↑

Properties

Label 2-460-460.279-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

Related objects

Downloads

Learn more

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: $\chi_{460} (279, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 460,\ (\ :0),\ -0.820 - 0.571i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6522871208\)
\(L(\frac12)\) \(\approx\) \(0.6522871208\)
\(L(1)\) 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 \( 1 + (0.142 + 0.989i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.95231896594442543304417039312, −10.52374711498487837500104934515, −9.711507692213016887055117214995, −8.756100503222714934814643291891, −7.79670281334869581583042273367, −6.95368387967443465659001506538, −6.32182607468501586481170632472, −5.18358439087945534557141520621, −3.82348024005423556856409720504, −2.80117992340952729784848563039, 0.849431941322208047246249641184, 3.03524404479934423098229268519, 3.97003534077669070132707488547, 4.59556288865392932182860149915, 6.15980952279369038731781004679, 7.27249669375938940638263985647, 8.571576948937312295583595521326, 9.341695528117120854576872546699, 10.08665054090634038098033664620, 10.75553458746209574155402537786

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