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Properties

Label 2-1140-1140.1139-c0-0-7
Degree $2$
Conductor $1140$
Sign $1$
Analytic cond. $0.568934$
Root an. cond. $0.754277$
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  − 2-s + 3-s + 4-s − 5-s − 6-s − 8-s + 9-s + 10-s + 12-s − 15-s + 16-s + 2·17-s − 18-s − 19-s − 20-s − 24-s + 25-s + 27-s + 30-s + 2·31-s − 32-s − 2·34-s + 36-s + 38-s + 40-s − 45-s + 48-s + ⋯
L(s)  = 1  − 2-s + 3-s + 4-s − 5-s − 6-s − 8-s + 9-s + 10-s + 12-s − 15-s + 16-s + 2·17-s − 18-s − 19-s − 20-s − 24-s + 25-s + 27-s + 30-s + 2·31-s − 32-s − 2·34-s + 36-s + 38-s + 40-s − 45-s + 48-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1140\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 19\)
Sign: $1$
Analytic conductor: \(0.568934\)
Root analytic conductor: \(0.754277\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1140} (1139, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1140,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8663987265\)
\(L(\frac12)\) \(\approx\) \(0.8663987265\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 - T \)
5 \( 1 + T \)
19 \( 1 + T \)
good7 \( 1 + T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + T^{2} \)
17 \( ( 1 - T )^{2} \)
23 \( ( 1 - T )( 1 + T ) \)
29 \( 1 + T^{2} \)
31 \( ( 1 - T )^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + T^{2} \)
47 \( ( 1 - T )( 1 + T ) \)
53 \( ( 1 - T )( 1 + T ) \)
59 \( ( 1 - T )( 1 + T ) \)
61 \( ( 1 + T )^{2} \)
67 \( ( 1 - T )( 1 + T ) \)
71 \( ( 1 - T )( 1 + T ) \)
73 \( ( 1 - T )( 1 + T ) \)
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

−9.991091784591913482802948329323, −9.044908906432032021485334744441, −8.236410968475553606018867051385, −7.87882944752597559770846808283, −7.10212501507218326758681080019, −6.13447526563256183699887901520, −4.62789065220292675396518566814, −3.48992837028671742901452593256, −2.76079689845360262519598353339, −1.28475125623379716808094780985, 1.28475125623379716808094780985, 2.76079689845360262519598353339, 3.48992837028671742901452593256, 4.62789065220292675396518566814, 6.13447526563256183699887901520, 7.10212501507218326758681080019, 7.87882944752597559770846808283, 8.236410968475553606018867051385, 9.044908906432032021485334744441, 9.991091784591913482802948329323

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