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Properties

Label 2-40e2-200.11-c0-0-2
Degree $2$
Conductor $1600$
Sign $0.790 + 0.612i$
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

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

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $0.790 + 0.612i$
Analytic conductor: \(0.798504\)
Root analytic conductor: \(0.893590\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (1311, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :0),\ 0.790 + 0.612i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8790973703\)
\(L(\frac12)\) \(\approx\) \(0.8790973703\)
\(L(1)\) 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

−9.384386592226789053726971748527, −8.906248548653584937728424893625, −7.84710650641998997001473933154, −7.11474946581388964399105293778, −6.52347748400643249944356308688, −5.05189164040826793547765312932, −4.22332973359119128743363745230, −4.10406886351164268100080291185, −2.57799399665655814024056689356, −0.798171941341680296420044569896, 1.41812033966594888991899230291, 2.65812602483812358060101301225, 3.80547869412704237945039475921, 4.56495705478741003235026362961, 5.88710288014835531906135789299, 6.42956243227564473999446142045, 7.34599149209054378889615897648, 8.036949970493345011302227447105, 8.663252638993537674353055199584, 9.872419593208841336675240633690

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