Properties

Label 2-50e2-100.59-c0-0-2
Degree $2$
Conductor $2500$
Sign $0.684 - 0.728i$
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 + (−0.309 − 0.951i)4-s + (0.951 + 0.309i)8-s + (0.809 − 0.587i)9-s + (0.951 + 1.30i)13-s + (−0.809 + 0.587i)16-s + (−0.587 − 0.190i)17-s + i·18-s − 1.61·26-s + (−0.190 − 0.587i)29-s i·32-s + (0.5 − 0.363i)34-s + (−0.809 − 0.587i)36-s + (0.363 + 0.5i)37-s + (1.30 − 0.951i)41-s + ⋯
L(s)  = 1  + (−0.587 + 0.809i)2-s + (−0.309 − 0.951i)4-s + (0.951 + 0.309i)8-s + (0.809 − 0.587i)9-s + (0.951 + 1.30i)13-s + (−0.809 + 0.587i)16-s + (−0.587 − 0.190i)17-s + i·18-s − 1.61·26-s + (−0.190 − 0.587i)29-s i·32-s + (0.5 − 0.363i)34-s + (−0.809 − 0.587i)36-s + (0.363 + 0.5i)37-s + (1.30 − 0.951i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.684 - 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.684 - 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2500\)    =    \(2^{2} \cdot 5^{4}\)
Sign: $0.684 - 0.728i$
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: \((2,\ 2500,\ (\ :0),\ 0.684 - 0.728i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9596693531\)
\(L(\frac12)\) \(\approx\) \(0.9596693531\)
\(L(1)\) 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 \( 1 + (-0.809 + 0.587i)T^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 + (-0.309 - 0.951i)T^{2} \)
13 \( 1 + (-0.951 - 1.30i)T + (-0.309 + 0.951i)T^{2} \)
17 \( 1 + (0.587 + 0.190i)T + (0.809 + 0.587i)T^{2} \)
19 \( 1 + (0.809 + 0.587i)T^{2} \)
23 \( 1 + (0.309 + 0.951i)T^{2} \)
29 \( 1 + (0.190 + 0.587i)T + (-0.809 + 0.587i)T^{2} \)
31 \( 1 + (0.809 + 0.587i)T^{2} \)
37 \( 1 + (-0.363 - 0.5i)T + (-0.309 + 0.951i)T^{2} \)
41 \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (-0.809 + 0.587i)T^{2} \)
53 \( 1 + (-1.53 + 0.5i)T + (0.809 - 0.587i)T^{2} \)
59 \( 1 + (-0.309 + 0.951i)T^{2} \)
61 \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \)
67 \( 1 + (-0.809 - 0.587i)T^{2} \)
71 \( 1 + (0.809 - 0.587i)T^{2} \)
73 \( 1 + (0.951 - 1.30i)T + (-0.309 - 0.951i)T^{2} \)
79 \( 1 + (0.809 - 0.587i)T^{2} \)
83 \( 1 + (-0.809 - 0.587i)T^{2} \)
89 \( 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2} \)
97 \( 1 + (-0.587 + 0.190i)T + (0.809 - 0.587i)T^{2} \)
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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

−9.078384237919109956829406438384, −8.567948820875594678599351710943, −7.58830408453701612413111884670, −6.85004440466613461652368538365, −6.39383604982125255962776850392, −5.52510234716843048877038324838, −4.39263511120083179412647862269, −3.91606213651111719508366354945, −2.19477488653037964372345342282, −1.10799447974058672047951640002, 1.05011273037368801703507147097, 2.13760148543456719917480459789, 3.16087552154813809890806011833, 4.02219669800612266713196779399, 4.85150707528879494676363011036, 5.89104972021015955691712935521, 6.94747097660788011730607683384, 7.74921776409781324221501610650, 8.270903056745153199843245102684, 9.073109017940789330124878489957

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