Properties

Label 2-50e2-100.59-c0-0-1
Degree $2$
Conductor $2500$
Sign $-0.992 - 0.125i$
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 + (0.5 + 1.53i)3-s + (0.309 + 0.951i)4-s + (−0.5 + 1.53i)6-s − 0.618·7-s + (−0.309 + 0.951i)8-s + (−1.30 + 0.951i)9-s + (−1.30 + 0.951i)12-s + (−0.500 − 0.363i)14-s + (−0.809 + 0.587i)16-s − 1.61·18-s + (−0.309 − 0.951i)21-s + (0.5 + 0.363i)23-s − 1.61·24-s + (−0.809 − 0.587i)27-s + (−0.190 − 0.587i)28-s + ⋯
L(s)  = 1  + (0.809 + 0.587i)2-s + (0.5 + 1.53i)3-s + (0.309 + 0.951i)4-s + (−0.5 + 1.53i)6-s − 0.618·7-s + (−0.309 + 0.951i)8-s + (−1.30 + 0.951i)9-s + (−1.30 + 0.951i)12-s + (−0.500 − 0.363i)14-s + (−0.809 + 0.587i)16-s − 1.61·18-s + (−0.309 − 0.951i)21-s + (0.5 + 0.363i)23-s − 1.61·24-s + (−0.809 − 0.587i)27-s + (−0.190 − 0.587i)28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 - 0.125i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 - 0.125i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2500\)    =    \(2^{2} \cdot 5^{4}\)
Sign: $-0.992 - 0.125i$
Analytic conductor: \(1.24766\)
Root analytic conductor: \(1.11698\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2500} (1499, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2500,\ (\ :0),\ -0.992 - 0.125i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.977445347\)
\(L(\frac12)\) \(\approx\) \(1.977445347\)
\(L(1)\) 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 \( 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2} \)
7 \( 1 + 0.618T + T^{2} \)
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 \( 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2} \)
29 \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \)
31 \( 1 + (0.809 + 0.587i)T^{2} \)
37 \( 1 + (-0.309 + 0.951i)T^{2} \)
41 \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \)
43 \( 1 - 1.61T + T^{2} \)
47 \( 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2} \)
53 \( 1 + (0.809 - 0.587i)T^{2} \)
59 \( 1 + (-0.309 + 0.951i)T^{2} \)
61 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
67 \( 1 + (0.618 - 1.90i)T + (-0.809 - 0.587i)T^{2} \)
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 \( 1 + (0.190 - 0.587i)T + (-0.809 - 0.587i)T^{2} \)
89 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
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

−9.337185565426151960265831030541, −8.859463013026715928483031552976, −7.898091544576949359452345339648, −7.21245284959818266361017934380, −6.04249410243797351577465301196, −5.57457571373889726278445499898, −4.46203862343632856159862758270, −4.09924138041649695139441526880, −3.18877364614556452628517955038, −2.49823782775422098636233583311, 0.960225447932536745197182587749, 2.00617053881978415967273458275, 2.84018867329524142305940854648, 3.55155787620817657507054437706, 4.74010947108876759133897983603, 5.79360395593900857721083594973, 6.42213669012807378572691820264, 7.09868666405295316275535525753, 7.71601474248302653145730681110, 8.849213853674028492945501352622

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