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Properties

Label 2-1191-1191.260-c0-0-0
Degree $2$
Conductor $1191$
Sign $0.906 - 0.421i$
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 + (−0.0475 + 0.998i)7-s + (0.0475 + 0.998i)9-s + (0.928 − 0.371i)12-s + (1.56 − 0.149i)13-s + (−0.654 − 0.755i)16-s + (−1.74 − 0.336i)19-s + (−0.723 + 0.690i)21-s + (−0.995 + 0.0950i)25-s + (−0.654 + 0.755i)27-s + (0.888 + 0.458i)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 + (−0.0475 + 0.998i)7-s + (0.0475 + 0.998i)9-s + (0.928 − 0.371i)12-s + (1.56 − 0.149i)13-s + (−0.654 − 0.755i)16-s + (−1.74 − 0.336i)19-s + (−0.723 + 0.690i)21-s + (−0.995 + 0.0950i)25-s + (−0.654 + 0.755i)27-s + (0.888 + 0.458i)28-s + (1.50 + 0.442i)31-s + (0.928 + 0.371i)36-s + (−0.308 − 1.27i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1191 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.906 - 0.421i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1191 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.906 - 0.421i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1191\)    =    \(3 \cdot 397\)
Sign: $0.906 - 0.421i$
Analytic conductor: \(0.594386\)
Root analytic conductor: \(0.770964\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1191} (260, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1191,\ (\ :0),\ 0.906 - 0.421i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.460183024\)
\(L(\frac12)\) \(\approx\) \(1.460183024\)
\(L(1)\) 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 \( 1 + (0.0475 - 0.998i)T + (-0.995 - 0.0950i)T^{2} \)
11 \( 1 + (-0.723 + 0.690i)T^{2} \)
13 \( 1 + (-1.56 + 0.149i)T + (0.981 - 0.189i)T^{2} \)
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 \( 1 + (1.44 + 0.137i)T + (0.981 + 0.189i)T^{2} \)
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

−10.07568771762601236931556402903, −8.991429341750819257817021825033, −8.757831992588776033751903878251, −7.74227273635942313624007166517, −6.38563426257764873156830412469, −5.90419403687573994607817971487, −4.88262823659771419292127082602, −3.89180504819339761170493317065, −2.68628560113101425939709922324, −1.81014645891886464549683845100, 1.48535877064368521460801869486, 2.66770881757197029396724566005, 3.77829433393517250367839814465, 4.19937511408853185940042566859, 6.25984687513681553537103018843, 6.56353490767783909911538482601, 7.57371325228060033129549375667, 8.278770615128220383100085387268, 8.650612236643195724661080754481, 9.876638272221831928266567685452

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