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Properties

Label 2-50e2-100.79-c0-0-0
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  + (0.309 − 0.951i)2-s + (−0.5 − 0.363i)3-s + (−0.809 − 0.587i)4-s + (−0.5 + 0.363i)6-s − 1.61·7-s + (−0.809 + 0.587i)8-s + (−0.190 − 0.587i)9-s + (0.190 + 0.587i)12-s + (−0.500 + 1.53i)14-s + (0.309 + 0.951i)16-s − 0.618·18-s + (0.809 + 0.587i)21-s + (−0.5 + 1.53i)23-s + 0.618·24-s + (−0.309 + 0.951i)27-s + (1.30 + 0.951i)28-s + ⋯
L(s)  = 1  + (0.309 − 0.951i)2-s + (−0.5 − 0.363i)3-s + (−0.809 − 0.587i)4-s + (−0.5 + 0.363i)6-s − 1.61·7-s + (−0.809 + 0.587i)8-s + (−0.190 − 0.587i)9-s + (0.190 + 0.587i)12-s + (−0.500 + 1.53i)14-s + (0.309 + 0.951i)16-s − 0.618·18-s + (0.809 + 0.587i)21-s + (−0.5 + 1.53i)23-s + 0.618·24-s + (−0.309 + 0.951i)27-s + (1.30 + 0.951i)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: $\chi_{2500} (1999, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2500,\ (\ :0),\ 0.637 - 0.770i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1547038809\)
\(L(\frac12)\) \(\approx\) \(0.1547038809\)
\(L(1)\) 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 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
7 \( 1 + 1.61T + T^{2} \)
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 \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \)
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 \( 1 - 0.618T + T^{2} \)
47 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
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 \( 1 + (1.61 - 1.17i)T + (0.309 - 0.951i)T^{2} \)
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 \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \)
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

−9.465381084049138238837142863626, −8.816091107816031871848806682945, −7.59034707351027836765961801597, −6.67109856615885111442917698745, −5.95594482080390037506365610999, −5.49351499201191168127081265547, −4.13660115052624248464680896495, −3.45798275297208945867002549464, −2.69945112830998560793613442524, −1.31329506776727121976079557250, 0.10226218936003877317716632035, 2.57085981066576402223042236850, 3.51816609724517987376546169782, 4.37639939261935441017678161702, 5.18435912036940852342520134532, 6.08077878203059521257669762957, 6.42759653189872476815767693084, 7.32791853162809149416096488965, 8.129029087979722495918571898673, 8.992524225984388873365467041634

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