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Properties

Label 2-1191-1191.503-c0-0-0
Degree $2$
Conductor $1191$
Sign $0.00556 + 0.999i$
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  + (0.235 − 0.971i)3-s + (0.415 − 0.909i)4-s + (0.888 − 0.458i)7-s + (−0.888 − 0.458i)9-s + (−0.786 − 0.618i)12-s + (1.07 + 1.51i)13-s + (−0.654 − 0.755i)16-s + (−0.0311 − 0.0899i)19-s + (−0.235 − 0.971i)21-s + (0.580 + 0.814i)25-s + (−0.654 + 0.755i)27-s + (−0.0475 − 0.998i)28-s + (−1.78 − 0.523i)31-s + (−0.786 + 0.618i)36-s + (−0.947 + 0.903i)37-s + ⋯
L(s)  = 1  + (0.235 − 0.971i)3-s + (0.415 − 0.909i)4-s + (0.888 − 0.458i)7-s + (−0.888 − 0.458i)9-s + (−0.786 − 0.618i)12-s + (1.07 + 1.51i)13-s + (−0.654 − 0.755i)16-s + (−0.0311 − 0.0899i)19-s + (−0.235 − 0.971i)21-s + (0.580 + 0.814i)25-s + (−0.654 + 0.755i)27-s + (−0.0475 − 0.998i)28-s + (−1.78 − 0.523i)31-s + (−0.786 + 0.618i)36-s + (−0.947 + 0.903i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1191 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00556 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1191 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00556 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1191\)    =    \(3 \cdot 397\)
Sign: $0.00556 + 0.999i$
Analytic conductor: \(0.594386\)
Root analytic conductor: \(0.770964\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1191} (503, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1191,\ (\ :0),\ 0.00556 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.337579458\)
\(L(\frac12)\) \(\approx\) \(1.337579458\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.235 + 0.971i)T \)
397 \( 1 - T \)
good2 \( 1 + (-0.415 + 0.909i)T^{2} \)
5 \( 1 + (-0.580 - 0.814i)T^{2} \)
7 \( 1 + (-0.888 + 0.458i)T + (0.580 - 0.814i)T^{2} \)
11 \( 1 + (-0.235 - 0.971i)T^{2} \)
13 \( 1 + (-1.07 - 1.51i)T + (-0.327 + 0.945i)T^{2} \)
17 \( 1 + (-0.415 - 0.909i)T^{2} \)
19 \( 1 + (0.0311 + 0.0899i)T + (-0.786 + 0.618i)T^{2} \)
23 \( 1 + (-0.235 - 0.971i)T^{2} \)
29 \( 1 + (0.995 - 0.0950i)T^{2} \)
31 \( 1 + (1.78 + 0.523i)T + (0.841 + 0.540i)T^{2} \)
37 \( 1 + (0.947 - 0.903i)T + (0.0475 - 0.998i)T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (1.49 - 0.961i)T + (0.415 - 0.909i)T^{2} \)
47 \( 1 + (-0.981 + 0.189i)T^{2} \)
53 \( 1 + (0.142 - 0.989i)T^{2} \)
59 \( 1 + (0.888 - 0.458i)T^{2} \)
61 \( 1 + (-0.273 + 0.384i)T + (-0.327 - 0.945i)T^{2} \)
67 \( 1 + (-0.601 + 0.573i)T + (0.0475 - 0.998i)T^{2} \)
71 \( 1 + (-0.841 - 0.540i)T^{2} \)
73 \( 1 + (-1.13 + 0.219i)T + (0.928 - 0.371i)T^{2} \)
79 \( 1 + (0.235 - 0.408i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (-0.415 + 0.909i)T^{2} \)
89 \( 1 + (0.327 + 0.945i)T^{2} \)
97 \( 1 + (-0.0934 + 1.96i)T + (-0.995 - 0.0950i)T^{2} \)
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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

−9.617893820607620127560009657883, −8.898608704671789201858890082866, −8.075320324267198629171659185136, −7.05629220109987037244349129776, −6.63832695105298183230004347851, −5.67605430718302694227308921409, −4.72723142765009507479931842797, −3.45372968089233704167197661773, −1.90476434875805935696096999864, −1.40123054673826580650066685564, 2.08866870053478799369419268941, 3.23391257656661866757414372922, 3.84763002261938494693523881286, 5.07277423128179086532176301141, 5.71313335637606789464465226778, 6.97561569579444925125186431862, 8.108171329250278842404178108972, 8.398913507271597858201975534889, 9.114784558100034025458859089539, 10.43006188042429052215908480816

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