Properties

Label 2-50e2-100.19-c0-0-4
Degree $2$
Conductor $2500$
Sign $0.844 + 0.535i$
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 + (0.809 − 0.587i)4-s + (0.587 − 0.809i)8-s + (−0.309 + 0.951i)9-s + (0.587 + 0.190i)13-s + (0.309 − 0.951i)16-s + (0.951 − 1.30i)17-s + 0.999i·18-s + 0.618·26-s + (−1.30 + 0.951i)29-s i·32-s + (0.499 − 1.53i)34-s + (0.309 + 0.951i)36-s + (1.53 + 0.5i)37-s + (0.190 − 0.587i)41-s + ⋯
L(s)  = 1  + (0.951 − 0.309i)2-s + (0.809 − 0.587i)4-s + (0.587 − 0.809i)8-s + (−0.309 + 0.951i)9-s + (0.587 + 0.190i)13-s + (0.309 − 0.951i)16-s + (0.951 − 1.30i)17-s + 0.999i·18-s + 0.618·26-s + (−1.30 + 0.951i)29-s i·32-s + (0.499 − 1.53i)34-s + (0.309 + 0.951i)36-s + (1.53 + 0.5i)37-s + (0.190 − 0.587i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.844 + 0.535i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.844 + 0.535i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2500\)    =    \(2^{2} \cdot 5^{4}\)
Sign: $0.844 + 0.535i$
Analytic conductor: \(1.24766\)
Root analytic conductor: \(1.11698\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2500} (499, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2500,\ (\ :0),\ 0.844 + 0.535i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.247843640\)
\(L(\frac12)\) \(\approx\) \(2.247843640\)
\(L(1)\) 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 \( 1 + (0.309 - 0.951i)T^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 + (0.809 - 0.587i)T^{2} \)
13 \( 1 + (-0.587 - 0.190i)T + (0.809 + 0.587i)T^{2} \)
17 \( 1 + (-0.951 + 1.30i)T + (-0.309 - 0.951i)T^{2} \)
19 \( 1 + (-0.309 - 0.951i)T^{2} \)
23 \( 1 + (-0.809 + 0.587i)T^{2} \)
29 \( 1 + (1.30 - 0.951i)T + (0.309 - 0.951i)T^{2} \)
31 \( 1 + (-0.309 - 0.951i)T^{2} \)
37 \( 1 + (-1.53 - 0.5i)T + (0.809 + 0.587i)T^{2} \)
41 \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (0.309 - 0.951i)T^{2} \)
53 \( 1 + (0.363 + 0.5i)T + (-0.309 + 0.951i)T^{2} \)
59 \( 1 + (0.809 + 0.587i)T^{2} \)
61 \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \)
67 \( 1 + (0.309 + 0.951i)T^{2} \)
71 \( 1 + (-0.309 + 0.951i)T^{2} \)
73 \( 1 + (0.587 - 0.190i)T + (0.809 - 0.587i)T^{2} \)
79 \( 1 + (-0.309 + 0.951i)T^{2} \)
83 \( 1 + (0.309 + 0.951i)T^{2} \)
89 \( 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2} \)
97 \( 1 + (0.951 + 1.30i)T + (-0.309 + 0.951i)T^{2} \)
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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

−9.212005797763672034787448451084, −8.059490514428628971290095254255, −7.43135419430000842565071412971, −6.62442109036484551620187371774, −5.62171261080230465408455876826, −5.18396476268936980214004123645, −4.26856163391458613995349159481, −3.29322901309741937730363633878, −2.51156753555057922610181773711, −1.37358505941803403156758882708, 1.49698339769286531086402846576, 2.81201456969045353424007726141, 3.68114378799668163629428273763, 4.21397714255248979253885898440, 5.46511811082813290352176007448, 6.03599250601619770518332428976, 6.54623312504236163957111081702, 7.71354421216587171413924538263, 8.100084456902377755872647803646, 9.106516237197797618422824147848

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