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Properties

Label 2-1191-1191.362-c0-0-0
Degree $2$
Conductor $1191$
Sign $0.983 - 0.181i$
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  + (−0.5 + 0.866i)3-s + 4-s + (0.5 − 0.866i)7-s + (−0.499 − 0.866i)9-s + (−0.5 + 0.866i)12-s + (0.5 − 0.866i)13-s + 16-s + (0.5 − 0.866i)19-s + (0.499 + 0.866i)21-s + (−0.5 + 0.866i)25-s + 0.999·27-s + (0.5 − 0.866i)28-s − 31-s + (−0.499 − 0.866i)36-s + (−1 + 1.73i)37-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)3-s + 4-s + (0.5 − 0.866i)7-s + (−0.499 − 0.866i)9-s + (−0.5 + 0.866i)12-s + (0.5 − 0.866i)13-s + 16-s + (0.5 − 0.866i)19-s + (0.499 + 0.866i)21-s + (−0.5 + 0.866i)25-s + 0.999·27-s + (0.5 − 0.866i)28-s − 31-s + (−0.499 − 0.866i)36-s + (−1 + 1.73i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1191 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 - 0.181i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1191 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 - 0.181i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1191\)    =    \(3 \cdot 397\)
Sign: $0.983 - 0.181i$
Analytic conductor: \(0.594386\)
Root analytic conductor: \(0.770964\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1191} (362, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1191,\ (\ :0),\ 0.983 - 0.181i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.188228842\)
\(L(\frac12)\) \(\approx\) \(1.188228842\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 - 0.866i)T \)
397 \( 1 - T \)
good2 \( 1 - T^{2} \)
5 \( 1 + (0.5 - 0.866i)T^{2} \)
7 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + T + T^{2} \)
37 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + T + T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
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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

−10.36580648817397519254262106260, −9.331845304976227116273634336828, −8.297608544503592803537409142696, −7.40468541004240576786324181119, −6.66936672746727633627829927549, −5.66697528411582546840597462380, −4.96821224203078443288670521255, −3.77683598915435837552072986548, −3.01277966235349210531708826476, −1.31745509513957046610727575364, 1.66760368303026855116449797013, 2.24263591636260292778326609599, 3.63932195691807034211097331263, 5.18270529614746290487231676135, 5.86874256513632101554633266792, 6.56687080506555226691507563310, 7.39596316037417131600434740534, 8.124357956856116367480144330839, 8.926281093854255611853910626774, 10.11423506100243364189140424238

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