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Properties

Label 2-920-920.859-c0-0-1
Degree $2$
Conductor $920$
Sign $0.969 + 0.246i$
Analytic cond. $0.459139$
Root an. cond. $0.677598$
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  + (0.142 − 0.989i)2-s + (−0.959 − 0.281i)4-s + (0.654 + 0.755i)5-s + (0.239 + 0.153i)7-s + (−0.415 + 0.909i)8-s + (−0.654 + 0.755i)9-s + (0.841 − 0.540i)10-s + (0.273 + 1.89i)11-s + (1.10 − 0.708i)13-s + (0.186 − 0.215i)14-s + (0.841 + 0.540i)16-s + (0.654 + 0.755i)18-s + (−0.797 − 0.234i)19-s + (−0.415 − 0.909i)20-s + 1.91·22-s + (−0.415 − 0.909i)23-s + ⋯
L(s)  = 1  + (0.142 − 0.989i)2-s + (−0.959 − 0.281i)4-s + (0.654 + 0.755i)5-s + (0.239 + 0.153i)7-s + (−0.415 + 0.909i)8-s + (−0.654 + 0.755i)9-s + (0.841 − 0.540i)10-s + (0.273 + 1.89i)11-s + (1.10 − 0.708i)13-s + (0.186 − 0.215i)14-s + (0.841 + 0.540i)16-s + (0.654 + 0.755i)18-s + (−0.797 − 0.234i)19-s + (−0.415 − 0.909i)20-s + 1.91·22-s + (−0.415 − 0.909i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 + 0.246i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 + 0.246i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(920\)    =    \(2^{3} \cdot 5 \cdot 23\)
Sign: $0.969 + 0.246i$
Analytic conductor: \(0.459139\)
Root analytic conductor: \(0.677598\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{920} (859, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 920,\ (\ :0),\ 0.969 + 0.246i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.067109478\)
\(L(\frac12)\) \(\approx\) \(1.067109478\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.142 + 0.989i)T \)
5 \( 1 + (-0.654 - 0.755i)T \)
23 \( 1 + (0.415 + 0.909i)T \)
good3 \( 1 + (0.654 - 0.755i)T^{2} \)
7 \( 1 + (-0.239 - 0.153i)T + (0.415 + 0.909i)T^{2} \)
11 \( 1 + (-0.273 - 1.89i)T + (-0.959 + 0.281i)T^{2} \)
13 \( 1 + (-1.10 + 0.708i)T + (0.415 - 0.909i)T^{2} \)
17 \( 1 + (-0.841 + 0.540i)T^{2} \)
19 \( 1 + (0.797 + 0.234i)T + (0.841 + 0.540i)T^{2} \)
29 \( 1 + (-0.841 + 0.540i)T^{2} \)
31 \( 1 + (0.654 + 0.755i)T^{2} \)
37 \( 1 + (-1.10 + 1.27i)T + (-0.142 - 0.989i)T^{2} \)
41 \( 1 + (0.544 + 0.627i)T + (-0.142 + 0.989i)T^{2} \)
43 \( 1 + (0.654 - 0.755i)T^{2} \)
47 \( 1 + 1.68T + T^{2} \)
53 \( 1 + (-1.10 - 0.708i)T + (0.415 + 0.909i)T^{2} \)
59 \( 1 + (-1.41 + 0.909i)T + (0.415 - 0.909i)T^{2} \)
61 \( 1 + (0.654 + 0.755i)T^{2} \)
67 \( 1 + (0.959 + 0.281i)T^{2} \)
71 \( 1 + (0.959 + 0.281i)T^{2} \)
73 \( 1 + (-0.841 - 0.540i)T^{2} \)
79 \( 1 + (-0.415 + 0.909i)T^{2} \)
83 \( 1 + (0.142 + 0.989i)T^{2} \)
89 \( 1 + (-0.698 - 1.53i)T + (-0.654 + 0.755i)T^{2} \)
97 \( 1 + (0.142 - 0.989i)T^{2} \)
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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

−10.37910753873441079012784910582, −9.720427955926258476676591560924, −8.746082335824729284469352274202, −7.941182186896539264996002522614, −6.72459110905741958446060221934, −5.73988869054692329086965119722, −4.86558311818923278764384341658, −3.82274809414409085969213239255, −2.50743502594252616792577332438, −1.91253752781147283459621724242, 1.14138029410244354729482275863, 3.29780529760729549188242692044, 4.15684984290513472924034542331, 5.36478100668832106732566468738, 6.18126523277285048074868270668, 6.42697758161668323952868929357, 8.114556127742554793364304213240, 8.553917580601985230111270993149, 9.112231867692913167647093760204, 10.01639211125747832550931778830

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