Properties

Label 2-50e2-100.59-c0-0-4
Degree $2$
Conductor $2500$
Sign $-0.992 - 0.125i$
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 + (−0.5 − 1.53i)3-s + (0.309 + 0.951i)4-s + (−0.5 + 1.53i)6-s + 0.618·7-s + (0.309 − 0.951i)8-s + (−1.30 + 0.951i)9-s + (1.30 − 0.951i)12-s + (−0.500 − 0.363i)14-s + (−0.809 + 0.587i)16-s + 1.61·18-s + (−0.309 − 0.951i)21-s + (−0.5 − 0.363i)23-s − 1.61·24-s + (0.809 + 0.587i)27-s + (0.190 + 0.587i)28-s + ⋯
L(s)  = 1  + (−0.809 − 0.587i)2-s + (−0.5 − 1.53i)3-s + (0.309 + 0.951i)4-s + (−0.5 + 1.53i)6-s + 0.618·7-s + (0.309 − 0.951i)8-s + (−1.30 + 0.951i)9-s + (1.30 − 0.951i)12-s + (−0.500 − 0.363i)14-s + (−0.809 + 0.587i)16-s + 1.61·18-s + (−0.309 − 0.951i)21-s + (−0.5 − 0.363i)23-s − 1.61·24-s + (0.809 + 0.587i)27-s + (0.190 + 0.587i)28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 - 0.125i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 - 0.125i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2500\)    =    \(2^{2} \cdot 5^{4}\)
Sign: $-0.992 - 0.125i$
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: \((2,\ 2500,\ (\ :0),\ -0.992 - 0.125i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5495875884\)
\(L(\frac12)\) \(\approx\) \(0.5495875884\)
\(L(1)\) 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 \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \)
7 \( 1 - 0.618T + T^{2} \)
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 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
29 \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \)
31 \( 1 + (0.809 + 0.587i)T^{2} \)
37 \( 1 + (-0.309 + 0.951i)T^{2} \)
41 \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \)
43 \( 1 + 1.61T + T^{2} \)
47 \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \)
53 \( 1 + (0.809 - 0.587i)T^{2} \)
59 \( 1 + (-0.309 + 0.951i)T^{2} \)
61 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
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 \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \)
89 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
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

−8.439383143053958352018869546685, −7.970284321699812641444748244687, −7.37190564555950334478913277367, −6.60006281714236366141676267425, −5.91519100068082864209217899615, −4.76434976190238893648256653769, −3.57870559537665670100833798410, −2.27050737583388664364618984720, −1.76415357208800149807363057969, −0.51602764847268626932077886377, 1.51541286343495239524025177349, 3.03417081999184028604173930513, 4.21342182419586493320455145483, 4.93527975255815716123480287763, 5.53751144888620573910112577479, 6.31542385665047452944571800113, 7.30238035993669576693396787361, 8.167646138181221618144939616928, 8.829117830064660071121801290564, 9.621630258559279268550307350265

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