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Properties

Label 2-460-460.59-c0-0-0
Degree $2$
Conductor $460$
Sign $-0.270 + 0.962i$
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  + (−0.654 − 0.755i)2-s + (−1.10 − 0.708i)3-s + (−0.142 + 0.989i)4-s + (0.415 − 0.909i)5-s + (0.186 + 1.29i)6-s + (1.84 + 0.540i)7-s + (0.841 − 0.540i)8-s + (0.297 + 0.650i)9-s + (−0.959 + 0.281i)10-s + (0.857 − 0.989i)12-s + (−0.797 − 1.74i)14-s + (−1.10 + 0.708i)15-s + (−0.959 − 0.281i)16-s + (0.297 − 0.650i)18-s + (0.841 + 0.540i)20-s + (−1.64 − 1.89i)21-s + ⋯
L(s)  = 1  + (−0.654 − 0.755i)2-s + (−1.10 − 0.708i)3-s + (−0.142 + 0.989i)4-s + (0.415 − 0.909i)5-s + (0.186 + 1.29i)6-s + (1.84 + 0.540i)7-s + (0.841 − 0.540i)8-s + (0.297 + 0.650i)9-s + (−0.959 + 0.281i)10-s + (0.857 − 0.989i)12-s + (−0.797 − 1.74i)14-s + (−1.10 + 0.708i)15-s + (−0.959 − 0.281i)16-s + (0.297 − 0.650i)18-s + (0.841 + 0.540i)20-s + (−1.64 − 1.89i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.270 + 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.270 + 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $-0.270 + 0.962i$
Analytic conductor: \(0.229569\)
Root analytic conductor: \(0.479134\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{460} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 460,\ (\ :0),\ -0.270 + 0.962i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5521925014\)
\(L(\frac12)\) \(\approx\) \(0.5521925014\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.654 + 0.755i)T \)
5 \( 1 + (-0.415 + 0.909i)T \)
23 \( 1 + (0.654 + 0.755i)T \)
good3 \( 1 + (1.10 + 0.708i)T + (0.415 + 0.909i)T^{2} \)
7 \( 1 + (-1.84 - 0.540i)T + (0.841 + 0.540i)T^{2} \)
11 \( 1 + (0.142 + 0.989i)T^{2} \)
13 \( 1 + (-0.841 + 0.540i)T^{2} \)
17 \( 1 + (0.959 - 0.281i)T^{2} \)
19 \( 1 + (0.959 + 0.281i)T^{2} \)
29 \( 1 + (0.118 + 0.822i)T + (-0.959 + 0.281i)T^{2} \)
31 \( 1 + (-0.415 + 0.909i)T^{2} \)
37 \( 1 + (0.654 - 0.755i)T^{2} \)
41 \( 1 + (0.797 - 1.74i)T + (-0.654 - 0.755i)T^{2} \)
43 \( 1 + (0.239 + 0.153i)T + (0.415 + 0.909i)T^{2} \)
47 \( 1 - 0.830T + T^{2} \)
53 \( 1 + (-0.841 - 0.540i)T^{2} \)
59 \( 1 + (-0.841 + 0.540i)T^{2} \)
61 \( 1 + (0.239 - 0.153i)T + (0.415 - 0.909i)T^{2} \)
67 \( 1 + (1.10 + 1.27i)T + (-0.142 + 0.989i)T^{2} \)
71 \( 1 + (0.142 - 0.989i)T^{2} \)
73 \( 1 + (0.959 + 0.281i)T^{2} \)
79 \( 1 + (-0.841 + 0.540i)T^{2} \)
83 \( 1 + (-0.698 - 1.53i)T + (-0.654 + 0.755i)T^{2} \)
89 \( 1 + (1.10 + 0.708i)T + (0.415 + 0.909i)T^{2} \)
97 \( 1 + (0.654 + 0.755i)T^{2} \)
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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

−11.23980266581803838882500480895, −10.34593587260553484197387914514, −9.167032719147522834011222014975, −8.306624391865728175277833891113, −7.69652117522326190847134803385, −6.27924339591144894785564483009, −5.20581380149905393445139996695, −4.42843015459289475312907962929, −2.15138670544972963534530417759, −1.21474585316427343560064881526, 1.76314745906567041021429655115, 4.18765402103705892117910309743, 5.21037384964522714526127243696, 5.78453116954310502253616435199, 7.01162729655797702049397008574, 7.72954538155011939679588576586, 8.816518527538309459270423194147, 10.07352237395629722897809785704, 10.56771810703043364029556376099, 11.17088403645791966046427769907

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