Properties

Label 2-500-100.79-c0-0-0
Degree $2$
Conductor $500$
Sign $0.815 + 0.578i$
Analytic cond. $0.249532$
Root an. cond. $0.499532$
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  + (−0.951 − 0.309i)2-s + (0.809 + 0.587i)4-s + (−0.587 − 0.809i)8-s + (−0.309 − 0.951i)9-s + (1.53 − 0.5i)13-s + (0.309 + 0.951i)16-s + (0.363 + 0.5i)17-s + 0.999i·18-s − 1.61·26-s + (0.5 + 0.363i)29-s i·32-s + (−0.190 − 0.587i)34-s + (0.309 − 0.951i)36-s + (0.587 − 0.190i)37-s + (−0.5 − 1.53i)41-s + ⋯
L(s)  = 1  + (−0.951 − 0.309i)2-s + (0.809 + 0.587i)4-s + (−0.587 − 0.809i)8-s + (−0.309 − 0.951i)9-s + (1.53 − 0.5i)13-s + (0.309 + 0.951i)16-s + (0.363 + 0.5i)17-s + 0.999i·18-s − 1.61·26-s + (0.5 + 0.363i)29-s i·32-s + (−0.190 − 0.587i)34-s + (0.309 − 0.951i)36-s + (0.587 − 0.190i)37-s + (−0.5 − 1.53i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.815 + 0.578i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.815 + 0.578i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(500\)    =    \(2^{2} \cdot 5^{3}\)
Sign: $0.815 + 0.578i$
Analytic conductor: \(0.249532\)
Root analytic conductor: \(0.499532\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{500} (399, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 500,\ (\ :0),\ 0.815 + 0.578i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6097516571\)
\(L(\frac12)\) \(\approx\) \(0.6097516571\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.951 + 0.309i)T \)
5 \( 1 \)
good3 \( 1 + (0.309 + 0.951i)T^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 + (0.809 + 0.587i)T^{2} \)
13 \( 1 + (-1.53 + 0.5i)T + (0.809 - 0.587i)T^{2} \)
17 \( 1 + (-0.363 - 0.5i)T + (-0.309 + 0.951i)T^{2} \)
19 \( 1 + (-0.309 + 0.951i)T^{2} \)
23 \( 1 + (-0.809 - 0.587i)T^{2} \)
29 \( 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2} \)
31 \( 1 + (-0.309 + 0.951i)T^{2} \)
37 \( 1 + (-0.587 + 0.190i)T + (0.809 - 0.587i)T^{2} \)
41 \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (0.309 + 0.951i)T^{2} \)
53 \( 1 + (0.951 - 1.30i)T + (-0.309 - 0.951i)T^{2} \)
59 \( 1 + (0.809 - 0.587i)T^{2} \)
61 \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \)
67 \( 1 + (0.309 - 0.951i)T^{2} \)
71 \( 1 + (-0.309 - 0.951i)T^{2} \)
73 \( 1 + (1.53 + 0.5i)T + (0.809 + 0.587i)T^{2} \)
79 \( 1 + (-0.309 - 0.951i)T^{2} \)
83 \( 1 + (0.309 - 0.951i)T^{2} \)
89 \( 1 + (0.190 - 0.587i)T + (-0.809 - 0.587i)T^{2} \)
97 \( 1 + (0.363 - 0.5i)T + (-0.309 - 0.951i)T^{2} \)
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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

−10.88931817019635962496444108485, −10.24597179086817327433599966012, −9.138063347214793890790482109716, −8.597411134646744528875752775525, −7.67234477715786868572793615888, −6.50973880711564053983261231600, −5.81910784176456719317638894415, −3.90535862606247224614709386924, −3.02160885439423441866354197822, −1.27399716263776711177466042119, 1.61333522460566084204331460652, 3.06875015261999150688202887396, 4.77313475693043116195313092183, 5.92341050515362157971284899219, 6.71670564344817266796950823836, 7.918874730200162577683677714885, 8.404792249482688628805979049705, 9.419570590992556373202107282856, 10.25391640092427965122692989174, 11.23484497350181169030966220824

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