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Properties

Label 2-1191-1191.824-c0-0-0
Degree $2$
Conductor $1191$
Sign $0.731 - 0.681i$
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  + (−0.786 + 0.618i)3-s + (−0.841 + 0.540i)4-s + (1.68 − 0.408i)7-s + (0.235 − 0.971i)9-s + (0.327 − 0.945i)12-s + (−0.173 + 0.336i)13-s + (0.415 − 0.909i)16-s + (0.839 − 1.17i)19-s + (−1.07 + 1.36i)21-s + (0.888 + 0.458i)25-s + (0.415 + 0.909i)27-s + (−1.19 + 1.25i)28-s + (−0.279 + 1.94i)31-s + (0.327 + 0.945i)36-s + (−0.771 + 0.308i)37-s + ⋯
L(s)  = 1  + (−0.786 + 0.618i)3-s + (−0.841 + 0.540i)4-s + (1.68 − 0.408i)7-s + (0.235 − 0.971i)9-s + (0.327 − 0.945i)12-s + (−0.173 + 0.336i)13-s + (0.415 − 0.909i)16-s + (0.839 − 1.17i)19-s + (−1.07 + 1.36i)21-s + (0.888 + 0.458i)25-s + (0.415 + 0.909i)27-s + (−1.19 + 1.25i)28-s + (−0.279 + 1.94i)31-s + (0.327 + 0.945i)36-s + (−0.771 + 0.308i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1191 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.731 - 0.681i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1191 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.731 - 0.681i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1191\)    =    \(3 \cdot 397\)
Sign: $0.731 - 0.681i$
Analytic conductor: \(0.594386\)
Root analytic conductor: \(0.770964\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1191} (824, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1191,\ (\ :0),\ 0.731 - 0.681i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8230784568\)
\(L(\frac12)\) \(\approx\) \(0.8230784568\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.786 - 0.618i)T \)
397 \( 1 - T \)
good2 \( 1 + (0.841 - 0.540i)T^{2} \)
5 \( 1 + (-0.888 - 0.458i)T^{2} \)
7 \( 1 + (-1.68 + 0.408i)T + (0.888 - 0.458i)T^{2} \)
11 \( 1 + (0.786 + 0.618i)T^{2} \)
13 \( 1 + (0.173 - 0.336i)T + (-0.580 - 0.814i)T^{2} \)
17 \( 1 + (0.841 + 0.540i)T^{2} \)
19 \( 1 + (-0.839 + 1.17i)T + (-0.327 - 0.945i)T^{2} \)
23 \( 1 + (0.786 + 0.618i)T^{2} \)
29 \( 1 + (-0.0475 - 0.998i)T^{2} \)
31 \( 1 + (0.279 - 1.94i)T + (-0.959 - 0.281i)T^{2} \)
37 \( 1 + (0.771 - 0.308i)T + (0.723 - 0.690i)T^{2} \)
41 \( 1 + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.452 + 0.132i)T + (0.841 - 0.540i)T^{2} \)
47 \( 1 + (0.995 - 0.0950i)T^{2} \)
53 \( 1 + (-0.654 - 0.755i)T^{2} \)
59 \( 1 + (0.235 + 0.971i)T^{2} \)
61 \( 1 + (-0.566 - 1.09i)T + (-0.580 + 0.814i)T^{2} \)
67 \( 1 + (1.56 - 0.625i)T + (0.723 - 0.690i)T^{2} \)
71 \( 1 + (-0.959 - 0.281i)T^{2} \)
73 \( 1 + (-1.76 + 0.168i)T + (0.981 - 0.189i)T^{2} \)
79 \( 1 + (0.786 + 1.36i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (-0.841 + 0.540i)T^{2} \)
89 \( 1 + (0.580 - 0.814i)T^{2} \)
97 \( 1 + (-1.44 + 1.37i)T + (0.0475 - 0.998i)T^{2} \)
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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

−10.18105204104591609341431930658, −9.031447262230448969524474706971, −8.709346455850391738837445462820, −7.52918252555183114770584729184, −6.92311319861544905257332482280, −5.33780416119972710983468329913, −4.93045154966282235430526786196, −4.27872237065334151917702716050, −3.16924641270383790485768948064, −1.22231599104511510944417701294, 1.11735714751445065116988286945, 2.14098193585063054783803207997, 4.05856663886731911673310717736, 5.04668868393526370489186732338, 5.43174955277991233009264550570, 6.29518778867131511206697788778, 7.67604212968501856279740134546, 8.019721578978938047474934766389, 8.967862900040828213470777393724, 9.985329413239007104060828732593

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