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Properties

Label 2-2527-133.104-c0-0-12
Degree $2$
Conductor $2527$
Sign $-0.994 - 0.101i$
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  + (−0.204 − 1.15i)2-s + (−0.358 + 0.130i)4-s + (0.5 − 0.866i)7-s + (−0.363 − 0.629i)8-s + (0.173 − 0.984i)9-s + (−0.809 − 1.40i)11-s + (−1.10 − 0.402i)14-s + (−0.946 + 0.794i)16-s − 1.17·18-s + (−1.45 + 1.22i)22-s + (−0.580 + 0.211i)23-s + (0.766 + 0.642i)25-s + (−0.0663 + 0.376i)28-s + (−0.330 + 1.87i)29-s + (0.556 + 0.467i)32-s + ⋯
L(s)  = 1  + (−0.204 − 1.15i)2-s + (−0.358 + 0.130i)4-s + (0.5 − 0.866i)7-s + (−0.363 − 0.629i)8-s + (0.173 − 0.984i)9-s + (−0.809 − 1.40i)11-s + (−1.10 − 0.402i)14-s + (−0.946 + 0.794i)16-s − 1.17·18-s + (−1.45 + 1.22i)22-s + (−0.580 + 0.211i)23-s + (0.766 + 0.642i)25-s + (−0.0663 + 0.376i)28-s + (−0.330 + 1.87i)29-s + (0.556 + 0.467i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2527 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 - 0.101i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2527 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 - 0.101i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2527\)    =    \(7 \cdot 19^{2}\)
Sign: $-0.994 - 0.101i$
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: \((2,\ 2527,\ (\ :0),\ -0.994 - 0.101i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.031595008\)
\(L(\frac12)\) \(\approx\) \(1.031595008\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 \)
good2 \( 1 + (0.204 + 1.15i)T + (-0.939 + 0.342i)T^{2} \)
3 \( 1 + (-0.173 + 0.984i)T^{2} \)
5 \( 1 + (-0.766 - 0.642i)T^{2} \)
11 \( 1 + (0.809 + 1.40i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.173 - 0.984i)T^{2} \)
17 \( 1 + (0.939 - 0.342i)T^{2} \)
23 \( 1 + (0.580 - 0.211i)T + (0.766 - 0.642i)T^{2} \)
29 \( 1 + (0.330 - 1.87i)T + (-0.939 - 0.342i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (-0.173 + 0.984i)T^{2} \)
43 \( 1 + (-1.52 - 0.553i)T + (0.766 + 0.642i)T^{2} \)
47 \( 1 + (0.939 + 0.342i)T^{2} \)
53 \( 1 + (1.10 - 0.402i)T + (0.766 - 0.642i)T^{2} \)
59 \( 1 + (0.939 - 0.342i)T^{2} \)
61 \( 1 + (-0.766 + 0.642i)T^{2} \)
67 \( 1 + (-0.204 + 1.15i)T + (-0.939 - 0.342i)T^{2} \)
71 \( 1 + (0.766 + 0.642i)T^{2} \)
73 \( 1 + (-0.173 + 0.984i)T^{2} \)
79 \( 1 + (-1.45 + 1.22i)T + (0.173 - 0.984i)T^{2} \)
83 \( 1 + (0.5 + 0.866i)T^{2} \)
89 \( 1 + (-0.173 - 0.984i)T^{2} \)
97 \( 1 + (0.939 - 0.342i)T^{2} \)
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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

−8.973020715063353235227317651042, −8.076489502438611894853505752359, −7.21309619980598802022186575872, −6.41938944888809936491532067977, −5.54249855323395341300014047068, −4.44418425500852060789430149298, −3.44347678814541575043624376123, −3.07623668800168457987919248326, −1.64219509313473012684296007021, −0.71915272128545242073617004136, 2.18543146460872041966070202200, 2.46127955277094101890909589773, 4.35325385889937347765596976049, 4.99465704815921439112677707440, 5.65560194333695133881461245701, 6.45945232806727938463848364423, 7.39717581077888487863022085488, 7.87046705197099425450764507506, 8.366493534457782825122490395441, 9.290602700804192745489566625774

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