Properties

Label 2-1191-1191.1001-c0-0-0
Degree $2$
Conductor $1191$
Sign $-0.313 + 0.949i$
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  + (−0.995 + 0.0950i)3-s + (0.142 − 0.989i)4-s + (0.327 − 1.70i)7-s + (0.981 − 0.189i)9-s + (−0.0475 + 0.998i)12-s + (0.340 + 0.850i)13-s + (−0.959 − 0.281i)16-s + (−0.473 − 0.451i)19-s + (−0.164 + 1.72i)21-s + (−0.928 + 0.371i)25-s + (−0.959 + 0.281i)27-s + (−1.63 − 0.566i)28-s + (−0.738 − 1.61i)31-s + (−0.0475 − 0.998i)36-s + (1.11 − 1.56i)37-s + ⋯
L(s)  = 1  + (−0.995 + 0.0950i)3-s + (0.142 − 0.989i)4-s + (0.327 − 1.70i)7-s + (0.981 − 0.189i)9-s + (−0.0475 + 0.998i)12-s + (0.340 + 0.850i)13-s + (−0.959 − 0.281i)16-s + (−0.473 − 0.451i)19-s + (−0.164 + 1.72i)21-s + (−0.928 + 0.371i)25-s + (−0.959 + 0.281i)27-s + (−1.63 − 0.566i)28-s + (−0.738 − 1.61i)31-s + (−0.0475 − 0.998i)36-s + (1.11 − 1.56i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1191 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.313 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1191 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.313 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1191\)    =    \(3 \cdot 397\)
Sign: $-0.313 + 0.949i$
Analytic conductor: \(0.594386\)
Root analytic conductor: \(0.770964\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1191} (1001, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1191,\ (\ :0),\ -0.313 + 0.949i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7481681638\)
\(L(\frac12)\) \(\approx\) \(0.7481681638\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.995 - 0.0950i)T \)
397 \( 1 - T \)
good2 \( 1 + (-0.142 + 0.989i)T^{2} \)
5 \( 1 + (0.928 - 0.371i)T^{2} \)
7 \( 1 + (-0.327 + 1.70i)T + (-0.928 - 0.371i)T^{2} \)
11 \( 1 + (0.995 + 0.0950i)T^{2} \)
13 \( 1 + (-0.340 - 0.850i)T + (-0.723 + 0.690i)T^{2} \)
17 \( 1 + (-0.142 - 0.989i)T^{2} \)
19 \( 1 + (0.473 + 0.451i)T + (0.0475 + 0.998i)T^{2} \)
23 \( 1 + (0.995 + 0.0950i)T^{2} \)
29 \( 1 + (0.786 + 0.618i)T^{2} \)
31 \( 1 + (0.738 + 1.61i)T + (-0.654 + 0.755i)T^{2} \)
37 \( 1 + (-1.11 + 1.56i)T + (-0.327 - 0.945i)T^{2} \)
41 \( 1 + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (-1.28 - 1.48i)T + (-0.142 + 0.989i)T^{2} \)
47 \( 1 + (-0.235 - 0.971i)T^{2} \)
53 \( 1 + (0.841 + 0.540i)T^{2} \)
59 \( 1 + (0.981 + 0.189i)T^{2} \)
61 \( 1 + (-0.0706 + 0.176i)T + (-0.723 - 0.690i)T^{2} \)
67 \( 1 + (-0.165 + 0.231i)T + (-0.327 - 0.945i)T^{2} \)
71 \( 1 + (-0.654 + 0.755i)T^{2} \)
73 \( 1 + (-0.437 - 1.80i)T + (-0.888 + 0.458i)T^{2} \)
79 \( 1 + (0.995 + 1.72i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.142 - 0.989i)T^{2} \)
89 \( 1 + (0.723 + 0.690i)T^{2} \)
97 \( 1 + (-0.154 - 0.445i)T + (-0.786 + 0.618i)T^{2} \)
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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

−9.777695227101618597373026742954, −9.342577434399784037669196997622, −7.73140250625982805043277388494, −7.11518515204739050195609461044, −6.29698445544983208355646722663, −5.59919128379106302926552952582, −4.36632326436244912184306504732, −4.11998368031029595526557741650, −1.91199600455429733036112696649, −0.76606977832280473787995230285, 1.90795356020371765507129609619, 3.01097174825983274007959039056, 4.22485890149881540877279700866, 5.30308113566050313187952727032, 5.91947841366677555438681252648, 6.78925068014706936415374383021, 7.86609921209262163440538201423, 8.433616667649606707094960887660, 9.278431992932691907604080627725, 10.36351476035827868809163797412

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