Properties

Label 2-50e2-100.11-c0-0-5
Degree $2$
Conductor $2500$
Sign $-0.728 + 0.684i$
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 + (0.309 − 0.951i)4-s + (−0.309 − 0.951i)8-s + (−0.809 − 0.587i)9-s + (−1.30 − 0.951i)13-s + (−0.809 − 0.587i)16-s + (−0.190 − 0.587i)17-s − 18-s − 1.61·26-s + (0.190 − 0.587i)29-s − 32-s + (−0.5 − 0.363i)34-s + (−0.809 + 0.587i)36-s + (0.5 + 0.363i)37-s + (1.30 + 0.951i)41-s + ⋯
L(s)  = 1  + (0.809 − 0.587i)2-s + (0.309 − 0.951i)4-s + (−0.309 − 0.951i)8-s + (−0.809 − 0.587i)9-s + (−1.30 − 0.951i)13-s + (−0.809 − 0.587i)16-s + (−0.190 − 0.587i)17-s − 18-s − 1.61·26-s + (0.190 − 0.587i)29-s − 32-s + (−0.5 − 0.363i)34-s + (−0.809 + 0.587i)36-s + (0.5 + 0.363i)37-s + (1.30 + 0.951i)41-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2500\)    =    \(2^{2} \cdot 5^{4}\)
Sign: $-0.728 + 0.684i$
Analytic conductor: \(1.24766\)
Root analytic conductor: \(1.11698\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2500} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2500,\ (\ :0),\ -0.728 + 0.684i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.469552068\)
\(L(\frac12)\) \(\approx\) \(1.469552068\)
\(L(1)\) 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 \( 1 + (0.809 + 0.587i)T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 + (-0.309 + 0.951i)T^{2} \)
13 \( 1 + (1.30 + 0.951i)T + (0.309 + 0.951i)T^{2} \)
17 \( 1 + (0.190 + 0.587i)T + (-0.809 + 0.587i)T^{2} \)
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 \( 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2} \)
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 \( 1 + (-0.5 + 1.53i)T + (-0.809 - 0.587i)T^{2} \)
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 \( 1 + (1.30 - 0.951i)T + (0.309 - 0.951i)T^{2} \)
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 \( 1 + (0.190 - 0.587i)T + (-0.809 - 0.587i)T^{2} \)
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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

−9.012778273656195249312756787179, −8.009065912854725802811762953853, −7.13921509636421032879954577086, −6.29719705657601986265894002770, −5.52362361463056173792463037062, −4.88963896468221950264321865224, −3.92753396762493846777370611865, −2.91495901844413406656902309694, −2.39227651126043525990292547829, −0.69999218743651607262078314993, 2.13063426063701713359973488934, 2.80374193750276773034243885836, 4.02780278679253793547775505645, 4.67505230967454342017837014212, 5.54085708956403222787278723645, 6.14925205868528148928978132087, 7.22517112789986821950921384255, 7.52734069540088603883847355445, 8.630621937921865682569906492123, 9.042816345950848547295866328787

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