Properties

Label 2-920-920.179-c0-0-1
Degree $2$
Conductor $920$
Sign $-0.899 + 0.436i$
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  + (−0.959 − 0.281i)2-s + (0.841 + 0.540i)4-s + (−0.142 + 0.989i)5-s + (−0.797 − 1.74i)7-s + (−0.654 − 0.755i)8-s + (−0.142 − 0.989i)9-s + (0.415 − 0.909i)10-s + (−1.61 + 0.474i)11-s + (−0.118 + 0.258i)13-s + (0.273 + 1.89i)14-s + (0.415 + 0.909i)16-s + (−0.142 + 0.989i)18-s + (−1.10 − 0.708i)19-s + (−0.654 + 0.755i)20-s + 1.68·22-s + (−0.654 + 0.755i)23-s + ⋯
L(s)  = 1  + (−0.959 − 0.281i)2-s + (0.841 + 0.540i)4-s + (−0.142 + 0.989i)5-s + (−0.797 − 1.74i)7-s + (−0.654 − 0.755i)8-s + (−0.142 − 0.989i)9-s + (0.415 − 0.909i)10-s + (−1.61 + 0.474i)11-s + (−0.118 + 0.258i)13-s + (0.273 + 1.89i)14-s + (0.415 + 0.909i)16-s + (−0.142 + 0.989i)18-s + (−1.10 − 0.708i)19-s + (−0.654 + 0.755i)20-s + 1.68·22-s + (−0.654 + 0.755i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.899 + 0.436i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.899 + 0.436i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(920\)    =    \(2^{3} \cdot 5 \cdot 23\)
Sign: $-0.899 + 0.436i$
Analytic conductor: \(0.459139\)
Root analytic conductor: \(0.677598\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{920} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 920,\ (\ :0),\ -0.899 + 0.436i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2105562012\)
\(L(\frac12)\) \(\approx\) \(0.2105562012\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.959 + 0.281i)T \)
5 \( 1 + (0.142 - 0.989i)T \)
23 \( 1 + (0.654 - 0.755i)T \)
good3 \( 1 + (0.142 + 0.989i)T^{2} \)
7 \( 1 + (0.797 + 1.74i)T + (-0.654 + 0.755i)T^{2} \)
11 \( 1 + (1.61 - 0.474i)T + (0.841 - 0.540i)T^{2} \)
13 \( 1 + (0.118 - 0.258i)T + (-0.654 - 0.755i)T^{2} \)
17 \( 1 + (-0.415 + 0.909i)T^{2} \)
19 \( 1 + (1.10 + 0.708i)T + (0.415 + 0.909i)T^{2} \)
29 \( 1 + (-0.415 + 0.909i)T^{2} \)
31 \( 1 + (0.142 - 0.989i)T^{2} \)
37 \( 1 + (0.118 + 0.822i)T + (-0.959 + 0.281i)T^{2} \)
41 \( 1 + (-0.186 + 1.29i)T + (-0.959 - 0.281i)T^{2} \)
43 \( 1 + (0.142 + 0.989i)T^{2} \)
47 \( 1 - 0.830T + T^{2} \)
53 \( 1 + (0.118 + 0.258i)T + (-0.654 + 0.755i)T^{2} \)
59 \( 1 + (-0.345 + 0.755i)T + (-0.654 - 0.755i)T^{2} \)
61 \( 1 + (0.142 - 0.989i)T^{2} \)
67 \( 1 + (-0.841 - 0.540i)T^{2} \)
71 \( 1 + (-0.841 - 0.540i)T^{2} \)
73 \( 1 + (-0.415 - 0.909i)T^{2} \)
79 \( 1 + (0.654 + 0.755i)T^{2} \)
83 \( 1 + (0.959 - 0.281i)T^{2} \)
89 \( 1 + (0.544 - 0.627i)T + (-0.142 - 0.989i)T^{2} \)
97 \( 1 + (0.959 + 0.281i)T^{2} \)
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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

−10.03034442485674897910534103236, −9.396465000704992716990529748743, −8.135740357889119787486322577537, −7.22297142411549570468911155049, −6.99178477541098972266664170055, −5.99989090032571317719954772787, −4.11765691834380892553123417439, −3.37854608390592429768864329275, −2.31658640916093717636201532659, −0.24966082803150411120429518507, 2.11110062747565028045917089065, 2.82633749512000037546165508585, 4.86593388676267918506415732272, 5.66335532364917489908201164741, 6.15371212417677617259915871800, 7.68660612326256150101341633781, 8.448940555391973222428538514499, 8.594027420964493763046719654290, 9.771798302812312999255294360070, 10.36396739006036080570383137380

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