Properties

Label 2-964-964.819-c0-0-0
Degree $2$
Conductor $964$
Sign $-0.347 + 0.937i$
Analytic cond. $0.481098$
Root an. cond. $0.693612$
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  + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s + (−1.73 + 0.564i)5-s − 0.999i·8-s + (−0.913 + 0.406i)9-s + (1.78 + 0.379i)10-s + (0.685 − 1.05i)13-s + (−0.5 + 0.866i)16-s + (0.243 − 1.53i)17-s + (0.994 + 0.104i)18-s + (−1.35 − 1.22i)20-s + (1.89 − 1.37i)25-s + (−1.12 + 0.571i)26-s + (−0.773 − 1.73i)29-s + (0.866 − 0.499i)32-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s + (−1.73 + 0.564i)5-s − 0.999i·8-s + (−0.913 + 0.406i)9-s + (1.78 + 0.379i)10-s + (0.685 − 1.05i)13-s + (−0.5 + 0.866i)16-s + (0.243 − 1.53i)17-s + (0.994 + 0.104i)18-s + (−1.35 − 1.22i)20-s + (1.89 − 1.37i)25-s + (−1.12 + 0.571i)26-s + (−0.773 − 1.73i)29-s + (0.866 − 0.499i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 964 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.347 + 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 964 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.347 + 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(964\)    =    \(2^{2} \cdot 241\)
Sign: $-0.347 + 0.937i$
Analytic conductor: \(0.481098\)
Root analytic conductor: \(0.693612\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{964} (819, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 964,\ (\ :0),\ -0.347 + 0.937i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3156354411\)
\(L(\frac12)\) \(\approx\) \(0.3156354411\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
241 \( 1 + T \)
good3 \( 1 + (0.913 - 0.406i)T^{2} \)
5 \( 1 + (1.73 - 0.564i)T + (0.809 - 0.587i)T^{2} \)
7 \( 1 + (0.207 + 0.978i)T^{2} \)
11 \( 1 + (-0.866 + 0.5i)T^{2} \)
13 \( 1 + (-0.685 + 1.05i)T + (-0.406 - 0.913i)T^{2} \)
17 \( 1 + (-0.243 + 1.53i)T + (-0.951 - 0.309i)T^{2} \)
19 \( 1 + (-0.866 + 0.5i)T^{2} \)
23 \( 1 + (-0.587 + 0.809i)T^{2} \)
29 \( 1 + (0.773 + 1.73i)T + (-0.669 + 0.743i)T^{2} \)
31 \( 1 + (-0.207 - 0.978i)T^{2} \)
37 \( 1 + (0.770 + 1.18i)T + (-0.406 + 0.913i)T^{2} \)
41 \( 1 + (-1.01 - 1.40i)T + (-0.309 + 0.951i)T^{2} \)
43 \( 1 + (-0.951 + 0.309i)T^{2} \)
47 \( 1 + (0.309 + 0.951i)T^{2} \)
53 \( 1 + (-0.809 - 0.0850i)T + (0.978 + 0.207i)T^{2} \)
59 \( 1 + (0.913 + 0.406i)T^{2} \)
61 \( 1 + (1.53 + 0.5i)T + (0.809 + 0.587i)T^{2} \)
67 \( 1 + (-0.978 + 0.207i)T^{2} \)
71 \( 1 + (0.743 + 0.669i)T^{2} \)
73 \( 1 + (0.896 - 1.76i)T + (-0.587 - 0.809i)T^{2} \)
79 \( 1 + (-0.809 + 0.587i)T^{2} \)
83 \( 1 + (-0.913 - 0.406i)T^{2} \)
89 \( 1 + (0.434 + 1.62i)T + (-0.866 + 0.5i)T^{2} \)
97 \( 1 + (-0.104 + 0.994i)T + (-0.978 - 0.207i)T^{2} \)
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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

−10.09490094547358517592885425556, −9.031414157261053379684966497404, −8.154291735759165357529967660924, −7.75033512011796867114361780441, −7.02539775547543583955386626018, −5.74914722386897586777181880959, −4.30158816435040269424703754421, −3.31739225414411484253127068154, −2.66632131201813639321400064538, −0.42626047175224010861918400919, 1.41319819404841582937990496285, 3.33852614597022256097534378719, 4.25337781287146049617024704511, 5.44314090987887125781755874917, 6.42938203488227925276376858352, 7.31328204841673613917600261937, 8.130207671190807026451152975075, 8.765904945645062786795522964738, 9.143397110384565432206116103638, 10.70760882423743966066006162465

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