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Properties

Label 2-50e2-100.31-c0-0-1
Degree $2$
Conductor $2500$
Sign $0.535 - 0.844i$
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 + (−0.809 + 0.587i)4-s + (−0.809 − 0.587i)8-s + (0.309 − 0.951i)9-s + (0.190 − 0.587i)13-s + (0.309 − 0.951i)16-s + (1.30 + 0.951i)17-s + 0.999·18-s + 0.618·26-s + (1.30 − 0.951i)29-s + 32-s + (−0.499 + 1.53i)34-s + (0.309 + 0.951i)36-s + (−0.5 + 1.53i)37-s + (0.190 − 0.587i)41-s + ⋯
L(s)  = 1  + (0.309 + 0.951i)2-s + (−0.809 + 0.587i)4-s + (−0.809 − 0.587i)8-s + (0.309 − 0.951i)9-s + (0.190 − 0.587i)13-s + (0.309 − 0.951i)16-s + (1.30 + 0.951i)17-s + 0.999·18-s + 0.618·26-s + (1.30 − 0.951i)29-s + 32-s + (−0.499 + 1.53i)34-s + (0.309 + 0.951i)36-s + (−0.5 + 1.53i)37-s + (0.190 − 0.587i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.535 - 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.535 - 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2500\)    =    \(2^{2} \cdot 5^{4}\)
Sign: $0.535 - 0.844i$
Analytic conductor: \(1.24766\)
Root analytic conductor: \(1.11698\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2500} (751, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2500,\ (\ :0),\ 0.535 - 0.844i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.354308221\)
\(L(\frac12)\) \(\approx\) \(1.354308221\)
\(L(1)\) 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 \( 1 + (-0.309 + 0.951i)T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 + (0.809 - 0.587i)T^{2} \)
13 \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \)
17 \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \)
19 \( 1 + (-0.309 - 0.951i)T^{2} \)
23 \( 1 + (0.809 - 0.587i)T^{2} \)
29 \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \)
31 \( 1 + (-0.309 - 0.951i)T^{2} \)
37 \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \)
41 \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (-0.309 + 0.951i)T^{2} \)
53 \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \)
59 \( 1 + (0.809 + 0.587i)T^{2} \)
61 \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \)
67 \( 1 + (-0.309 - 0.951i)T^{2} \)
71 \( 1 + (-0.309 + 0.951i)T^{2} \)
73 \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \)
79 \( 1 + (-0.309 + 0.951i)T^{2} \)
83 \( 1 + (-0.309 - 0.951i)T^{2} \)
89 \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \)
97 \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \)
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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

−8.988883107248748624877235481832, −8.319461856251164029620716774085, −7.70457792855588158424804488448, −6.82374820934425663575350869223, −6.12904923794926219593187198082, −5.54300650437359506044810565373, −4.50581275880461499129304215114, −3.71189609442901170933963957361, −2.95374727689042182857232603560, −1.07378338559188825633651364852, 1.18976783125266095511919614404, 2.26680092853660939200980642338, 3.16504215310281493827556082027, 4.11864284953562135152126630277, 4.99060614554309244578703277541, 5.50362124000553685207637829266, 6.63291453588320583816204477210, 7.55180941069283531573547665174, 8.346529134837210488429628082943, 9.198921904593590848385027142022

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