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Properties

Label 2-50e2-100.79-c0-0-1
Degree $2$
Conductor $2500$
Sign $0.929 + 0.368i$
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  + (−0.309 + 0.951i)2-s + (−1.30 − 0.951i)3-s + (−0.809 − 0.587i)4-s + (1.30 − 0.951i)6-s − 0.618·7-s + (0.809 − 0.587i)8-s + (0.500 + 1.53i)9-s + (0.499 + 1.53i)12-s + (0.190 − 0.587i)14-s + (0.309 + 0.951i)16-s − 1.61·18-s + (0.809 + 0.587i)21-s + (−0.190 + 0.587i)23-s − 1.61·24-s + (0.309 − 0.951i)27-s + (0.5 + 0.363i)28-s + ⋯
L(s)  = 1  + (−0.309 + 0.951i)2-s + (−1.30 − 0.951i)3-s + (−0.809 − 0.587i)4-s + (1.30 − 0.951i)6-s − 0.618·7-s + (0.809 − 0.587i)8-s + (0.500 + 1.53i)9-s + (0.499 + 1.53i)12-s + (0.190 − 0.587i)14-s + (0.309 + 0.951i)16-s − 1.61·18-s + (0.809 + 0.587i)21-s + (−0.190 + 0.587i)23-s − 1.61·24-s + (0.309 − 0.951i)27-s + (0.5 + 0.363i)28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 + 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 + 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2500\)    =    \(2^{2} \cdot 5^{4}\)
Sign: $0.929 + 0.368i$
Analytic conductor: \(1.24766\)
Root analytic conductor: \(1.11698\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2500} (1999, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2500,\ (\ :0),\ 0.929 + 0.368i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4681656492\)
\(L(\frac12)\) \(\approx\) \(0.4681656492\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−8.909004968832851136355292259457, −8.053213654991646835518784997496, −7.26560271533105660366828322762, −6.70867637645829227635651644074, −6.17404689567621861619154803652, −5.41147387494631502569469772070, −4.78830838677638834278201643373, −3.52081820727396470738447883201, −1.81780882985048552637600785011, −0.59419872341747865676429970734, 0.880397737145067512000591798564, 2.54323362755843291156891669940, 3.54479439617205875246132175120, 4.42710052881032456329789226494, 4.92010890859246411725800940399, 5.98130236904461519546577517603, 6.58477440620241367059185558409, 7.81083196204754245025410632102, 8.662288538921565485678920526899, 9.612891559583808603436029473208

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