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Properties

Label 2-50e2-100.59-c0-0-0
Degree $2$
Conductor $2500$
Sign $0.876 - 0.481i$
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 + (0.190 + 0.587i)3-s + (0.309 + 0.951i)4-s + (0.190 − 0.587i)6-s − 1.61·7-s + (0.309 − 0.951i)8-s + (0.5 − 0.363i)9-s + (−0.5 + 0.363i)12-s + (1.30 + 0.951i)14-s + (−0.809 + 0.587i)16-s − 0.618·18-s + (−0.309 − 0.951i)21-s + (1.30 + 0.951i)23-s + 0.618·24-s + (0.809 + 0.587i)27-s + (−0.500 − 1.53i)28-s + ⋯
L(s)  = 1  + (−0.809 − 0.587i)2-s + (0.190 + 0.587i)3-s + (0.309 + 0.951i)4-s + (0.190 − 0.587i)6-s − 1.61·7-s + (0.309 − 0.951i)8-s + (0.5 − 0.363i)9-s + (−0.5 + 0.363i)12-s + (1.30 + 0.951i)14-s + (−0.809 + 0.587i)16-s − 0.618·18-s + (−0.309 − 0.951i)21-s + (1.30 + 0.951i)23-s + 0.618·24-s + (0.809 + 0.587i)27-s + (−0.500 − 1.53i)28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.876 - 0.481i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.876 - 0.481i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2500\)    =    \(2^{2} \cdot 5^{4}\)
Sign: $0.876 - 0.481i$
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: \((2,\ 2500,\ (\ :0),\ 0.876 - 0.481i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7295367910\)
\(L(\frac12)\) \(\approx\) \(0.7295367910\)
\(L(1)\) 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 \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \)
7 \( 1 + 1.61T + T^{2} \)
11 \( 1 + (-0.309 - 0.951i)T^{2} \)
13 \( 1 + (-0.309 + 0.951i)T^{2} \)
17 \( 1 + (0.809 + 0.587i)T^{2} \)
19 \( 1 + (0.809 + 0.587i)T^{2} \)
23 \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \)
29 \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \)
31 \( 1 + (0.809 + 0.587i)T^{2} \)
37 \( 1 + (-0.309 + 0.951i)T^{2} \)
41 \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \)
43 \( 1 - 0.618T + T^{2} \)
47 \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \)
53 \( 1 + (0.809 - 0.587i)T^{2} \)
59 \( 1 + (-0.309 + 0.951i)T^{2} \)
61 \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \)
67 \( 1 + (-0.618 + 1.90i)T + (-0.809 - 0.587i)T^{2} \)
71 \( 1 + (0.809 - 0.587i)T^{2} \)
73 \( 1 + (-0.309 - 0.951i)T^{2} \)
79 \( 1 + (0.809 - 0.587i)T^{2} \)
83 \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \)
89 \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \)
97 \( 1 + (0.809 - 0.587i)T^{2} \)
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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

−9.304452858731608836631716804175, −8.839276967599651032326293172249, −7.70629494478535921134290504264, −6.90131278633930664625707407848, −6.42569545697976257439607982438, −5.12503931356421975676221739015, −3.93088397965880431387320748440, −3.39846691240787527656508361330, −2.64698304962521730189400228383, −1.10339695037839386158293914189, 0.75149585213663157042123248289, 2.15275931680747985772103235157, 3.03629575700486161031951516084, 4.35078400151346468009951839520, 5.42684274384342620847585226531, 6.32045039261643648421527118447, 6.89611904397199265642881798643, 7.32363589750857508442372645291, 8.340018827025775311591630974537, 8.915135217261769705177078284372

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