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Properties

Label 2-920-920.739-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: $\chi_{920} (739, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 920,\ (\ :0),\ -0.969 + 0.246i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4370999486\)
\(L(\frac12)\) \(\approx\) \(0.4370999486\)
\(L(1)\) 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

−10.16733939972452661522623528741, −8.965150877757386641561609531375, −8.544605198930792092555643931141, −7.58112048164175280521936407236, −6.36916390045024420909167665778, −5.51947322394204190523054399284, −4.12557895490280651812835955506, −3.22171281286362722472820971209, −2.67102488293210396609783467266, −0.43818975481077492959816249273, 1.95105220681315665939950730667, 3.85992441429275422697915500469, 4.83955808028787415015144280763, 5.15543989179493294617398856541, 6.72714199500559727605956729831, 7.25728168195682861968895846243, 8.036964312971543739300912583117, 8.913132950324522745020703106849, 9.597624061274166994811401058513, 10.35166744443448740938518376218

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