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Properties

Label 2-920-920.219-c0-0-1
Degree $2$
Conductor $920$
Sign $-0.451 + 0.892i$
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  + (0.841 − 0.540i)2-s + (0.415 − 0.909i)4-s + (−0.959 + 0.281i)5-s + (−1.10 − 1.27i)7-s + (−0.142 − 0.989i)8-s + (−0.959 − 0.281i)9-s + (−0.654 + 0.755i)10-s + (0.698 + 0.449i)11-s + (1.25 − 1.45i)13-s + (−1.61 − 0.474i)14-s + (−0.654 − 0.755i)16-s + (−0.959 + 0.281i)18-s + (−0.118 + 0.258i)19-s + (−0.142 + 0.989i)20-s + 0.830·22-s + (−0.142 + 0.989i)23-s + ⋯
L(s)  = 1  + (0.841 − 0.540i)2-s + (0.415 − 0.909i)4-s + (−0.959 + 0.281i)5-s + (−1.10 − 1.27i)7-s + (−0.142 − 0.989i)8-s + (−0.959 − 0.281i)9-s + (−0.654 + 0.755i)10-s + (0.698 + 0.449i)11-s + (1.25 − 1.45i)13-s + (−1.61 − 0.474i)14-s + (−0.654 − 0.755i)16-s + (−0.959 + 0.281i)18-s + (−0.118 + 0.258i)19-s + (−0.142 + 0.989i)20-s + 0.830·22-s + (−0.142 + 0.989i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.451 + 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.451 + 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(920\)    =    \(2^{3} \cdot 5 \cdot 23\)
Sign: $-0.451 + 0.892i$
Analytic conductor: \(0.459139\)
Root analytic conductor: \(0.677598\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{920} (219, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 920,\ (\ :0),\ -0.451 + 0.892i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.153755321\)
\(L(\frac12)\) \(\approx\) \(1.153755321\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.841 + 0.540i)T \)
5 \( 1 + (0.959 - 0.281i)T \)
23 \( 1 + (0.142 - 0.989i)T \)
good3 \( 1 + (0.959 + 0.281i)T^{2} \)
7 \( 1 + (1.10 + 1.27i)T + (-0.142 + 0.989i)T^{2} \)
11 \( 1 + (-0.698 - 0.449i)T + (0.415 + 0.909i)T^{2} \)
13 \( 1 + (-1.25 + 1.45i)T + (-0.142 - 0.989i)T^{2} \)
17 \( 1 + (0.654 - 0.755i)T^{2} \)
19 \( 1 + (0.118 - 0.258i)T + (-0.654 - 0.755i)T^{2} \)
29 \( 1 + (0.654 - 0.755i)T^{2} \)
31 \( 1 + (0.959 - 0.281i)T^{2} \)
37 \( 1 + (-1.25 - 0.368i)T + (0.841 + 0.540i)T^{2} \)
41 \( 1 + (-0.273 + 0.0801i)T + (0.841 - 0.540i)T^{2} \)
43 \( 1 + (0.959 + 0.281i)T^{2} \)
47 \( 1 + 1.30T + T^{2} \)
53 \( 1 + (-1.25 - 1.45i)T + (-0.142 + 0.989i)T^{2} \)
59 \( 1 + (-0.857 + 0.989i)T + (-0.142 - 0.989i)T^{2} \)
61 \( 1 + (0.959 - 0.281i)T^{2} \)
67 \( 1 + (-0.415 + 0.909i)T^{2} \)
71 \( 1 + (-0.415 + 0.909i)T^{2} \)
73 \( 1 + (0.654 + 0.755i)T^{2} \)
79 \( 1 + (0.142 + 0.989i)T^{2} \)
83 \( 1 + (-0.841 - 0.540i)T^{2} \)
89 \( 1 + (-0.186 + 1.29i)T + (-0.959 - 0.281i)T^{2} \)
97 \( 1 + (-0.841 + 0.540i)T^{2} \)
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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

−10.26552165233801474696190053321, −9.470932978344490009547059884926, −8.243550223859760139918172394534, −7.28061192513082526899390870612, −6.43382239512272478921131754811, −5.73283361727178987565889485597, −4.27402596512784925374935939004, −3.54591958813462068173228489755, −3.07288007134165207556450816412, −0.892832564119778869891732179869, 2.47124130789604597714801844899, 3.47760560217073519609881914528, 4.24867822527182365083757373702, 5.46509972820044801958703505196, 6.30777534400221026761717991463, 6.75594213629159727421757228369, 8.220954521585278943983445974033, 8.696188518147392233707098575901, 9.266929284961828068526432227468, 11.03953378295640786568350865735

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