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Properties

Label 2-50e2-100.31-c0-0-3
Degree $2$
Conductor $2500$
Sign $0.637 + 0.770i$
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  + (0.951 − 0.309i)2-s + (−0.363 − 0.5i)3-s + (0.809 − 0.587i)4-s + (−0.5 − 0.363i)6-s + 1.61i·7-s + (0.587 − 0.809i)8-s + (0.190 − 0.587i)9-s + (−0.587 − 0.190i)12-s + (0.500 + 1.53i)14-s + (0.309 − 0.951i)16-s − 0.618i·18-s + (0.809 − 0.587i)21-s + (1.53 − 0.5i)23-s − 0.618·24-s + (−0.951 + 0.309i)27-s + (0.951 + 1.30i)28-s + ⋯
L(s)  = 1  + (0.951 − 0.309i)2-s + (−0.363 − 0.5i)3-s + (0.809 − 0.587i)4-s + (−0.5 − 0.363i)6-s + 1.61i·7-s + (0.587 − 0.809i)8-s + (0.190 − 0.587i)9-s + (−0.587 − 0.190i)12-s + (0.500 + 1.53i)14-s + (0.309 − 0.951i)16-s − 0.618i·18-s + (0.809 − 0.587i)21-s + (1.53 − 0.5i)23-s − 0.618·24-s + (−0.951 + 0.309i)27-s + (0.951 + 1.30i)28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.637 + 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.637 + 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2500\)    =    \(2^{2} \cdot 5^{4}\)
Sign: $0.637 + 0.770i$
Analytic conductor: \(1.24766\)
Root analytic conductor: \(1.11698\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2500} (751, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2500,\ (\ :0),\ 0.637 + 0.770i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.038281062\)
\(L(\frac12)\) \(\approx\) \(2.038281062\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.951 + 0.309i)T \)
5 \( 1 \)
good3 \( 1 + (0.363 + 0.5i)T + (-0.309 + 0.951i)T^{2} \)
7 \( 1 - 1.61iT - T^{2} \)
11 \( 1 + (0.809 - 0.587i)T^{2} \)
13 \( 1 + (-0.809 - 0.587i)T^{2} \)
17 \( 1 + (0.309 + 0.951i)T^{2} \)
19 \( 1 + (-0.309 - 0.951i)T^{2} \)
23 \( 1 + (-1.53 + 0.5i)T + (0.809 - 0.587i)T^{2} \)
29 \( 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2} \)
31 \( 1 + (-0.309 - 0.951i)T^{2} \)
37 \( 1 + (-0.809 - 0.587i)T^{2} \)
41 \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \)
43 \( 1 - 0.618iT - T^{2} \)
47 \( 1 + (-0.363 - 0.5i)T + (-0.309 + 0.951i)T^{2} \)
53 \( 1 + (0.309 - 0.951i)T^{2} \)
59 \( 1 + (0.809 + 0.587i)T^{2} \)
61 \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \)
67 \( 1 + (1.17 - 1.61i)T + (-0.309 - 0.951i)T^{2} \)
71 \( 1 + (-0.309 + 0.951i)T^{2} \)
73 \( 1 + (-0.809 + 0.587i)T^{2} \)
79 \( 1 + (-0.309 + 0.951i)T^{2} \)
83 \( 1 + (0.951 - 1.30i)T + (-0.309 - 0.951i)T^{2} \)
89 \( 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2} \)
97 \( 1 + (0.309 - 0.951i)T^{2} \)
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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

−9.134243028353035364487516267976, −8.209882019883114131649714945682, −7.10717537580601376829572731898, −6.50197103788852191967605258161, −5.80803069247723096205514599945, −5.21494761690833408035255513524, −4.27734163056205190914135954593, −3.10586889761170867483202245624, −2.43411741747403841401146720769, −1.27152748627914436400486702395, 1.45099762011790078038706254524, 2.91801339883104498661046554793, 3.81471589214732982108705302363, 4.53745996726259665182123444342, 5.06198705789343238629797757092, 5.99901059458384041059656834966, 7.02931525235320627353719705252, 7.34337659954529163508196517041, 8.163910923326706358926327529069, 9.256131188551315450049071296363

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