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Properties

Label 2-460-115.22-c1-0-1
Degree $2$
Conductor $460$
Sign $-0.847 - 0.531i$
Analytic cond. $3.67311$
Root an. cond. $1.91653$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 + i)3-s + (−0.386 + 2.20i)5-s + (0.386 − 0.386i)7-s + i·9-s + 0.772i·11-s + (2.70 − 2.70i)13-s + (−1.81 − 2.58i)15-s + (−4.79 + 4.79i)17-s − 5.95·19-s + 0.772i·21-s + (−3.87 + 2.82i)23-s + (−4.70 − 1.70i)25-s + (−4 − 4i)27-s + 2.29i·29-s + 9.10·31-s + ⋯
L(s)  = 1  + (−0.577 + 0.577i)3-s + (−0.172 + 0.984i)5-s + (0.146 − 0.146i)7-s + 0.333i·9-s + 0.232i·11-s + (0.749 − 0.749i)13-s + (−0.468 − 0.668i)15-s + (−1.16 + 1.16i)17-s − 1.36·19-s + 0.168i·21-s + (−0.807 + 0.589i)23-s + (−0.940 − 0.340i)25-s + (−0.769 − 0.769i)27-s + 0.426i·29-s + 1.63·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.847 - 0.531i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.847 - 0.531i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $-0.847 - 0.531i$
Analytic conductor: \(3.67311\)
Root analytic conductor: \(1.91653\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{460} (137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 460,\ (\ :1/2),\ -0.847 - 0.531i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.218150 + 0.758741i\)
\(L(\frac12)\) \(\approx\) \(0.218150 + 0.758741i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (0.386 - 2.20i)T \)
23 \( 1 + (3.87 - 2.82i)T \)
good3 \( 1 + (1 - i)T - 3iT^{2} \)
7 \( 1 + (-0.386 + 0.386i)T - 7iT^{2} \)
11 \( 1 - 0.772iT - 11T^{2} \)
13 \( 1 + (-2.70 + 2.70i)T - 13iT^{2} \)
17 \( 1 + (4.79 - 4.79i)T - 17iT^{2} \)
19 \( 1 + 5.95T + 19T^{2} \)
29 \( 1 - 2.29iT - 29T^{2} \)
31 \( 1 - 9.10T + 31T^{2} \)
37 \( 1 + (4.79 - 4.79i)T - 37iT^{2} \)
41 \( 1 - 7.70T + 41T^{2} \)
43 \( 1 + (4.13 + 4.13i)T + 43iT^{2} \)
47 \( 1 + (6.40 + 6.40i)T + 47iT^{2} \)
53 \( 1 + (-5.56 - 5.56i)T + 53iT^{2} \)
59 \( 1 - 3.70iT - 59T^{2} \)
61 \( 1 - 7.49iT - 61T^{2} \)
67 \( 1 + (-6.33 + 6.33i)T - 67iT^{2} \)
71 \( 1 - 2.29T + 71T^{2} \)
73 \( 1 + (2.40 - 2.40i)T - 73iT^{2} \)
79 \( 1 - 13.2T + 79T^{2} \)
83 \( 1 + (0.386 + 0.386i)T + 83iT^{2} \)
89 \( 1 - 16.8T + 89T^{2} \)
97 \( 1 + (-10.6 + 10.6i)T - 97iT^{2} \)
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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

−11.11192537094483814500658672867, −10.55921540312190102815958323512, −10.12502613008045390436821271436, −8.603453774165910780220642694134, −7.85570656651291434156609417180, −6.57278898110856874294827739036, −5.93265155262810627591515778281, −4.59022329943581041641818492040, −3.71547888989345970484720092892, −2.18607161634898865016992946037, 0.51070447740054458583285326032, 2.05772276677572188733281530455, 4.00283528446624759119570319188, 4.89118462483169780890352076522, 6.18488844353047836587441505899, 6.70226710613350149309929787422, 8.105212140152921016505256967755, 8.802248142198307542706929976686, 9.629040569954693370210253481816, 11.02639798340769235469835662640

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