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Properties

Label 2-1191-1191.1190-c0-0-8
Degree $2$
Conductor $1191$
Sign $1$
Analytic cond. $0.594386$
Root an. cond. $0.770964$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.93·2-s − 3-s + 2.73·4-s − 0.517·5-s − 1.93·6-s + 3.34·8-s + 9-s − 0.999·10-s − 2.73·12-s + 0.517·15-s + 3.73·16-s − 1.41·17-s + 1.93·18-s + 19-s − 1.41·20-s − 3.34·24-s − 0.732·25-s − 27-s + 0.999·30-s − 31-s + 3.86·32-s − 2.73·34-s + 2.73·36-s + 1.93·38-s − 1.73·40-s + 1.41·41-s − 0.517·45-s + ⋯
L(s)  = 1  + 1.93·2-s − 3-s + 2.73·4-s − 0.517·5-s − 1.93·6-s + 3.34·8-s + 9-s − 0.999·10-s − 2.73·12-s + 0.517·15-s + 3.73·16-s − 1.41·17-s + 1.93·18-s + 19-s − 1.41·20-s − 3.34·24-s − 0.732·25-s − 27-s + 0.999·30-s − 31-s + 3.86·32-s − 2.73·34-s + 2.73·36-s + 1.93·38-s − 1.73·40-s + 1.41·41-s − 0.517·45-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1191\)    =    \(3 \cdot 397\)
Sign: $1$
Analytic conductor: \(0.594386\)
Root analytic conductor: \(0.770964\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1191} (1190, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1191,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.336685346\)
\(L(\frac12)\) \(\approx\) \(2.336685346\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
397 \( 1 + T \)
good2 \( 1 - 1.93T + T^{2} \)
5 \( 1 + 0.517T + T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + 1.41T + T^{2} \)
19 \( 1 - T + T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + T + T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 - 1.41T + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + 1.93T + T^{2} \)
59 \( 1 + 1.93T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + 1.73T + T^{2} \)
71 \( 1 + 1.41T + T^{2} \)
73 \( 1 - T + T^{2} \)
79 \( 1 + 1.73T + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - 0.517T + T^{2} \)
97 \( 1 - 1.73T + T^{2} \)
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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

−10.57688575317036840523599726214, −9.331515386955938378135892592754, −7.66730441021853596663718474481, −7.20165220117132312963967109716, −6.22934594369213000324451759453, −5.69171842019854498984168504218, −4.67733729562518867337520387394, −4.20625525530942914781382447372, −3.14510258706625496672280430292, −1.79367700281003494054032234104, 1.79367700281003494054032234104, 3.14510258706625496672280430292, 4.20625525530942914781382447372, 4.67733729562518867337520387394, 5.69171842019854498984168504218, 6.22934594369213000324451759453, 7.20165220117132312963967109716, 7.66730441021853596663718474481, 9.331515386955938378135892592754, 10.57688575317036840523599726214

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