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Properties

Label 2-2527-133.104-c0-0-6
Degree $2$
Conductor $2527$
Sign $0.999 - 0.0231i$
Analytic cond. $1.26113$
Root an. cond. $1.12300$
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  + (0.173 + 0.984i)2-s + (−0.642 − 0.766i)3-s + (0.342 − 0.939i)5-s + (0.642 − 0.766i)6-s + (0.866 + 0.5i)7-s + (0.5 + 0.866i)8-s + (0.984 + 0.173i)10-s + (0.642 − 0.766i)13-s + (−0.342 + 0.939i)14-s + (−0.939 + 0.342i)15-s + (−0.766 + 0.642i)16-s + (0.984 − 0.173i)17-s + (−0.173 − 0.984i)21-s + (−0.939 + 0.342i)23-s + (0.342 − 0.939i)24-s + ⋯
L(s)  = 1  + (0.173 + 0.984i)2-s + (−0.642 − 0.766i)3-s + (0.342 − 0.939i)5-s + (0.642 − 0.766i)6-s + (0.866 + 0.5i)7-s + (0.5 + 0.866i)8-s + (0.984 + 0.173i)10-s + (0.642 − 0.766i)13-s + (−0.342 + 0.939i)14-s + (−0.939 + 0.342i)15-s + (−0.766 + 0.642i)16-s + (0.984 − 0.173i)17-s + (−0.173 − 0.984i)21-s + (−0.939 + 0.342i)23-s + (0.342 − 0.939i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2527 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0231i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2527 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0231i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2527\)    =    \(7 \cdot 19^{2}\)
Sign: $0.999 - 0.0231i$
Analytic conductor: \(1.26113\)
Root analytic conductor: \(1.12300\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2527} (1833, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2527,\ (\ :0),\ 0.999 - 0.0231i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.494989787\)
\(L(\frac12)\) \(\approx\) \(1.494989787\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.866 - 0.5i)T \)
19 \( 1 \)
good2 \( 1 + (-0.173 - 0.984i)T + (-0.939 + 0.342i)T^{2} \)
3 \( 1 + (0.642 + 0.766i)T + (-0.173 + 0.984i)T^{2} \)
5 \( 1 + (-0.342 + 0.939i)T + (-0.766 - 0.642i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.642 + 0.766i)T + (-0.173 - 0.984i)T^{2} \)
17 \( 1 + (-0.984 + 0.173i)T + (0.939 - 0.342i)T^{2} \)
23 \( 1 + (0.939 - 0.342i)T + (0.766 - 0.642i)T^{2} \)
29 \( 1 + (0.173 - 0.984i)T + (-0.939 - 0.342i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (0.642 + 0.766i)T + (-0.173 + 0.984i)T^{2} \)
43 \( 1 + (0.939 + 0.342i)T + (0.766 + 0.642i)T^{2} \)
47 \( 1 + (0.984 + 0.173i)T + (0.939 + 0.342i)T^{2} \)
53 \( 1 + (-0.939 + 0.342i)T + (0.766 - 0.642i)T^{2} \)
59 \( 1 + (0.984 - 0.173i)T + (0.939 - 0.342i)T^{2} \)
61 \( 1 + (0.342 + 0.939i)T + (-0.766 + 0.642i)T^{2} \)
67 \( 1 + (-0.173 + 0.984i)T + (-0.939 - 0.342i)T^{2} \)
71 \( 1 + (-0.939 - 0.342i)T + (0.766 + 0.642i)T^{2} \)
73 \( 1 + (-0.642 - 0.766i)T + (-0.173 + 0.984i)T^{2} \)
79 \( 1 + (-0.766 + 0.642i)T + (0.173 - 0.984i)T^{2} \)
83 \( 1 + (0.5 + 0.866i)T^{2} \)
89 \( 1 + (0.642 - 0.766i)T + (-0.173 - 0.984i)T^{2} \)
97 \( 1 + (-0.984 + 0.173i)T + (0.939 - 0.342i)T^{2} \)
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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

−8.712561028481432062501449235971, −8.149267751618803808336222617287, −7.54386949978185688774326386915, −6.67866923287986163642635805640, −5.93987184647194502334119150712, −5.31922902695572350033566144707, −5.02908773156376429408339086673, −3.54481871644914335914723155213, −1.92959838148373369915521601912, −1.22560295283093110368559893154, 1.46771892933088065230824016785, 2.35180684635270349982532340150, 3.48009069014650549752955204954, 4.16561102669127988359780657487, 4.88076756208853874160232957727, 5.97915926179516335234653076698, 6.64734811817770174021714125035, 7.57586916058551984019179095897, 8.284515987407899608850953668973, 9.649429291709831761856766814033

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