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Properties

Label 2-40e2-100.31-c0-0-0
Degree $2$
Conductor $1600$
Sign $-0.368 - 0.929i$
Analytic cond. $0.798504$
Root an. cond. $0.893590$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.587 + 0.809i)3-s + (−0.309 + 0.951i)5-s + 0.618i·7-s + (−0.190 + 0.587i)13-s + (−0.951 + 0.309i)15-s + (−0.951 + 1.30i)19-s + (−0.500 + 0.363i)21-s + (−0.951 + 0.309i)23-s + (−0.809 − 0.587i)25-s + (0.951 − 0.309i)27-s + (1.30 − 0.951i)29-s + (0.587 − 0.809i)31-s + (−0.587 − 0.190i)35-s + (0.309 − 0.951i)37-s + (−0.587 + 0.190i)39-s + ⋯
L(s)  = 1  + (0.587 + 0.809i)3-s + (−0.309 + 0.951i)5-s + 0.618i·7-s + (−0.190 + 0.587i)13-s + (−0.951 + 0.309i)15-s + (−0.951 + 1.30i)19-s + (−0.500 + 0.363i)21-s + (−0.951 + 0.309i)23-s + (−0.809 − 0.587i)25-s + (0.951 − 0.309i)27-s + (1.30 − 0.951i)29-s + (0.587 − 0.809i)31-s + (−0.587 − 0.190i)35-s + (0.309 − 0.951i)37-s + (−0.587 + 0.190i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.368 - 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.368 - 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $-0.368 - 0.929i$
Analytic conductor: \(0.798504\)
Root analytic conductor: \(0.893590\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (831, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :0),\ -0.368 - 0.929i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.203547662\)
\(L(\frac12)\) \(\approx\) \(1.203547662\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (0.309 - 0.951i)T \)
good3 \( 1 + (-0.587 - 0.809i)T + (-0.309 + 0.951i)T^{2} \)
7 \( 1 - 0.618iT - T^{2} \)
11 \( 1 + (0.809 - 0.587i)T^{2} \)
13 \( 1 + (0.190 - 0.587i)T + (-0.809 - 0.587i)T^{2} \)
17 \( 1 + (0.309 + 0.951i)T^{2} \)
19 \( 1 + (0.951 - 1.30i)T + (-0.309 - 0.951i)T^{2} \)
23 \( 1 + (0.951 - 0.309i)T + (0.809 - 0.587i)T^{2} \)
29 \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \)
31 \( 1 + (-0.587 + 0.809i)T + (-0.309 - 0.951i)T^{2} \)
37 \( 1 + (-0.309 + 0.951i)T + (-0.809 - 0.587i)T^{2} \)
41 \( 1 + (-0.809 - 0.587i)T^{2} \)
43 \( 1 - 1.61iT - T^{2} \)
47 \( 1 + (-0.363 - 0.5i)T + (-0.309 + 0.951i)T^{2} \)
53 \( 1 + (0.809 - 0.587i)T + (0.309 - 0.951i)T^{2} \)
59 \( 1 + (1.53 + 0.5i)T + (0.809 + 0.587i)T^{2} \)
61 \( 1 + (0.309 + 0.951i)T + (-0.809 + 0.587i)T^{2} \)
67 \( 1 + (-0.309 - 0.951i)T^{2} \)
71 \( 1 + (0.363 + 0.5i)T + (-0.309 + 0.951i)T^{2} \)
73 \( 1 + (-0.309 - 0.951i)T + (-0.809 + 0.587i)T^{2} \)
79 \( 1 + (-0.951 - 1.30i)T + (-0.309 + 0.951i)T^{2} \)
83 \( 1 + (-0.587 + 0.809i)T + (-0.309 - 0.951i)T^{2} \)
89 \( 1 + (-0.809 + 0.587i)T^{2} \)
97 \( 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.809001227361923247004439598759, −9.224272471307720745956932795526, −8.194567102398282497923174291592, −7.73924018735133681392093578133, −6.35551578589057793637075630985, −6.09784496662361021465195119874, −4.50667966400483159496300568566, −3.97879710914419698497383099839, −2.99392619603187657659906976570, −2.12933265333839861037575657740, 0.906349613690973228259059539696, 2.11868694300867345995657015303, 3.21307347883301428800597200976, 4.46675729343005338010062467282, 4.99702864640554786271150938351, 6.32660077880599792239766290920, 7.12208267590452753699829002608, 7.82864318467304611372112415787, 8.534897087310013864931274552520, 8.975510094366421031631099485596

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