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Properties

Label 2-460-460.239-c0-0-0
Degree $2$
Conductor $460$
Sign $0.559 - 0.828i$
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  + (−0.959 + 0.281i)2-s + (1.25 + 1.45i)3-s + (0.841 − 0.540i)4-s + (−0.142 − 0.989i)5-s + (−1.61 − 1.03i)6-s + (0.345 − 0.755i)7-s + (−0.654 + 0.755i)8-s + (−0.381 + 2.65i)9-s + (0.415 + 0.909i)10-s + (1.84 + 0.540i)12-s + (−0.118 + 0.822i)14-s + (1.25 − 1.45i)15-s + (0.415 − 0.909i)16-s + (−0.381 − 2.65i)18-s + (−0.654 − 0.755i)20-s + (1.52 − 0.449i)21-s + ⋯
L(s)  = 1  + (−0.959 + 0.281i)2-s + (1.25 + 1.45i)3-s + (0.841 − 0.540i)4-s + (−0.142 − 0.989i)5-s + (−1.61 − 1.03i)6-s + (0.345 − 0.755i)7-s + (−0.654 + 0.755i)8-s + (−0.381 + 2.65i)9-s + (0.415 + 0.909i)10-s + (1.84 + 0.540i)12-s + (−0.118 + 0.822i)14-s + (1.25 − 1.45i)15-s + (0.415 − 0.909i)16-s + (−0.381 − 2.65i)18-s + (−0.654 − 0.755i)20-s + (1.52 − 0.449i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.559 - 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.559 - 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $0.559 - 0.828i$
Analytic conductor: \(0.229569\)
Root analytic conductor: \(0.479134\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{460} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 460,\ (\ :0),\ 0.559 - 0.828i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8281799683\)
\(L(\frac12)\) \(\approx\) \(0.8281799683\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.959 - 0.281i)T \)
5 \( 1 + (0.142 + 0.989i)T \)
23 \( 1 + (0.959 - 0.281i)T \)
good3 \( 1 + (-1.25 - 1.45i)T + (-0.142 + 0.989i)T^{2} \)
7 \( 1 + (-0.345 + 0.755i)T + (-0.654 - 0.755i)T^{2} \)
11 \( 1 + (-0.841 - 0.540i)T^{2} \)
13 \( 1 + (0.654 - 0.755i)T^{2} \)
17 \( 1 + (-0.415 - 0.909i)T^{2} \)
19 \( 1 + (-0.415 + 0.909i)T^{2} \)
29 \( 1 + (0.239 + 0.153i)T + (0.415 + 0.909i)T^{2} \)
31 \( 1 + (0.142 + 0.989i)T^{2} \)
37 \( 1 + (0.959 + 0.281i)T^{2} \)
41 \( 1 + (0.118 + 0.822i)T + (-0.959 + 0.281i)T^{2} \)
43 \( 1 + (1.10 + 1.27i)T + (-0.142 + 0.989i)T^{2} \)
47 \( 1 + 0.284T + T^{2} \)
53 \( 1 + (0.654 + 0.755i)T^{2} \)
59 \( 1 + (0.654 - 0.755i)T^{2} \)
61 \( 1 + (1.10 - 1.27i)T + (-0.142 - 0.989i)T^{2} \)
67 \( 1 + (-1.25 + 0.368i)T + (0.841 - 0.540i)T^{2} \)
71 \( 1 + (-0.841 + 0.540i)T^{2} \)
73 \( 1 + (-0.415 + 0.909i)T^{2} \)
79 \( 1 + (0.654 - 0.755i)T^{2} \)
83 \( 1 + (-0.186 + 1.29i)T + (-0.959 - 0.281i)T^{2} \)
89 \( 1 + (-1.25 - 1.45i)T + (-0.142 + 0.989i)T^{2} \)
97 \( 1 + (0.959 - 0.281i)T^{2} \)
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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

−10.91483425537526987423312165401, −10.19537309865978822326717032335, −9.522710747798696199751936780274, −8.755556068806536079785110445606, −8.136987451593417341837442212695, −7.39156779526518463546868810782, −5.55989719576445126113024418349, −4.56350235431036066847800588024, −3.57263244467458726209747405409, −1.97729578106915619157977245097, 1.78484961541532501920619003560, 2.61989487524056147674429014415, 3.52415500206230296535913446515, 6.18339278268621029179183478186, 6.80005337923226235243679443240, 7.82889138959312838208494163486, 8.188029410847674472327900817327, 9.159501505886341486083140523919, 9.993010713726863114378531035350, 11.29868146232255924473062359527

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