645516118252116849557836, 3.97684252973236866686426127814, 4.07133480292001590559103890102, 4.82158112378960269952407416452, 4.99645635141766035807604281441, 5.23836378639303074355942443853, 5.53979235609469217217540497385, 6.18375373971857637843275841181, 6.20242912785246667214964559843, 6.70632109226115886085407918240, 6.77086773895658191399135554089, 7.20000621384617264478028679674, 7.61833166595229432950179686904, 7.71151566435928984158117542335

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