Properties

Label 2-50e2-100.91-c0-0-0
Degree $2$
Conductor $2500$
Sign $-0.876 + 0.481i$
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 + (−0.587 + 0.190i)3-s + (−0.309 − 0.951i)4-s + (0.190 − 0.587i)6-s + 1.61i·7-s + (0.951 + 0.309i)8-s + (−0.5 + 0.363i)9-s + (0.363 + 0.5i)12-s + (−1.30 − 0.951i)14-s + (−0.809 + 0.587i)16-s − 0.618i·18-s + (−0.309 − 0.951i)21-s + (−0.951 + 1.30i)23-s − 0.618·24-s + (0.587 − 0.809i)27-s + (1.53 − 0.500i)28-s + ⋯
L(s)  = 1  + (−0.587 + 0.809i)2-s + (−0.587 + 0.190i)3-s + (−0.309 − 0.951i)4-s + (0.190 − 0.587i)6-s + 1.61i·7-s + (0.951 + 0.309i)8-s + (−0.5 + 0.363i)9-s + (0.363 + 0.5i)12-s + (−1.30 − 0.951i)14-s + (−0.809 + 0.587i)16-s − 0.618i·18-s + (−0.309 − 0.951i)21-s + (−0.951 + 1.30i)23-s − 0.618·24-s + (0.587 − 0.809i)27-s + (1.53 − 0.500i)28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.876 + 0.481i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.876 + 0.481i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2500\)    =    \(2^{2} \cdot 5^{4}\)
Sign: $-0.876 + 0.481i$
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: \((2,\ 2500,\ (\ :0),\ -0.876 + 0.481i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3622726954\)
\(L(\frac12)\) \(\approx\) \(0.3622726954\)
\(L(1)\) 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 \( 1 + (0.587 - 0.190i)T + (0.809 - 0.587i)T^{2} \)
7 \( 1 - 1.61iT - T^{2} \)
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 \( 1 + (0.951 - 1.30i)T + (-0.309 - 0.951i)T^{2} \)
29 \( 1 + (0.190 + 0.587i)T + (-0.809 + 0.587i)T^{2} \)
31 \( 1 + (0.809 + 0.587i)T^{2} \)
37 \( 1 + (0.309 - 0.951i)T^{2} \)
41 \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \)
43 \( 1 - 0.618iT - T^{2} \)
47 \( 1 + (-0.587 + 0.190i)T + (0.809 - 0.587i)T^{2} \)
53 \( 1 + (-0.809 + 0.587i)T^{2} \)
59 \( 1 + (-0.309 + 0.951i)T^{2} \)
61 \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \)
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 \( 1 + (1.53 + 0.5i)T + (0.809 + 0.587i)T^{2} \)
89 \( 1 + (1.30 + 0.951i)T + (0.309 + 0.951i)T^{2} \)
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

−9.470588703898915161078551844748, −8.643205747983097859091347639519, −8.179689443354622864875091979119, −7.31096942348632801452330722096, −6.23170166920619614438345675637, −5.71904517048376746642695875892, −5.30534784312345641148415691542, −4.30202060023720230905546737158, −2.78129991147965456166618915443, −1.74926181396743575189722097491, 0.32500793612336343149295070688, 1.41813181501329240009208764361, 2.76788593266052022347166623852, 3.79816719399189894255585496450, 4.37132083364064005931700686779, 5.48299985122091005827565842320, 6.67072828722449316470340746538, 7.09339002052025088065480015706, 8.037676404925900824665020162807, 8.664248548195685529570771369745

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