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Properties

Label 2-1191-1191.110-c0-0-0
Degree $2$
Conductor $1191$
Sign $0.816 - 0.577i$
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  + (0.0475 + 0.998i)3-s + (−0.654 − 0.755i)4-s + (0.995 + 0.0950i)7-s + (−0.995 + 0.0950i)9-s + (0.723 − 0.690i)12-s + (0.462 − 0.0892i)13-s + (−0.142 + 0.989i)16-s + (1.07 + 0.431i)19-s + (−0.0475 + 0.998i)21-s + (0.981 − 0.189i)25-s + (−0.142 − 0.989i)27-s + (−0.580 − 0.814i)28-s + (0.396 + 0.254i)31-s + (0.723 + 0.690i)36-s + (0.252 − 0.130i)37-s + ⋯
L(s)  = 1  + (0.0475 + 0.998i)3-s + (−0.654 − 0.755i)4-s + (0.995 + 0.0950i)7-s + (−0.995 + 0.0950i)9-s + (0.723 − 0.690i)12-s + (0.462 − 0.0892i)13-s + (−0.142 + 0.989i)16-s + (1.07 + 0.431i)19-s + (−0.0475 + 0.998i)21-s + (0.981 − 0.189i)25-s + (−0.142 − 0.989i)27-s + (−0.580 − 0.814i)28-s + (0.396 + 0.254i)31-s + (0.723 + 0.690i)36-s + (0.252 − 0.130i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1191 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1191 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1191\)    =    \(3 \cdot 397\)
Sign: $0.816 - 0.577i$
Analytic conductor: \(0.594386\)
Root analytic conductor: \(0.770964\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1191} (110, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1191,\ (\ :0),\ 0.816 - 0.577i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.028263087\)
\(L(\frac12)\) \(\approx\) \(1.028263087\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.0475 - 0.998i)T \)
397 \( 1 - T \)
good2 \( 1 + (0.654 + 0.755i)T^{2} \)
5 \( 1 + (-0.981 + 0.189i)T^{2} \)
7 \( 1 + (-0.995 - 0.0950i)T + (0.981 + 0.189i)T^{2} \)
11 \( 1 + (-0.0475 + 0.998i)T^{2} \)
13 \( 1 + (-0.462 + 0.0892i)T + (0.928 - 0.371i)T^{2} \)
17 \( 1 + (0.654 - 0.755i)T^{2} \)
19 \( 1 + (-1.07 - 0.431i)T + (0.723 + 0.690i)T^{2} \)
23 \( 1 + (-0.0475 + 0.998i)T^{2} \)
29 \( 1 + (0.327 - 0.945i)T^{2} \)
31 \( 1 + (-0.396 - 0.254i)T + (0.415 + 0.909i)T^{2} \)
37 \( 1 + (-0.252 + 0.130i)T + (0.580 - 0.814i)T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.827 - 1.81i)T + (-0.654 - 0.755i)T^{2} \)
47 \( 1 + (0.786 + 0.618i)T^{2} \)
53 \( 1 + (0.959 + 0.281i)T^{2} \)
59 \( 1 + (0.995 + 0.0950i)T^{2} \)
61 \( 1 + (-0.0934 - 0.0180i)T + (0.928 + 0.371i)T^{2} \)
67 \( 1 + (-1.16 + 0.600i)T + (0.580 - 0.814i)T^{2} \)
71 \( 1 + (-0.415 - 0.909i)T^{2} \)
73 \( 1 + (1.54 + 1.21i)T + (0.235 + 0.971i)T^{2} \)
79 \( 1 + (0.0475 - 0.0824i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.654 + 0.755i)T^{2} \)
89 \( 1 + (-0.928 - 0.371i)T^{2} \)
97 \( 1 + (0.911 - 1.28i)T + (-0.327 - 0.945i)T^{2} \)
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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

−9.945807196290660657670015750407, −9.359783159825894811443668862377, −8.506195665155793913882350676324, −7.950109913956617471777047063899, −6.46641663149693175425924527563, −5.50329600445033326521966851724, −4.92617376498288662567362251987, −4.18477291730929986028139821676, −3.03068033990282501716924848455, −1.38817053429679278699736488111, 1.17832718735144983556012179747, 2.58962441239634730596251002030, 3.63251447210355441698184914872, 4.80986713868821819913208444026, 5.55702595699030132055612935565, 6.86502863089922835080618020155, 7.44641132044809362120953423026, 8.323255717083883702525199558768, 8.683416283211420510896708355354, 9.673131310736083511877905518890

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