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Properties

Label 2-460-460.119-c0-0-0
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

\(L(\frac{1}{2})\) \(\approx\) \(0.8220629733\)
\(L(\frac12)\) \(\approx\) \(0.8220629733\)
\(L(1)\) 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

−11.59653185245431302017663168841, −10.78522047083123168175280991038, −9.713904148478878372304240654688, −8.915360432354618883884136987641, −7.70413684107038814351674998259, −6.77176947573617887181606481624, −5.67359554868459904413176750886, −5.50992257734782715243948842171, −4.07636416428298516283650858348, −2.60198901249585320706787028576, 1.15179526671110268792710479049, 2.65487663904368049588644007873, 4.18051391972138378195854566797, 5.30209095772964960157147324413, 5.86927373259338228072401588111, 6.95407994775750603139056059703, 8.663120080371824562975659310521, 9.283702618854526392541620644464, 10.39231888752591442913980441628, 10.94493479476526537486897894322

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