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Properties

Label 2-1191-1191.686-c0-0-0
Degree $2$
Conductor $1191$
Sign $0.997 - 0.0744i$
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  + (0.580 + 0.814i)3-s + (0.142 − 0.989i)4-s + (1.63 − 0.566i)7-s + (−0.327 + 0.945i)9-s + (0.888 − 0.458i)12-s + (−1.23 + 1.57i)13-s + (−0.959 − 0.281i)16-s + (0.462 − 1.90i)19-s + (1.41 + 1.00i)21-s + (0.786 + 0.618i)25-s + (−0.959 + 0.281i)27-s + (−0.327 − 1.70i)28-s + (0.0395 + 0.0865i)31-s + (0.888 + 0.458i)36-s + (−1.91 − 0.182i)37-s + ⋯
L(s)  = 1  + (0.580 + 0.814i)3-s + (0.142 − 0.989i)4-s + (1.63 − 0.566i)7-s + (−0.327 + 0.945i)9-s + (0.888 − 0.458i)12-s + (−1.23 + 1.57i)13-s + (−0.959 − 0.281i)16-s + (0.462 − 1.90i)19-s + (1.41 + 1.00i)21-s + (0.786 + 0.618i)25-s + (−0.959 + 0.281i)27-s + (−0.327 − 1.70i)28-s + (0.0395 + 0.0865i)31-s + (0.888 + 0.458i)36-s + (−1.91 − 0.182i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1191 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0744i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1191 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0744i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1191\)    =    \(3 \cdot 397\)
Sign: $0.997 - 0.0744i$
Analytic conductor: \(0.594386\)
Root analytic conductor: \(0.770964\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1191} (686, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1191,\ (\ :0),\ 0.997 - 0.0744i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.441578257\)
\(L(\frac12)\) \(\approx\) \(1.441578257\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.580 - 0.814i)T \)
397 \( 1 - T \)
good2 \( 1 + (-0.142 + 0.989i)T^{2} \)
5 \( 1 + (-0.786 - 0.618i)T^{2} \)
7 \( 1 + (-1.63 + 0.566i)T + (0.786 - 0.618i)T^{2} \)
11 \( 1 + (-0.580 + 0.814i)T^{2} \)
13 \( 1 + (1.23 - 1.57i)T + (-0.235 - 0.971i)T^{2} \)
17 \( 1 + (-0.142 - 0.989i)T^{2} \)
19 \( 1 + (-0.462 + 1.90i)T + (-0.888 - 0.458i)T^{2} \)
23 \( 1 + (-0.580 + 0.814i)T^{2} \)
29 \( 1 + (-0.928 + 0.371i)T^{2} \)
31 \( 1 + (-0.0395 - 0.0865i)T + (-0.654 + 0.755i)T^{2} \)
37 \( 1 + (1.91 + 0.182i)T + (0.981 + 0.189i)T^{2} \)
41 \( 1 + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.428 + 0.494i)T + (-0.142 + 0.989i)T^{2} \)
47 \( 1 + (-0.723 + 0.690i)T^{2} \)
53 \( 1 + (0.841 + 0.540i)T^{2} \)
59 \( 1 + (-0.327 - 0.945i)T^{2} \)
61 \( 1 + (-1.00 - 1.28i)T + (-0.235 + 0.971i)T^{2} \)
67 \( 1 + (0.283 + 0.0270i)T + (0.981 + 0.189i)T^{2} \)
71 \( 1 + (-0.654 + 0.755i)T^{2} \)
73 \( 1 + (1.13 - 1.08i)T + (0.0475 - 0.998i)T^{2} \)
79 \( 1 + (-0.580 + 1.00i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.142 - 0.989i)T^{2} \)
89 \( 1 + (0.235 - 0.971i)T^{2} \)
97 \( 1 + (1.42 + 0.273i)T + (0.928 + 0.371i)T^{2} \)
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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

−9.989096675176104449963652419073, −9.096461984337967158399357327616, −8.689032817071117988160604914956, −7.30534783488353762997155515550, −6.97712851859216743281842750367, −5.17681718838640504428936975340, −4.94745878922649085613068416020, −4.17170587089843275561672325870, −2.54374667639633853645931489779, −1.61992630781522830491126593692, 1.68066703876015595368781172973, 2.63502722120872506675950571301, 3.53286886649685822885099320772, 4.85961022377496667565182992236, 5.68963854650844775905004250048, 6.97478407212224244102941560717, 7.80862933251774539154444507452, 8.108741233645788020397830649957, 8.669686086008928573663926935299, 9.877690822566046068880176444986

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