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Properties

Label 2-50e2-100.31-c0-0-4
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

\(L(\frac{1}{2})\) \(\approx\) \(0.8142159939\)
\(L(\frac12)\) \(\approx\) \(0.8142159939\)
\(L(1)\) 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

−9.153325329533513510829227954136, −8.105188975223415867399326028794, −7.69802562038385982802050103185, −6.77281575466231143706749815353, −6.27979233183552976514865895008, −5.05670184341011107487522044858, −4.02642854632506720358379902359, −3.39937507200025083265685515587, −1.97799285139323930590400927942, −0.71018602798468856769516035871, 1.55960895735471721871116825581, 2.38406122563998450689499898681, 2.96022010774513746667941485020, 4.34704978080347132322989726240, 5.54661851190633755322309183354, 6.28157561580941330616919772697, 7.10183109389334130456490599031, 8.106956444688785780933155711717, 8.305408904224305718277361094135, 9.104684851093010004017949004334

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