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Properties

Label 2-1191-1191.137-c0-0-0
Degree $2$
Conductor $1191$
Sign $-0.779 - 0.626i$
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  + (0.723 + 0.690i)3-s + (−0.415 + 0.909i)4-s + (−1.73 − 0.0824i)7-s + (0.0475 + 0.998i)9-s + (−0.928 + 0.371i)12-s + (0.117 + 1.23i)13-s + (−0.654 − 0.755i)16-s + (−1.74 − 0.336i)19-s + (−1.19 − 1.25i)21-s + (0.995 − 0.0950i)25-s + (−0.654 + 0.755i)27-s + (0.793 − 1.53i)28-s + (1.50 + 0.442i)31-s + (−0.928 − 0.371i)36-s + (0.308 + 1.27i)37-s + ⋯
L(s)  = 1  + (0.723 + 0.690i)3-s + (−0.415 + 0.909i)4-s + (−1.73 − 0.0824i)7-s + (0.0475 + 0.998i)9-s + (−0.928 + 0.371i)12-s + (0.117 + 1.23i)13-s + (−0.654 − 0.755i)16-s + (−1.74 − 0.336i)19-s + (−1.19 − 1.25i)21-s + (0.995 − 0.0950i)25-s + (−0.654 + 0.755i)27-s + (0.793 − 1.53i)28-s + (1.50 + 0.442i)31-s + (−0.928 − 0.371i)36-s + (0.308 + 1.27i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1191 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.779 - 0.626i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1191 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.779 - 0.626i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1191\)    =    \(3 \cdot 397\)
Sign: $-0.779 - 0.626i$
Analytic conductor: \(0.594386\)
Root analytic conductor: \(0.770964\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1191} (137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1191,\ (\ :0),\ -0.779 - 0.626i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8274390456\)
\(L(\frac12)\) \(\approx\) \(0.8274390456\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.723 - 0.690i)T \)
397 \( 1 - T \)
good2 \( 1 + (0.415 - 0.909i)T^{2} \)
5 \( 1 + (-0.995 + 0.0950i)T^{2} \)
7 \( 1 + (1.73 + 0.0824i)T + (0.995 + 0.0950i)T^{2} \)
11 \( 1 + (-0.723 + 0.690i)T^{2} \)
13 \( 1 + (-0.117 - 1.23i)T + (-0.981 + 0.189i)T^{2} \)
17 \( 1 + (0.415 + 0.909i)T^{2} \)
19 \( 1 + (1.74 + 0.336i)T + (0.928 + 0.371i)T^{2} \)
23 \( 1 + (-0.723 + 0.690i)T^{2} \)
29 \( 1 + (-0.580 - 0.814i)T^{2} \)
31 \( 1 + (-1.50 - 0.442i)T + (0.841 + 0.540i)T^{2} \)
37 \( 1 + (-0.308 - 1.27i)T + (-0.888 + 0.458i)T^{2} \)
41 \( 1 + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.0800 - 0.0514i)T + (0.415 - 0.909i)T^{2} \)
47 \( 1 + (0.327 - 0.945i)T^{2} \)
53 \( 1 + (-0.142 + 0.989i)T^{2} \)
59 \( 1 + (0.0475 - 0.998i)T^{2} \)
61 \( 1 + (0.131 - 1.37i)T + (-0.981 - 0.189i)T^{2} \)
67 \( 1 + (0.195 + 0.807i)T + (-0.888 + 0.458i)T^{2} \)
71 \( 1 + (0.841 + 0.540i)T^{2} \)
73 \( 1 + (-0.651 + 1.88i)T + (-0.786 - 0.618i)T^{2} \)
79 \( 1 + (-0.723 - 1.25i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (-0.415 + 0.909i)T^{2} \)
89 \( 1 + (0.981 + 0.189i)T^{2} \)
97 \( 1 + (0.581 - 0.299i)T + (0.580 - 0.814i)T^{2} \)
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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

−10.05949008048503305593680741270, −9.292630624508761070203882576770, −8.822078771062689531370084262961, −8.094094542855996903899883545307, −6.86609934286722560856305490817, −6.42083347355582060249838419348, −4.72534355874076679665829613750, −4.12753515496976546725586695806, −3.22700940119806078060686312863, −2.49708152266330188670175426500, 0.65238113681850315714511191809, 2.31505568506258091268304537424, 3.25546794620634499486172700108, 4.28342445472627337572720646511, 5.73647281143110514658240453176, 6.31115085489865059053796964414, 6.96195089640727311928194405756, 8.203141516221089744919733574532, 8.823294305502245563045058981077, 9.662067242568405878189706187093

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