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Properties

Label 2-1191-1191.1190-c0-0-5
Degree $2$
Conductor $1191$
Sign $1$
Analytic cond. $0.594386$
Root an. cond. $0.770964$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 4-s + 9-s − 12-s + 16-s + 2·19-s − 25-s + 27-s + 2·31-s − 36-s − 2·37-s − 2·43-s + 48-s + 49-s + 2·57-s − 64-s − 2·67-s + 2·73-s − 75-s − 2·76-s − 2·79-s + 81-s + 2·93-s − 2·97-s + 100-s − 108-s − 2·111-s + ⋯
L(s)  = 1  + 3-s − 4-s + 9-s − 12-s + 16-s + 2·19-s − 25-s + 27-s + 2·31-s − 36-s − 2·37-s − 2·43-s + 48-s + 49-s + 2·57-s − 64-s − 2·67-s + 2·73-s − 75-s − 2·76-s − 2·79-s + 81-s + 2·93-s − 2·97-s + 100-s − 108-s − 2·111-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: yes
Arithmetic: yes
Character: $\chi_{1191} (1190, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1191,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.248033783\)
\(L(\frac12)\) \(\approx\) \(1.248033783\)
\(L(1)\) 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 \( 1 + T^{2} \)
5 \( 1 + T^{2} \)
7 \( ( 1 - T )( 1 + T ) \)
11 \( ( 1 - T )( 1 + T ) \)
13 \( ( 1 - T )( 1 + T ) \)
17 \( 1 + T^{2} \)
19 \( ( 1 - T )^{2} \)
23 \( ( 1 - T )( 1 + T ) \)
29 \( ( 1 - T )( 1 + T ) \)
31 \( ( 1 - T )^{2} \)
37 \( ( 1 + T )^{2} \)
41 \( 1 + T^{2} \)
43 \( ( 1 + T )^{2} \)
47 \( ( 1 - T )( 1 + T ) \)
53 \( 1 + T^{2} \)
59 \( 1 + T^{2} \)
61 \( ( 1 - T )( 1 + T ) \)
67 \( ( 1 + T )^{2} \)
71 \( 1 + T^{2} \)
73 \( ( 1 - T )^{2} \)
79 \( ( 1 + T )^{2} \)
83 \( ( 1 - T )( 1 + T ) \)
89 \( 1 + T^{2} \)
97 \( ( 1 + T )^{2} \)
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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

−9.893606757532386027497521558517, −9.109203232965583024233153922153, −8.380670648781126712699683970530, −7.74764024087367885466045725478, −6.85032702069394696444899381008, −5.52734735128488485823562562866, −4.71105527454989489968231811918, −3.71896309203292980158128140327, −2.96661582784196102496507769473, −1.40310922148516324268238960221, 1.40310922148516324268238960221, 2.96661582784196102496507769473, 3.71896309203292980158128140327, 4.71105527454989489968231811918, 5.52734735128488485823562562866, 6.85032702069394696444899381008, 7.74764024087367885466045725478, 8.380670648781126712699683970530, 9.109203232965583024233153922153, 9.893606757532386027497521558517

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