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Properties

Label 2-920-920.259-c0-0-0
Degree $2$
Conductor $920$
Sign $0.117 - 0.993i$
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  + (−0.415 + 0.909i)2-s + (−0.654 − 0.755i)4-s + (−0.841 + 0.540i)5-s + (0.118 − 0.822i)7-s + (0.959 − 0.281i)8-s + (0.841 + 0.540i)9-s + (−0.142 − 0.989i)10-s + (−0.544 − 1.19i)11-s + (0.239 + 1.66i)13-s + (0.698 + 0.449i)14-s + (−0.142 + 0.989i)16-s + (−0.841 + 0.540i)18-s + (1.25 + 1.45i)19-s + (0.959 + 0.281i)20-s + 1.30·22-s + (0.959 + 0.281i)23-s + ⋯
L(s)  = 1  + (−0.415 + 0.909i)2-s + (−0.654 − 0.755i)4-s + (−0.841 + 0.540i)5-s + (0.118 − 0.822i)7-s + (0.959 − 0.281i)8-s + (0.841 + 0.540i)9-s + (−0.142 − 0.989i)10-s + (−0.544 − 1.19i)11-s + (0.239 + 1.66i)13-s + (0.698 + 0.449i)14-s + (−0.142 + 0.989i)16-s + (−0.841 + 0.540i)18-s + (1.25 + 1.45i)19-s + (0.959 + 0.281i)20-s + 1.30·22-s + (0.959 + 0.281i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.117 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.117 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(920\)    =    \(2^{3} \cdot 5 \cdot 23\)
Sign: $0.117 - 0.993i$
Analytic conductor: \(0.459139\)
Root analytic conductor: \(0.677598\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{920} (259, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 920,\ (\ :0),\ 0.117 - 0.993i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7120138332\)
\(L(\frac12)\) \(\approx\) \(0.7120138332\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.415 - 0.909i)T \)
5 \( 1 + (0.841 - 0.540i)T \)
23 \( 1 + (-0.959 - 0.281i)T \)
good3 \( 1 + (-0.841 - 0.540i)T^{2} \)
7 \( 1 + (-0.118 + 0.822i)T + (-0.959 - 0.281i)T^{2} \)
11 \( 1 + (0.544 + 1.19i)T + (-0.654 + 0.755i)T^{2} \)
13 \( 1 + (-0.239 - 1.66i)T + (-0.959 + 0.281i)T^{2} \)
17 \( 1 + (0.142 + 0.989i)T^{2} \)
19 \( 1 + (-1.25 - 1.45i)T + (-0.142 + 0.989i)T^{2} \)
29 \( 1 + (0.142 + 0.989i)T^{2} \)
31 \( 1 + (-0.841 + 0.540i)T^{2} \)
37 \( 1 + (-0.239 - 0.153i)T + (0.415 + 0.909i)T^{2} \)
41 \( 1 + (1.61 - 1.03i)T + (0.415 - 0.909i)T^{2} \)
43 \( 1 + (-0.841 - 0.540i)T^{2} \)
47 \( 1 - 0.284T + T^{2} \)
53 \( 1 + (-0.239 + 1.66i)T + (-0.959 - 0.281i)T^{2} \)
59 \( 1 + (-0.0405 - 0.281i)T + (-0.959 + 0.281i)T^{2} \)
61 \( 1 + (-0.841 + 0.540i)T^{2} \)
67 \( 1 + (0.654 + 0.755i)T^{2} \)
71 \( 1 + (0.654 + 0.755i)T^{2} \)
73 \( 1 + (0.142 - 0.989i)T^{2} \)
79 \( 1 + (0.959 - 0.281i)T^{2} \)
83 \( 1 + (-0.415 - 0.909i)T^{2} \)
89 \( 1 + (-0.273 - 0.0801i)T + (0.841 + 0.540i)T^{2} \)
97 \( 1 + (-0.415 + 0.909i)T^{2} \)
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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

−10.37349655324038526998652975855, −9.657516604894092516316523017043, −8.510862755548202198742885532551, −7.84641491293440071304599986675, −7.16217463158190080722792467011, −6.53793961660520211888347081860, −5.30652226508433078883291986092, −4.30930825279518474277164075997, −3.49026441129056096493313685343, −1.38382651797861231223427782509, 0.990948216681463801329753978947, 2.59656417724190333240811885509, 3.53174658956549234641272534352, 4.75073032129363861225136730827, 5.26840472583165514533302748247, 7.16372633209154939809547520320, 7.63344666752142119880235585061, 8.663204434520478371182759511677, 9.249657499825763607520443008553, 10.11634986149476230898474897638

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