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Properties

Label 2-50e2-500.91-c0-0-0
Degree $2$
Conductor $2500$
Sign $0.711 + 0.702i$
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  + (−0.728 − 0.684i)2-s + (0.0627 + 0.998i)4-s + (0.637 − 0.770i)8-s + (−0.425 − 0.904i)9-s + (0.824 + 1.75i)13-s + (−0.992 + 0.125i)16-s + (0.0534 − 0.849i)17-s + (−0.309 + 0.951i)18-s + (0.598 − 1.84i)26-s + (1.84 − 0.730i)29-s + (0.809 + 0.587i)32-s + (−0.620 + 0.582i)34-s + (0.876 − 0.481i)36-s + (−0.371 + 0.0469i)37-s + (−0.0235 + 0.123i)41-s + ⋯
L(s)  = 1  + (−0.728 − 0.684i)2-s + (0.0627 + 0.998i)4-s + (0.637 − 0.770i)8-s + (−0.425 − 0.904i)9-s + (0.824 + 1.75i)13-s + (−0.992 + 0.125i)16-s + (0.0534 − 0.849i)17-s + (−0.309 + 0.951i)18-s + (0.598 − 1.84i)26-s + (1.84 − 0.730i)29-s + (0.809 + 0.587i)32-s + (−0.620 + 0.582i)34-s + (0.876 − 0.481i)36-s + (−0.371 + 0.0469i)37-s + (−0.0235 + 0.123i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.711 + 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.711 + 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2500\)    =    \(2^{2} \cdot 5^{4}\)
Sign: $0.711 + 0.702i$
Analytic conductor: \(1.24766\)
Root analytic conductor: \(1.11698\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2500} (1451, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2500,\ (\ :0),\ 0.711 + 0.702i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8407303932\)
\(L(\frac12)\) \(\approx\) \(0.8407303932\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.728 + 0.684i)T \)
5 \( 1 \)
good3 \( 1 + (0.425 + 0.904i)T^{2} \)
7 \( 1 + (0.809 - 0.587i)T^{2} \)
11 \( 1 + (-0.0627 - 0.998i)T^{2} \)
13 \( 1 + (-0.824 - 1.75i)T + (-0.637 + 0.770i)T^{2} \)
17 \( 1 + (-0.0534 + 0.849i)T + (-0.992 - 0.125i)T^{2} \)
19 \( 1 + (0.425 - 0.904i)T^{2} \)
23 \( 1 + (0.929 - 0.368i)T^{2} \)
29 \( 1 + (-1.84 + 0.730i)T + (0.728 - 0.684i)T^{2} \)
31 \( 1 + (0.992 + 0.125i)T^{2} \)
37 \( 1 + (0.371 - 0.0469i)T + (0.968 - 0.248i)T^{2} \)
41 \( 1 + (0.0235 - 0.123i)T + (-0.929 - 0.368i)T^{2} \)
43 \( 1 + (-0.309 + 0.951i)T^{2} \)
47 \( 1 + (0.187 - 0.982i)T^{2} \)
53 \( 1 + (-1.80 - 0.462i)T + (0.876 + 0.481i)T^{2} \)
59 \( 1 + (-0.535 - 0.844i)T^{2} \)
61 \( 1 + (0.200 + 1.05i)T + (-0.929 + 0.368i)T^{2} \)
67 \( 1 + (-0.728 - 0.684i)T^{2} \)
71 \( 1 + (0.187 - 0.982i)T^{2} \)
73 \( 1 + (-1.11 + 0.614i)T + (0.535 - 0.844i)T^{2} \)
79 \( 1 + (0.425 + 0.904i)T^{2} \)
83 \( 1 + (0.425 - 0.904i)T^{2} \)
89 \( 1 + (-1.27 + 0.702i)T + (0.535 - 0.844i)T^{2} \)
97 \( 1 + (-1.62 + 0.645i)T + (0.728 - 0.684i)T^{2} \)
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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

−9.032438680746707177012711647907, −8.579334460734014390432067655621, −7.63329352621746171390415157848, −6.66831725812153133932754808180, −6.29984935185668617974054407384, −4.80411777911758022883778907208, −3.99642680117117698386112469770, −3.16662221771960472787401647547, −2.16176441869538905080199325174, −0.957359308938235313964507426259, 1.03831079036850339634791597982, 2.33969879390601902662971769891, 3.44266424544714408610725603800, 4.79167118023486746645242423679, 5.48656175369545581286840617356, 6.10727627387746632844460937184, 6.99324054497004119804883572597, 7.922493907782021654139826830409, 8.353034622375917817650815894339, 8.842058227002002947955083924147

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