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Properties

Label 2-1191-1191.560-c0-0-0
Degree $2$
Conductor $1191$
Sign $0.123 + 0.992i$
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  + (−0.327 − 0.945i)3-s + (0.959 − 0.281i)4-s + (1.07 − 1.36i)7-s + (−0.786 + 0.618i)9-s + (−0.580 − 0.814i)12-s + (0.184 + 0.0448i)13-s + (0.841 − 0.540i)16-s + (−1.65 + 0.850i)19-s + (−1.63 − 0.566i)21-s + (−0.235 + 0.971i)25-s + (0.841 + 0.540i)27-s + (0.643 − 1.60i)28-s + (1.30 + 1.50i)31-s + (−0.580 + 0.814i)36-s + (−1.65 + 0.318i)37-s + ⋯
L(s)  = 1  + (−0.327 − 0.945i)3-s + (0.959 − 0.281i)4-s + (1.07 − 1.36i)7-s + (−0.786 + 0.618i)9-s + (−0.580 − 0.814i)12-s + (0.184 + 0.0448i)13-s + (0.841 − 0.540i)16-s + (−1.65 + 0.850i)19-s + (−1.63 − 0.566i)21-s + (−0.235 + 0.971i)25-s + (0.841 + 0.540i)27-s + (0.643 − 1.60i)28-s + (1.30 + 1.50i)31-s + (−0.580 + 0.814i)36-s + (−1.65 + 0.318i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1191 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.123 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1191 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.123 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1191\)    =    \(3 \cdot 397\)
Sign: $0.123 + 0.992i$
Analytic conductor: \(0.594386\)
Root analytic conductor: \(0.770964\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1191} (560, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1191,\ (\ :0),\ 0.123 + 0.992i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.252866613\)
\(L(\frac12)\) \(\approx\) \(1.252866613\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.327 + 0.945i)T \)
397 \( 1 - T \)
good2 \( 1 + (-0.959 + 0.281i)T^{2} \)
5 \( 1 + (0.235 - 0.971i)T^{2} \)
7 \( 1 + (-1.07 + 1.36i)T + (-0.235 - 0.971i)T^{2} \)
11 \( 1 + (0.327 - 0.945i)T^{2} \)
13 \( 1 + (-0.184 - 0.0448i)T + (0.888 + 0.458i)T^{2} \)
17 \( 1 + (-0.959 - 0.281i)T^{2} \)
19 \( 1 + (1.65 - 0.850i)T + (0.580 - 0.814i)T^{2} \)
23 \( 1 + (0.327 - 0.945i)T^{2} \)
29 \( 1 + (-0.723 - 0.690i)T^{2} \)
31 \( 1 + (-1.30 - 1.50i)T + (-0.142 + 0.989i)T^{2} \)
37 \( 1 + (1.65 - 0.318i)T + (0.928 - 0.371i)T^{2} \)
41 \( 1 + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.223 + 1.55i)T + (-0.959 + 0.281i)T^{2} \)
47 \( 1 + (-0.0475 - 0.998i)T^{2} \)
53 \( 1 + (0.415 - 0.909i)T^{2} \)
59 \( 1 + (-0.786 - 0.618i)T^{2} \)
61 \( 1 + (1.83 - 0.445i)T + (0.888 - 0.458i)T^{2} \)
67 \( 1 + (-1.88 + 0.363i)T + (0.928 - 0.371i)T^{2} \)
71 \( 1 + (-0.142 + 0.989i)T^{2} \)
73 \( 1 + (-0.0224 - 0.470i)T + (-0.995 + 0.0950i)T^{2} \)
79 \( 1 + (0.327 - 0.566i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.959 - 0.281i)T^{2} \)
89 \( 1 + (-0.888 + 0.458i)T^{2} \)
97 \( 1 + (0.0883 - 0.0353i)T + (0.723 - 0.690i)T^{2} \)
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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

−10.28115414678735816553445696372, −8.552155804757439432737177387616, −8.010430719084535199923371009734, −7.08392409550013640085476796018, −6.73715511353745862632454544426, −5.69676202345955855029697694832, −4.75654836902816376914627666272, −3.49520277341051445797180795672, −1.99986209628792604472648204882, −1.30482457263549992950813514192, 2.07024043898759107144297437644, 2.84042389137820108258507608783, 4.20923227622396089097335357297, 5.00567453540514632742309915646, 6.02226350614866112159679931195, 6.49939522644599150389057372467, 7.986985940263739280952233707285, 8.465240673602050371975478500741, 9.272163549248563404366900496830, 10.35574799076323150264011168548

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