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Properties

Label 2-500-500.111-c0-0-0
Degree $2$
Conductor $500$
Sign $0.994 + 0.100i$
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  + (−0.425 − 0.904i)2-s + (−0.637 + 0.770i)4-s + (0.0627 + 0.998i)5-s + (0.968 + 0.248i)8-s + (−0.992 − 0.125i)9-s + (0.876 − 0.481i)10-s + (1.84 + 0.233i)13-s + (−0.187 − 0.982i)16-s + (1.26 + 1.52i)17-s + (0.309 + 0.951i)18-s + (−0.809 − 0.587i)20-s + (−0.992 + 0.125i)25-s + (−0.574 − 1.76i)26-s + (−0.200 − 0.316i)29-s + (−0.809 + 0.587i)32-s + ⋯
L(s)  = 1  + (−0.425 − 0.904i)2-s + (−0.637 + 0.770i)4-s + (0.0627 + 0.998i)5-s + (0.968 + 0.248i)8-s + (−0.992 − 0.125i)9-s + (0.876 − 0.481i)10-s + (1.84 + 0.233i)13-s + (−0.187 − 0.982i)16-s + (1.26 + 1.52i)17-s + (0.309 + 0.951i)18-s + (−0.809 − 0.587i)20-s + (−0.992 + 0.125i)25-s + (−0.574 − 1.76i)26-s + (−0.200 − 0.316i)29-s + (−0.809 + 0.587i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.100i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.100i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(500\)    =    \(2^{2} \cdot 5^{3}\)
Sign: $0.994 + 0.100i$
Analytic conductor: \(0.249532\)
Root analytic conductor: \(0.499532\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{500} (111, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 500,\ (\ :0),\ 0.994 + 0.100i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6859157206\)
\(L(\frac12)\) \(\approx\) \(0.6859157206\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.425 + 0.904i)T \)
5 \( 1 + (-0.0627 - 0.998i)T \)
good3 \( 1 + (0.992 + 0.125i)T^{2} \)
7 \( 1 + (0.809 + 0.587i)T^{2} \)
11 \( 1 + (0.637 - 0.770i)T^{2} \)
13 \( 1 + (-1.84 - 0.233i)T + (0.968 + 0.248i)T^{2} \)
17 \( 1 + (-1.26 - 1.52i)T + (-0.187 + 0.982i)T^{2} \)
19 \( 1 + (0.992 - 0.125i)T^{2} \)
23 \( 1 + (-0.535 - 0.844i)T^{2} \)
29 \( 1 + (0.200 + 0.316i)T + (-0.425 + 0.904i)T^{2} \)
31 \( 1 + (0.187 - 0.982i)T^{2} \)
37 \( 1 + (0.328 + 1.72i)T + (-0.929 + 0.368i)T^{2} \)
41 \( 1 + (1.11 - 0.614i)T + (0.535 - 0.844i)T^{2} \)
43 \( 1 + (-0.309 - 0.951i)T^{2} \)
47 \( 1 + (-0.876 + 0.481i)T^{2} \)
53 \( 1 + (0.996 + 0.394i)T + (0.728 + 0.684i)T^{2} \)
59 \( 1 + (-0.0627 - 0.998i)T^{2} \)
61 \( 1 + (-0.110 - 0.0604i)T + (0.535 + 0.844i)T^{2} \)
67 \( 1 + (0.425 + 0.904i)T^{2} \)
71 \( 1 + (-0.876 + 0.481i)T^{2} \)
73 \( 1 + (-1.41 + 1.32i)T + (0.0627 - 0.998i)T^{2} \)
79 \( 1 + (0.992 + 0.125i)T^{2} \)
83 \( 1 + (0.992 - 0.125i)T^{2} \)
89 \( 1 + (0.620 - 0.582i)T + (0.0627 - 0.998i)T^{2} \)
97 \( 1 + (-0.781 - 1.23i)T + (-0.425 + 0.904i)T^{2} \)
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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

−11.00596884690921562417173294105, −10.51108402728292976591943863760, −9.499371034486438713353395799990, −8.469293870661842078400384216777, −7.908247158567114055913614333208, −6.48618357035376565572239047134, −5.63081982093952190061210228899, −3.77750325161044938778932445502, −3.28127941674208694054285763752, −1.77834810822693312369575499424, 1.18729550736458619462429373059, 3.42887521655129635760673972908, 4.93287309706349136058902162925, 5.57734291052898859971001402031, 6.48010247820051446953245533396, 7.81652938389986269980755314339, 8.437750918161318920300887421101, 9.115702222054423151730659993786, 9.987394852429527493507871408320, 11.10633028852852023622392221031

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