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Properties

Label 2-500-100.59-c0-0-1
Degree $2$
Conductor $500$
Sign $0.113 + 0.993i$
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  + (0.587 − 0.809i)2-s + (−0.309 − 0.951i)4-s + (−0.951 − 0.309i)8-s + (0.809 − 0.587i)9-s + (0.363 + 0.5i)13-s + (−0.809 + 0.587i)16-s + (−1.53 − 0.5i)17-s − i·18-s + 0.618·26-s + (0.5 + 1.53i)29-s + i·32-s + (−1.30 + 0.951i)34-s + (−0.809 − 0.587i)36-s + (0.951 + 1.30i)37-s + (−0.5 + 0.363i)41-s + ⋯
L(s)  = 1  + (0.587 − 0.809i)2-s + (−0.309 − 0.951i)4-s + (−0.951 − 0.309i)8-s + (0.809 − 0.587i)9-s + (0.363 + 0.5i)13-s + (−0.809 + 0.587i)16-s + (−1.53 − 0.5i)17-s − i·18-s + 0.618·26-s + (0.5 + 1.53i)29-s + i·32-s + (−1.30 + 0.951i)34-s + (−0.809 − 0.587i)36-s + (0.951 + 1.30i)37-s + (−0.5 + 0.363i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.113 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.113 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(500\)    =    \(2^{2} \cdot 5^{3}\)
Sign: $0.113 + 0.993i$
Analytic conductor: \(0.249532\)
Root analytic conductor: \(0.499532\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{500} (299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 500,\ (\ :0),\ 0.113 + 0.993i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.132292615\)
\(L(\frac12)\) \(\approx\) \(1.132292615\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−11.11186153040731635121268535947, −10.16491238504858916015812892946, −9.360501634761487931551469773632, −8.584649499171081999679384329472, −6.94253286307170467190946832444, −6.34318568952491199003375412975, −4.92181556655166736244648475433, −4.18138019345118271356680949538, −2.98893205220437398064675246322, −1.53932250961726095328118610814, 2.37563363174862717024295042731, 3.93456442801136042716346287352, 4.67688022754285680200155746501, 5.85311237040924469954682677941, 6.70555393708153919780878436089, 7.66520216386631015953572771647, 8.410849869079441102516618011727, 9.409463776406769703161789200193, 10.54060290474907860121218863535, 11.41742989088817812572410886999

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