Properties

Label 2-50e2-100.91-c0-0-3
Degree $2$
Conductor $2500$
Sign $0.728 + 0.684i$
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  + (−0.809 − 0.587i)2-s + (0.309 + 0.951i)4-s + (0.309 − 0.951i)8-s + (−0.809 + 0.587i)9-s + (1.30 − 0.951i)13-s + (−0.809 + 0.587i)16-s + (0.190 − 0.587i)17-s + 18-s − 1.61·26-s + (0.190 + 0.587i)29-s + 32-s + (−0.5 + 0.363i)34-s + (−0.809 − 0.587i)36-s + (−0.5 + 0.363i)37-s + (1.30 − 0.951i)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 + (1.30 − 0.951i)13-s + (−0.809 + 0.587i)16-s + (0.190 − 0.587i)17-s + 18-s − 1.61·26-s + (0.190 + 0.587i)29-s + 32-s + (−0.5 + 0.363i)34-s + (−0.809 − 0.587i)36-s + (−0.5 + 0.363i)37-s + (1.30 − 0.951i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.728 + 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.728 + 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2500\)    =    \(2^{2} \cdot 5^{4}\)
Sign: $0.728 + 0.684i$
Analytic conductor: \(1.24766\)
Root analytic conductor: \(1.11698\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2500} (2251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2500,\ (\ :0),\ 0.728 + 0.684i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8043669551\)
\(L(\frac12)\) \(\approx\) \(0.8043669551\)
\(L(1)\) 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 \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \)
17 \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \)
19 \( 1 + (0.809 + 0.587i)T^{2} \)
23 \( 1 + (-0.309 - 0.951i)T^{2} \)
29 \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \)
31 \( 1 + (0.809 + 0.587i)T^{2} \)
37 \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \)
41 \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.809 - 0.587i)T^{2} \)
53 \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \)
59 \( 1 + (-0.309 + 0.951i)T^{2} \)
61 \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \)
67 \( 1 + (0.809 + 0.587i)T^{2} \)
71 \( 1 + (0.809 - 0.587i)T^{2} \)
73 \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \)
79 \( 1 + (0.809 - 0.587i)T^{2} \)
83 \( 1 + (0.809 + 0.587i)T^{2} \)
89 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
97 \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \)
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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

−8.836794948844011131875188692137, −8.459485724253044539205796933030, −7.73799725874098532654204772274, −6.93156540062914761065020880410, −5.92656706472065396111437987045, −5.14382458355951634825515537200, −3.86946682190427220348141451577, −3.11328672026137483943424580955, −2.23444745983071046506044731110, −0.900456715950755990029111354991, 1.06815228181934673297886953534, 2.25898291840861027747833050121, 3.52906757319584919597598803973, 4.50679504468833932241103769960, 5.73440878559893036806229518181, 6.15227828822242522152735180767, 6.84163730965286167912515110580, 7.83772600348834377436165991896, 8.488103395688779964656832981053, 9.082144407673108099980334059526

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