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Properties

Label 2-40e2-100.31-c0-0-1
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

\(L(\frac{1}{2})\) \(\approx\) \(0.8398855740\)
\(L(\frac12)\) \(\approx\) \(0.8398855740\)
\(L(1)\) 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

−9.521867148370727658243742037445, −8.595823714532381180555686933365, −7.36964491718163959748907770483, −7.09194443256375601558206659818, −6.53274409577203277470058497460, −5.50011382813438281639614260636, −4.41364455571503391694933072613, −3.40048980547358237103845485568, −2.32745217497129139081008180874, −0.810968582850708200782224513428, 1.35817696112181883502391481191, 2.97841933972140303446279061096, 4.03286646745425696811016095452, 5.02539183658707146599799327079, 5.32959474685605602911565120298, 6.25236653853839616776583536915, 7.60678883117688226848152681318, 8.166007018244835889792040129004, 9.093988772961717591560050547608, 9.769404273756052486963282669916

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