Properties

Label 2-1191-1191.587-c0-0-0
Degree $2$
Conductor $1191$
Sign $-0.882 + 0.469i$
Analytic cond. $0.594386$
Root an. cond. $0.770964$
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  + (−0.995 + 0.0950i)3-s + (−0.142 + 0.989i)4-s + (−0.981 − 0.189i)7-s + (0.981 − 0.189i)9-s + (0.0475 − 0.998i)12-s + (−1.65 + 0.660i)13-s + (−0.959 − 0.281i)16-s + (−0.473 − 0.451i)19-s + (0.995 + 0.0950i)21-s + (0.928 − 0.371i)25-s + (−0.959 + 0.281i)27-s + (0.327 − 0.945i)28-s + (−0.738 − 1.61i)31-s + (0.0475 + 0.998i)36-s + (−1.11 + 1.56i)37-s + ⋯
L(s)  = 1  + (−0.995 + 0.0950i)3-s + (−0.142 + 0.989i)4-s + (−0.981 − 0.189i)7-s + (0.981 − 0.189i)9-s + (0.0475 − 0.998i)12-s + (−1.65 + 0.660i)13-s + (−0.959 − 0.281i)16-s + (−0.473 − 0.451i)19-s + (0.995 + 0.0950i)21-s + (0.928 − 0.371i)25-s + (−0.959 + 0.281i)27-s + (0.327 − 0.945i)28-s + (−0.738 − 1.61i)31-s + (0.0475 + 0.998i)36-s + (−1.11 + 1.56i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1191 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.882 + 0.469i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1191 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.882 + 0.469i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1191\)    =    \(3 \cdot 397\)
Sign: $-0.882 + 0.469i$
Analytic conductor: \(0.594386\)
Root analytic conductor: \(0.770964\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1191} (587, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1191,\ (\ :0),\ -0.882 + 0.469i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.05379893345\)
\(L(\frac12)\) \(\approx\) \(0.05379893345\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.995 - 0.0950i)T \)
397 \( 1 - T \)
good2 \( 1 + (0.142 - 0.989i)T^{2} \)
5 \( 1 + (-0.928 + 0.371i)T^{2} \)
7 \( 1 + (0.981 + 0.189i)T + (0.928 + 0.371i)T^{2} \)
11 \( 1 + (0.995 + 0.0950i)T^{2} \)
13 \( 1 + (1.65 - 0.660i)T + (0.723 - 0.690i)T^{2} \)
17 \( 1 + (0.142 + 0.989i)T^{2} \)
19 \( 1 + (0.473 + 0.451i)T + (0.0475 + 0.998i)T^{2} \)
23 \( 1 + (0.995 + 0.0950i)T^{2} \)
29 \( 1 + (0.786 + 0.618i)T^{2} \)
31 \( 1 + (0.738 + 1.61i)T + (-0.654 + 0.755i)T^{2} \)
37 \( 1 + (1.11 - 1.56i)T + (-0.327 - 0.945i)T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (1.28 + 1.48i)T + (-0.142 + 0.989i)T^{2} \)
47 \( 1 + (-0.235 - 0.971i)T^{2} \)
53 \( 1 + (-0.841 - 0.540i)T^{2} \)
59 \( 1 + (-0.981 - 0.189i)T^{2} \)
61 \( 1 + (1.84 + 0.739i)T + (0.723 + 0.690i)T^{2} \)
67 \( 1 + (0.165 - 0.231i)T + (-0.327 - 0.945i)T^{2} \)
71 \( 1 + (0.654 - 0.755i)T^{2} \)
73 \( 1 + (-0.437 - 1.80i)T + (-0.888 + 0.458i)T^{2} \)
79 \( 1 + (-0.995 - 1.72i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.142 - 0.989i)T^{2} \)
89 \( 1 + (-0.723 - 0.690i)T^{2} \)
97 \( 1 + (0.154 + 0.445i)T + (-0.786 + 0.618i)T^{2} \)
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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

−10.33046654788429063979428337348, −9.685208092924316868952362446770, −8.953129993341206305236271297182, −7.78886737324817682559018961623, −6.86822576004870449717279738299, −6.62570888590811615609596521571, −5.19507032504111143550463246306, −4.47580172419869243363265811392, −3.52140194035693050524052481856, −2.32082798726675602730080454022, 0.05056778604595899224270438857, 1.73950041327614017357660135756, 3.18612788855541061560695760285, 4.69740376779057225157350552459, 5.23487745363462233726269150265, 6.09240736537010390493587055169, 6.79716784019536180325420565236, 7.54459874743266388127539134252, 9.006295513108270945282995064857, 9.640661580191857388000832244266

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