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Properties

Label 2-460-460.399-c0-0-0
Degree $2$
Conductor $460$
Sign $0.820 - 0.571i$
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  + (−0.142 + 0.989i)2-s + (−0.118 − 0.258i)3-s + (−0.959 − 0.281i)4-s + (−0.654 − 0.755i)5-s + (0.273 − 0.0801i)6-s + (1.41 + 0.909i)7-s + (0.415 − 0.909i)8-s + (0.601 − 0.694i)9-s + (0.841 − 0.540i)10-s + (0.0405 + 0.281i)12-s + (−1.10 + 1.27i)14-s + (−0.118 + 0.258i)15-s + (0.841 + 0.540i)16-s + (0.601 + 0.694i)18-s + (0.415 + 0.909i)20-s + (0.0681 − 0.474i)21-s + ⋯
L(s)  = 1  + (−0.142 + 0.989i)2-s + (−0.118 − 0.258i)3-s + (−0.959 − 0.281i)4-s + (−0.654 − 0.755i)5-s + (0.273 − 0.0801i)6-s + (1.41 + 0.909i)7-s + (0.415 − 0.909i)8-s + (0.601 − 0.694i)9-s + (0.841 − 0.540i)10-s + (0.0405 + 0.281i)12-s + (−1.10 + 1.27i)14-s + (−0.118 + 0.258i)15-s + (0.841 + 0.540i)16-s + (0.601 + 0.694i)18-s + (0.415 + 0.909i)20-s + (0.0681 − 0.474i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.820 - 0.571i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.820 - 0.571i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $0.820 - 0.571i$
Analytic conductor: \(0.229569\)
Root analytic conductor: \(0.479134\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{460} (399, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 460,\ (\ :0),\ 0.820 - 0.571i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7572003214\)
\(L(\frac12)\) \(\approx\) \(0.7572003214\)
\(L(1)\) 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 \( 1 + (0.142 - 0.989i)T \)
good3 \( 1 + (0.118 + 0.258i)T + (-0.654 + 0.755i)T^{2} \)
7 \( 1 + (-1.41 - 0.909i)T + (0.415 + 0.909i)T^{2} \)
11 \( 1 + (0.959 - 0.281i)T^{2} \)
13 \( 1 + (-0.415 + 0.909i)T^{2} \)
17 \( 1 + (-0.841 + 0.540i)T^{2} \)
19 \( 1 + (-0.841 - 0.540i)T^{2} \)
29 \( 1 + (-1.25 + 0.368i)T + (0.841 - 0.540i)T^{2} \)
31 \( 1 + (0.654 + 0.755i)T^{2} \)
37 \( 1 + (0.142 + 0.989i)T^{2} \)
41 \( 1 + (1.10 + 1.27i)T + (-0.142 + 0.989i)T^{2} \)
43 \( 1 + (0.797 + 1.74i)T + (-0.654 + 0.755i)T^{2} \)
47 \( 1 + 1.30T + T^{2} \)
53 \( 1 + (-0.415 - 0.909i)T^{2} \)
59 \( 1 + (-0.415 + 0.909i)T^{2} \)
61 \( 1 + (0.797 - 1.74i)T + (-0.654 - 0.755i)T^{2} \)
67 \( 1 + (0.118 - 0.822i)T + (-0.959 - 0.281i)T^{2} \)
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 \( 1 + (0.544 - 0.627i)T + (-0.142 - 0.989i)T^{2} \)
89 \( 1 + (0.118 + 0.258i)T + (-0.654 + 0.755i)T^{2} \)
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

−11.76443813046396262623384623067, −10.29340332966671370893548916271, −9.137052763542604688115934892744, −8.516380908641240509994102086007, −7.79909327464302942882346254284, −6.88914746840062918143540833498, −5.61188774979793984934676571713, −4.90843587380567943952959074312, −3.89613566747901658409205003847, −1.45711501731144619573076515107, 1.63502924948714106559242296047, 3.15517778507255571453765322417, 4.48664013917577981519257492030, 4.76597103182224887006392461251, 6.76249103356811031355501793222, 7.975631267896280998008604955072, 8.200538965137483657506722749779, 9.863994636927661209651495281164, 10.51014478654752597182309561490, 11.10403428662982762314185903470

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