Properties

Label 2-460-460.119-c0-0-1
Degree $2$
Conductor $460$
Sign $0.635 + 0.771i$
Analytic cond. $0.229569$
Root an. cond. $0.479134$
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 + (0.797 − 0.234i)3-s + (−0.654 + 0.755i)4-s + (0.841 + 0.540i)5-s + (−0.544 − 0.627i)6-s + (−0.0405 − 0.281i)7-s + (0.959 + 0.281i)8-s + (−0.260 + 0.167i)9-s + (0.142 − 0.989i)10-s + (−0.345 + 0.755i)12-s + (−0.239 + 0.153i)14-s + (0.797 + 0.234i)15-s + (−0.142 − 0.989i)16-s + (0.260 + 0.167i)18-s + (−0.959 + 0.281i)20-s + (−0.0982 − 0.215i)21-s + ⋯
L(s)  = 1  + (−0.415 − 0.909i)2-s + (0.797 − 0.234i)3-s + (−0.654 + 0.755i)4-s + (0.841 + 0.540i)5-s + (−0.544 − 0.627i)6-s + (−0.0405 − 0.281i)7-s + (0.959 + 0.281i)8-s + (−0.260 + 0.167i)9-s + (0.142 − 0.989i)10-s + (−0.345 + 0.755i)12-s + (−0.239 + 0.153i)14-s + (0.797 + 0.234i)15-s + (−0.142 − 0.989i)16-s + (0.260 + 0.167i)18-s + (−0.959 + 0.281i)20-s + (−0.0982 − 0.215i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.635 + 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.635 + 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $0.635 + 0.771i$
Analytic conductor: \(0.229569\)
Root analytic conductor: \(0.479134\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{460} (119, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 460,\ (\ :0),\ 0.635 + 0.771i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9156207480\)
\(L(\frac12)\) \(\approx\) \(0.9156207480\)
\(L(1)\) 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 \( 1 + (0.415 + 0.909i)T \)
good3 \( 1 + (-0.797 + 0.234i)T + (0.841 - 0.540i)T^{2} \)
7 \( 1 + (0.0405 + 0.281i)T + (-0.959 + 0.281i)T^{2} \)
11 \( 1 + (0.654 + 0.755i)T^{2} \)
13 \( 1 + (0.959 + 0.281i)T^{2} \)
17 \( 1 + (0.142 - 0.989i)T^{2} \)
19 \( 1 + (0.142 + 0.989i)T^{2} \)
29 \( 1 + (1.10 + 1.27i)T + (-0.142 + 0.989i)T^{2} \)
31 \( 1 + (-0.841 - 0.540i)T^{2} \)
37 \( 1 + (-0.415 + 0.909i)T^{2} \)
41 \( 1 + (0.239 + 0.153i)T + (0.415 + 0.909i)T^{2} \)
43 \( 1 + (1.25 - 0.368i)T + (0.841 - 0.540i)T^{2} \)
47 \( 1 + 1.68T + T^{2} \)
53 \( 1 + (0.959 - 0.281i)T^{2} \)
59 \( 1 + (0.959 + 0.281i)T^{2} \)
61 \( 1 + (-1.25 - 0.368i)T + (0.841 + 0.540i)T^{2} \)
67 \( 1 + (-0.797 - 1.74i)T + (-0.654 + 0.755i)T^{2} \)
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 \( 1 + (-1.61 + 1.03i)T + (0.415 - 0.909i)T^{2} \)
89 \( 1 + (0.797 - 0.234i)T + (0.841 - 0.540i)T^{2} \)
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

−11.03681308059800758543969283493, −10.15681809447826327542706096642, −9.519306006638442336569401208722, −8.563145588480550003253698761997, −7.81410735549305902474322566865, −6.72375255795949475246923912138, −5.34875749239491039102613365839, −3.87099769901459644457773296430, −2.76637376910511324667357269487, −1.91352392781156901862574849936, 1.85253555252842241386988273216, 3.55227430695356547925973151471, 5.01244197742317207939267254419, 5.78034273270603811153484906285, 6.79910417142291441246842416073, 8.028251359377525447111527967943, 8.706489693949095080765070779187, 9.438660940717777525934566087365, 9.947452276459989187369494191557, 11.19298607653153340266802678199

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