Properties

Label 4-825e2-1.1-c5e2-0-1
Degree $4$
Conductor $680625$
Sign $1$
Analytic cond. $17507.6$
Root an. cond. $11.5028$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 13·2-s − 18·3-s + 71·4-s + 234·6-s − 146·7-s − 65·8-s + 243·9-s − 242·11-s − 1.27e3·12-s + 130·13-s + 1.89e3·14-s − 1.72e3·16-s + 728·17-s − 3.15e3·18-s − 828·19-s + 2.62e3·21-s + 3.14e3·22-s + 238·23-s + 1.17e3·24-s − 1.69e3·26-s − 2.91e3·27-s − 1.03e4·28-s + 696·29-s − 1.04e4·31-s + 1.26e4·32-s + 4.35e3·33-s − 9.46e3·34-s + ⋯
L(s)  = 1  − 2.29·2-s − 1.15·3-s + 2.21·4-s + 2.65·6-s − 1.12·7-s − 0.359·8-s + 9-s − 0.603·11-s − 2.56·12-s + 0.213·13-s + 2.58·14-s − 1.68·16-s + 0.610·17-s − 2.29·18-s − 0.526·19-s + 1.30·21-s + 1.38·22-s + 0.0938·23-s + 0.414·24-s − 0.490·26-s − 0.769·27-s − 2.49·28-s + 0.153·29-s − 1.95·31-s + 2.17·32-s + 0.696·33-s − 1.40·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 680625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 680625 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(680625\)    =    \(3^{2} \cdot 5^{4} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(17507.6\)
Root analytic conductor: \(11.5028\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 680625,\ (\ :5/2, 5/2),\ 1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) 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$C_1$ \( ( 1 + p^{2} T )^{2} \)
5 \( 1 \)
11$C_1$ \( ( 1 + p^{2} T )^{2} \)
good2$D_{4}$ \( 1 + 13 T + 49 p T^{2} + 13 p^{5} T^{3} + p^{10} T^{4} \)
7$D_{4}$ \( 1 + 146 T + 7230 T^{2} + 146 p^{5} T^{3} + p^{10} T^{4} \)
13$D_{4}$ \( 1 - 10 p T + 701634 T^{2} - 10 p^{6} T^{3} + p^{10} T^{4} \)
17$D_{4}$ \( 1 - 728 T + 1830542 T^{2} - 728 p^{5} T^{3} + p^{10} T^{4} \)
19$D_{4}$ \( 1 + 828 T - 865114 T^{2} + 828 p^{5} T^{3} + p^{10} T^{4} \)
23$D_{4}$ \( 1 - 238 T + 11988422 T^{2} - 238 p^{5} T^{3} + p^{10} T^{4} \)
29$D_{4}$ \( 1 - 24 p T + 22778374 T^{2} - 24 p^{6} T^{3} + p^{10} T^{4} \)
31$D_{4}$ \( 1 + 10480 T + 63595902 T^{2} + 10480 p^{5} T^{3} + p^{10} T^{4} \)
37$D_{4}$ \( 1 - 1908 T + 74176718 T^{2} - 1908 p^{5} T^{3} + p^{10} T^{4} \)
41$D_{4}$ \( 1 - 36484 T + 564482438 T^{2} - 36484 p^{5} T^{3} + p^{10} T^{4} \)
43$D_{4}$ \( 1 + 9768 T + 237972854 T^{2} + 9768 p^{5} T^{3} + p^{10} T^{4} \)
47$D_{4}$ \( 1 + 43742 T + 935775830 T^{2} + 43742 p^{5} T^{3} + p^{10} T^{4} \)
53$D_{4}$ \( 1 - 12174 T + 457325722 T^{2} - 12174 p^{5} T^{3} + p^{10} T^{4} \)
59$D_{4}$ \( 1 + 2788 T + 1399448534 T^{2} + 2788 p^{5} T^{3} + p^{10} T^{4} \)
61$D_{4}$ \( 1 + 25302 T + 1815376826 T^{2} + 25302 p^{5} T^{3} + p^{10} T^{4} \)
67$D_{4}$ \( 1 - 40520 T + 1779236982 T^{2} - 40520 p^{5} T^{3} + p^{10} T^{4} \)
71$D_{4}$ \( 1 - 31386 T + 3698331094 T^{2} - 31386 p^{5} T^{3} + p^{10} T^{4} \)
73$D_{4}$ \( 1 - 46780 T + 4437463638 T^{2} - 46780 p^{5} T^{3} + p^{10} T^{4} \)
79$D_{4}$ \( 1 + 16850 T + 5148027246 T^{2} + 16850 p^{5} T^{3} + p^{10} T^{4} \)
83$D_{4}$ \( 1 + 79440 T + 5477266486 T^{2} + 79440 p^{5} T^{3} + p^{10} T^{4} \)
89$D_{4}$ \( 1 + 54204 T + 6019532470 T^{2} + 54204 p^{5} T^{3} + p^{10} T^{4} \)
97$D_{4}$ \( 1 - 241568 T + 30950947518 T^{2} - 241568 p^{5} T^{3} + p^{10} T^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.207225527815318482736899596114, −9.183083562493198373111822261516, −8.424386915179532802702516916830, −8.057035185994303022535504120838, −7.68218227638774868274934247614, −7.27443820280870338075471698933, −6.81364505083086098657355616177, −6.44594471416899298005470468298, −5.75532845437825442897583374720, −5.64926142722997867984714934937, −4.81725415842755290482740904813, −4.44332843685958960586013585373, −3.68245444943480817872939951451, −3.22607153611287515436809177572, −2.32437877421733121314570263391, −1.82496914453194142174581907831, −1.08269092503715194022116730788, −0.73154538696126853521692391709, 0, 0, 0.73154538696126853521692391709, 1.08269092503715194022116730788, 1.82496914453194142174581907831, 2.32437877421733121314570263391, 3.22607153611287515436809177572, 3.68245444943480817872939951451, 4.44332843685958960586013585373, 4.81725415842755290482740904813, 5.64926142722997867984714934937, 5.75532845437825442897583374720, 6.44594471416899298005470468298, 6.81364505083086098657355616177, 7.27443820280870338075471698933, 7.68218227638774868274934247614, 8.057035185994303022535504120838, 8.424386915179532802702516916830, 9.183083562493198373111822261516, 9.207225527815318482736899596114

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