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Properties

Label 2-460-460.439-c0-0-1
Degree $2$
Conductor $460$
Sign $0.684 + 0.728i$
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  + (0.841 + 0.540i)2-s + (−0.239 − 1.66i)3-s + (0.415 + 0.909i)4-s + (−0.959 − 0.281i)5-s + (0.698 − 1.53i)6-s + (0.857 − 0.989i)7-s + (−0.142 + 0.989i)8-s + (−1.75 + 0.515i)9-s + (−0.654 − 0.755i)10-s + (1.41 − 0.909i)12-s + (1.25 − 0.368i)14-s + (−0.239 + 1.66i)15-s + (−0.654 + 0.755i)16-s + (−1.75 − 0.515i)18-s + (−0.142 − 0.989i)20-s + (−1.85 − 1.19i)21-s + ⋯
L(s)  = 1  + (0.841 + 0.540i)2-s + (−0.239 − 1.66i)3-s + (0.415 + 0.909i)4-s + (−0.959 − 0.281i)5-s + (0.698 − 1.53i)6-s + (0.857 − 0.989i)7-s + (−0.142 + 0.989i)8-s + (−1.75 + 0.515i)9-s + (−0.654 − 0.755i)10-s + (1.41 − 0.909i)12-s + (1.25 − 0.368i)14-s + (−0.239 + 1.66i)15-s + (−0.654 + 0.755i)16-s + (−1.75 − 0.515i)18-s + (−0.142 − 0.989i)20-s + (−1.85 − 1.19i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.684 + 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.684 + 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $0.684 + 0.728i$
Analytic conductor: \(0.229569\)
Root analytic conductor: \(0.479134\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{460} (439, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 460,\ (\ :0),\ 0.684 + 0.728i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.139267152\)
\(L(\frac12)\) \(\approx\) \(1.139267152\)
\(L(1)\) 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 \( 1 + (-0.841 - 0.540i)T \)
good3 \( 1 + (0.239 + 1.66i)T + (-0.959 + 0.281i)T^{2} \)
7 \( 1 + (-0.857 + 0.989i)T + (-0.142 - 0.989i)T^{2} \)
11 \( 1 + (-0.415 + 0.909i)T^{2} \)
13 \( 1 + (0.142 - 0.989i)T^{2} \)
17 \( 1 + (0.654 + 0.755i)T^{2} \)
19 \( 1 + (0.654 - 0.755i)T^{2} \)
29 \( 1 + (0.797 - 1.74i)T + (-0.654 - 0.755i)T^{2} \)
31 \( 1 + (0.959 + 0.281i)T^{2} \)
37 \( 1 + (-0.841 + 0.540i)T^{2} \)
41 \( 1 + (-1.25 - 0.368i)T + (0.841 + 0.540i)T^{2} \)
43 \( 1 + (0.118 + 0.822i)T + (-0.959 + 0.281i)T^{2} \)
47 \( 1 + 1.91T + T^{2} \)
53 \( 1 + (0.142 + 0.989i)T^{2} \)
59 \( 1 + (0.142 - 0.989i)T^{2} \)
61 \( 1 + (0.118 - 0.822i)T + (-0.959 - 0.281i)T^{2} \)
67 \( 1 + (0.239 + 0.153i)T + (0.415 + 0.909i)T^{2} \)
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 \( 1 + (-0.273 + 0.0801i)T + (0.841 - 0.540i)T^{2} \)
89 \( 1 + (0.239 + 1.66i)T + (-0.959 + 0.281i)T^{2} \)
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

−11.35543124631909122794608486538, −10.99204697528130721553520463430, −8.758147003596952286056470826358, −7.923391144507074391305669112829, −7.33416779369024792667977691728, −6.84805266441687961843338503100, −5.52346846019849407586524232169, −4.54315406489463633890640452151, −3.26122238474860645996683708302, −1.49485736739102726127018663985, 2.61020528821056679396064806042, 3.72679691420953821507245666642, 4.58034795077814508859690684697, 5.23465340519305615323499632496, 6.29585603412893663190870228542, 7.904287913703103322988437176856, 8.999756263858731835234842914837, 9.824396039466979623274747061544, 10.85474686265037919263834147475, 11.32836663904701064015067008797

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