Properties

Label 2-500-100.91-c0-0-0
Degree $2$
Conductor $500$
Sign $0.425 - 0.904i$
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  + (0.809 + 0.587i)2-s + (0.309 + 0.951i)4-s + (−0.309 + 0.951i)8-s + (−0.809 + 0.587i)9-s + (0.5 − 0.363i)13-s + (−0.809 + 0.587i)16-s + (0.5 − 1.53i)17-s − 18-s + 0.618·26-s + (−0.5 − 1.53i)29-s − 32-s + (1.30 − 0.951i)34-s + (−0.809 − 0.587i)36-s + (−1.30 + 0.951i)37-s + (−0.5 + 0.363i)41-s + ⋯
L(s)  = 1  + (0.809 + 0.587i)2-s + (0.309 + 0.951i)4-s + (−0.309 + 0.951i)8-s + (−0.809 + 0.587i)9-s + (0.5 − 0.363i)13-s + (−0.809 + 0.587i)16-s + (0.5 − 1.53i)17-s − 18-s + 0.618·26-s + (−0.5 − 1.53i)29-s − 32-s + (1.30 − 0.951i)34-s + (−0.809 − 0.587i)36-s + (−1.30 + 0.951i)37-s + (−0.5 + 0.363i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.425 - 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.425 - 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(500\)    =    \(2^{2} \cdot 5^{3}\)
Sign: $0.425 - 0.904i$
Analytic conductor: \(0.249532\)
Root analytic conductor: \(0.499532\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{500} (451, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 500,\ (\ :0),\ 0.425 - 0.904i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.276169010\)
\(L(\frac12)\) \(\approx\) \(1.276169010\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.809 - 0.587i)T \)
5 \( 1 \)
good3 \( 1 + (0.809 - 0.587i)T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 + (-0.309 - 0.951i)T^{2} \)
13 \( 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2} \)
17 \( 1 + (-0.5 + 1.53i)T + (-0.809 - 0.587i)T^{2} \)
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 \( 1 + (1.30 - 0.951i)T + (0.309 - 0.951i)T^{2} \)
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 \( 1 + (0.190 + 0.587i)T + (-0.809 + 0.587i)T^{2} \)
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 \( 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2} \)
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 \( 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2} \)
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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

−11.59254151002989031291715689987, −10.63389285136999010913993163707, −9.379790965994361439111353849395, −8.325218650228243729722808444338, −7.66781577251715711708457081064, −6.61071024389396147058275865079, −5.58644150523634675874638064398, −4.90127302739369193307540039603, −3.54382219627713142150204716001, −2.50481158892233231583573835780, 1.68175807421680079578803111696, 3.22864378461474225646871697236, 3.98718938395307868810508543941, 5.38674994216360083097844497251, 6.08020554531281454284609123230, 7.07292977975925877799934291376, 8.534945737512449926868684487286, 9.262960130249037410894942683946, 10.48892355580520236459691459006, 10.95785539591166485099844598629

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