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Properties

Label 2-1191-1191.902-c0-0-0
Degree $2$
Conductor $1191$
Sign $-0.357 - 0.934i$
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  + (0.580 + 0.814i)3-s + (−0.142 + 0.989i)4-s + (0.327 + 0.945i)7-s + (−0.327 + 0.945i)9-s + (−0.888 + 0.458i)12-s + (−0.0748 − 0.0588i)13-s + (−0.959 − 0.281i)16-s + (0.462 − 1.90i)19-s + (−0.580 + 0.814i)21-s + (−0.786 − 0.618i)25-s + (−0.959 + 0.281i)27-s + (−0.981 + 0.189i)28-s + (0.0395 + 0.0865i)31-s + (−0.888 − 0.458i)36-s + (1.91 + 0.182i)37-s + ⋯
L(s)  = 1  + (0.580 + 0.814i)3-s + (−0.142 + 0.989i)4-s + (0.327 + 0.945i)7-s + (−0.327 + 0.945i)9-s + (−0.888 + 0.458i)12-s + (−0.0748 − 0.0588i)13-s + (−0.959 − 0.281i)16-s + (0.462 − 1.90i)19-s + (−0.580 + 0.814i)21-s + (−0.786 − 0.618i)25-s + (−0.959 + 0.281i)27-s + (−0.981 + 0.189i)28-s + (0.0395 + 0.0865i)31-s + (−0.888 − 0.458i)36-s + (1.91 + 0.182i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1191 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.357 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1191 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.357 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1191\)    =    \(3 \cdot 397\)
Sign: $-0.357 - 0.934i$
Analytic conductor: \(0.594386\)
Root analytic conductor: \(0.770964\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1191} (902, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1191,\ (\ :0),\ -0.357 - 0.934i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.220446068\)
\(L(\frac12)\) \(\approx\) \(1.220446068\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.580 - 0.814i)T \)
397 \( 1 - T \)
good2 \( 1 + (0.142 - 0.989i)T^{2} \)
5 \( 1 + (0.786 + 0.618i)T^{2} \)
7 \( 1 + (-0.327 - 0.945i)T + (-0.786 + 0.618i)T^{2} \)
11 \( 1 + (-0.580 + 0.814i)T^{2} \)
13 \( 1 + (0.0748 + 0.0588i)T + (0.235 + 0.971i)T^{2} \)
17 \( 1 + (0.142 + 0.989i)T^{2} \)
19 \( 1 + (-0.462 + 1.90i)T + (-0.888 - 0.458i)T^{2} \)
23 \( 1 + (-0.580 + 0.814i)T^{2} \)
29 \( 1 + (-0.928 + 0.371i)T^{2} \)
31 \( 1 + (-0.0395 - 0.0865i)T + (-0.654 + 0.755i)T^{2} \)
37 \( 1 + (-1.91 - 0.182i)T + (0.981 + 0.189i)T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.428 - 0.494i)T + (-0.142 + 0.989i)T^{2} \)
47 \( 1 + (-0.723 + 0.690i)T^{2} \)
53 \( 1 + (-0.841 - 0.540i)T^{2} \)
59 \( 1 + (0.327 + 0.945i)T^{2} \)
61 \( 1 + (0.911 - 0.717i)T + (0.235 - 0.971i)T^{2} \)
67 \( 1 + (-0.283 - 0.0270i)T + (0.981 + 0.189i)T^{2} \)
71 \( 1 + (0.654 - 0.755i)T^{2} \)
73 \( 1 + (1.13 - 1.08i)T + (0.0475 - 0.998i)T^{2} \)
79 \( 1 + (0.580 - 1.00i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.142 - 0.989i)T^{2} \)
89 \( 1 + (-0.235 + 0.971i)T^{2} \)
97 \( 1 + (-1.42 - 0.273i)T + (0.928 + 0.371i)T^{2} \)
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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

−9.945676331787800107038827420620, −9.222715222712257517327285225224, −8.676253111734098882844858580041, −7.964630146595182198287927128339, −7.17273517179699607554783596854, −5.86668328184424545076229908873, −4.81862480835454314647451386125, −4.19652317818262626403729510528, −2.94311811709556211047621386382, −2.43355284283967069753834984978, 1.10735571530196574360001190970, 2.02155377616556610873458066062, 3.51758672381126258711255093582, 4.43999893747186323487394692408, 5.73282031352582731348754988621, 6.28862529516681926569165039767, 7.47275688362598724981326175182, 7.80676045280159486188483311061, 8.943358905000993443916764762447, 9.722028658825292574160184295645

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