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Properties

Label 2-40e2-200.91-c0-0-2
Degree $2$
Conductor $1600$
Sign $0.612 + 0.790i$
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  + (−0.190 − 0.587i)3-s + (0.951 − 0.309i)5-s − i·7-s + (0.5 − 0.363i)9-s + (0.809 + 0.587i)11-s + (0.587 + 0.809i)13-s + (−0.363 − 0.5i)15-s + (−0.190 + 0.587i)17-s + (−0.5 + 1.53i)19-s + (−0.587 + 0.190i)21-s + (0.587 − 0.809i)23-s + (0.809 − 0.587i)25-s + (−0.809 − 0.587i)27-s + (−0.951 + 0.309i)29-s + (−0.951 − 0.309i)31-s + ⋯
L(s)  = 1  + (−0.190 − 0.587i)3-s + (0.951 − 0.309i)5-s − i·7-s + (0.5 − 0.363i)9-s + (0.809 + 0.587i)11-s + (0.587 + 0.809i)13-s + (−0.363 − 0.5i)15-s + (−0.190 + 0.587i)17-s + (−0.5 + 1.53i)19-s + (−0.587 + 0.190i)21-s + (0.587 − 0.809i)23-s + (0.809 − 0.587i)25-s + (−0.809 − 0.587i)27-s + (−0.951 + 0.309i)29-s + (−0.951 − 0.309i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.612 + 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.612 + 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $0.612 + 0.790i$
Analytic conductor: \(0.798504\)
Root analytic conductor: \(0.893590\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (991, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :0),\ 0.612 + 0.790i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.344407149\)
\(L(\frac12)\) \(\approx\) \(1.344407149\)
\(L(1)\) 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 \( 1 + (-0.951 + 0.309i)T \)
good3 \( 1 + (0.190 + 0.587i)T + (-0.809 + 0.587i)T^{2} \)
7 \( 1 + iT - T^{2} \)
11 \( 1 + (-0.809 - 0.587i)T + (0.309 + 0.951i)T^{2} \)
13 \( 1 + (-0.587 - 0.809i)T + (-0.309 + 0.951i)T^{2} \)
17 \( 1 + (0.190 - 0.587i)T + (-0.809 - 0.587i)T^{2} \)
19 \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \)
23 \( 1 + (-0.587 + 0.809i)T + (-0.309 - 0.951i)T^{2} \)
29 \( 1 + (0.951 - 0.309i)T + (0.809 - 0.587i)T^{2} \)
31 \( 1 + (0.951 + 0.309i)T + (0.809 + 0.587i)T^{2} \)
37 \( 1 + (0.951 + 1.30i)T + (-0.309 + 0.951i)T^{2} \)
41 \( 1 + (0.809 - 0.587i)T + (0.309 - 0.951i)T^{2} \)
43 \( 1 - T + T^{2} \)
47 \( 1 + (0.951 - 0.309i)T + (0.809 - 0.587i)T^{2} \)
53 \( 1 + (0.809 - 0.587i)T^{2} \)
59 \( 1 + (0.309 - 0.951i)T^{2} \)
61 \( 1 + (0.363 - 0.5i)T + (-0.309 - 0.951i)T^{2} \)
67 \( 1 + (0.309 - 0.951i)T + (-0.809 - 0.587i)T^{2} \)
71 \( 1 + (-0.951 + 0.309i)T + (0.809 - 0.587i)T^{2} \)
73 \( 1 + (0.309 + 0.951i)T^{2} \)
79 \( 1 + (0.809 - 0.587i)T^{2} \)
83 \( 1 + (-0.309 + 0.951i)T + (-0.809 - 0.587i)T^{2} \)
89 \( 1 + (0.309 + 0.951i)T^{2} \)
97 \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \)
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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

−9.424229577657609096146672121086, −8.865482718531532220297419762389, −7.74751419660897702842998572218, −6.86753468341896200753426846183, −6.45408517441311662654374177926, −5.59178470639441381394185887322, −4.25423979370695114079836207738, −3.81220001636805018053349117424, −1.87987041888850405286480673610, −1.38801853677988037524679972142, 1.63439771681993302826216149007, 2.77513657758626602574663045965, 3.70809966413531325502237877032, 5.10396660829772425070218045272, 5.40976704629954460840045870320, 6.43085598203888535267259818445, 7.11993395628793595700247743370, 8.369853202386518978804488746090, 9.309523065524580407305186796385, 9.374172183960187038895467787779

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