Properties

Label 2-50e2-100.11-c0-0-2
Degree $2$
Conductor $2500$
Sign $0.992 - 0.125i$
Analytic cond. $1.24766$
Root an. cond. $1.11698$
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 + (−1.53 − 0.5i)3-s + (−0.309 + 0.951i)4-s + (−0.5 − 1.53i)6-s − 0.618i·7-s + (−0.951 + 0.309i)8-s + (1.30 + 0.951i)9-s + (0.951 − 1.30i)12-s + (0.500 − 0.363i)14-s + (−0.809 − 0.587i)16-s + 1.61i·18-s + (−0.309 + 0.951i)21-s + (−0.363 − 0.5i)23-s + 1.61·24-s + (−0.587 − 0.809i)27-s + (0.587 + 0.190i)28-s + ⋯
L(s)  = 1  + (0.587 + 0.809i)2-s + (−1.53 − 0.5i)3-s + (−0.309 + 0.951i)4-s + (−0.5 − 1.53i)6-s − 0.618i·7-s + (−0.951 + 0.309i)8-s + (1.30 + 0.951i)9-s + (0.951 − 1.30i)12-s + (0.500 − 0.363i)14-s + (−0.809 − 0.587i)16-s + 1.61i·18-s + (−0.309 + 0.951i)21-s + (−0.363 − 0.5i)23-s + 1.61·24-s + (−0.587 − 0.809i)27-s + (0.587 + 0.190i)28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.125i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.125i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2500\)    =    \(2^{2} \cdot 5^{4}\)
Sign: $0.992 - 0.125i$
Analytic conductor: \(1.24766\)
Root analytic conductor: \(1.11698\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2500} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2500,\ (\ :0),\ 0.992 - 0.125i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8351012570\)
\(L(\frac12)\) \(\approx\) \(0.8351012570\)
\(L(1)\) 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 \( 1 + (1.53 + 0.5i)T + (0.809 + 0.587i)T^{2} \)
7 \( 1 + 0.618iT - T^{2} \)
11 \( 1 + (-0.309 + 0.951i)T^{2} \)
13 \( 1 + (0.309 + 0.951i)T^{2} \)
17 \( 1 + (-0.809 + 0.587i)T^{2} \)
19 \( 1 + (0.809 - 0.587i)T^{2} \)
23 \( 1 + (0.363 + 0.5i)T + (-0.309 + 0.951i)T^{2} \)
29 \( 1 + (-0.5 + 1.53i)T + (-0.809 - 0.587i)T^{2} \)
31 \( 1 + (0.809 - 0.587i)T^{2} \)
37 \( 1 + (0.309 + 0.951i)T^{2} \)
41 \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \)
43 \( 1 + 1.61iT - T^{2} \)
47 \( 1 + (-1.53 - 0.5i)T + (0.809 + 0.587i)T^{2} \)
53 \( 1 + (-0.809 - 0.587i)T^{2} \)
59 \( 1 + (-0.309 - 0.951i)T^{2} \)
61 \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \)
67 \( 1 + (-1.90 + 0.618i)T + (0.809 - 0.587i)T^{2} \)
71 \( 1 + (0.809 + 0.587i)T^{2} \)
73 \( 1 + (0.309 - 0.951i)T^{2} \)
79 \( 1 + (0.809 + 0.587i)T^{2} \)
83 \( 1 + (0.587 - 0.190i)T + (0.809 - 0.587i)T^{2} \)
89 \( 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2} \)
97 \( 1 + (-0.809 - 0.587i)T^{2} \)
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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

−8.961606690057294216861552537988, −7.917851153111174399446483456217, −7.38352182824106144091003633849, −6.61082808566534714245662773596, −6.07165106817189966703430796939, −5.41712970354310008669973435020, −4.53330768371920553982158128893, −3.92797736135726459682657788441, −2.43230910503552835816862843487, −0.70424549784273213239820526894, 1.05050473353857816995691392772, 2.39559070974866594647875387314, 3.57807358097184774797934981632, 4.44216034535009297279470184156, 5.17472415395217950920013344882, 5.73668458550283964444848829289, 6.33946407962144978150390410438, 7.27103343700753195499013740995, 8.657816282111984193168235964510, 9.387352743719960961251605637523

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