Properties

Label 2-1134-1.1-c1-0-20
Degree $2$
Conductor $1134$
Sign $-1$
Analytic cond. $9.05503$
Root an. cond. $3.00915$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s − 3.73·5-s − 7-s + 8-s − 3.73·10-s + 4.19·11-s + 0.464·13-s − 14-s + 16-s − 7·17-s − 2.73·19-s − 3.73·20-s + 4.19·22-s − 6.19·23-s + 8.92·25-s + 0.464·26-s − 28-s − 8.46·29-s − 2.19·31-s + 32-s − 7·34-s + 3.73·35-s − 6.66·37-s − 2.73·38-s − 3.73·40-s − 9.46·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s − 1.66·5-s − 0.377·7-s + 0.353·8-s − 1.18·10-s + 1.26·11-s + 0.128·13-s − 0.267·14-s + 0.250·16-s − 1.69·17-s − 0.626·19-s − 0.834·20-s + 0.894·22-s − 1.29·23-s + 1.78·25-s + 0.0910·26-s − 0.188·28-s − 1.57·29-s − 0.394·31-s + 0.176·32-s − 1.20·34-s + 0.630·35-s − 1.09·37-s − 0.443·38-s − 0.590·40-s − 1.47·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1134\)    =    \(2 \cdot 3^{4} \cdot 7\)
Sign: $-1$
Analytic conductor: \(9.05503\)
Root analytic conductor: \(3.00915\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1134,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 + T \)
good5 \( 1 + 3.73T + 5T^{2} \)
11 \( 1 - 4.19T + 11T^{2} \)
13 \( 1 - 0.464T + 13T^{2} \)
17 \( 1 + 7T + 17T^{2} \)
19 \( 1 + 2.73T + 19T^{2} \)
23 \( 1 + 6.19T + 23T^{2} \)
29 \( 1 + 8.46T + 29T^{2} \)
31 \( 1 + 2.19T + 31T^{2} \)
37 \( 1 + 6.66T + 37T^{2} \)
41 \( 1 + 9.46T + 41T^{2} \)
43 \( 1 - 5.46T + 43T^{2} \)
47 \( 1 + 1.26T + 47T^{2} \)
53 \( 1 - 2.53T + 53T^{2} \)
59 \( 1 - 6.19T + 59T^{2} \)
61 \( 1 + 9.92T + 61T^{2} \)
67 \( 1 + 3.26T + 67T^{2} \)
71 \( 1 + 13.4T + 71T^{2} \)
73 \( 1 - 11.7T + 73T^{2} \)
79 \( 1 - 15.1T + 79T^{2} \)
83 \( 1 - 14.5T + 83T^{2} \)
89 \( 1 - 3.92T + 89T^{2} \)
97 \( 1 + 2.92T + 97T^{2} \)
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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

−9.207135692349532420199214553518, −8.567860918077030616389573513678, −7.58938048544217216140004445826, −6.83580903220873777734978824021, −6.17325726335194061442348403962, −4.79036484471245917661873888650, −3.91150210175583802167136505912, −3.61666009803687336262678884369, −2.00739084460189131272843904705, 0, 2.00739084460189131272843904705, 3.61666009803687336262678884369, 3.91150210175583802167136505912, 4.79036484471245917661873888650, 6.17325726335194061442348403962, 6.83580903220873777734978824021, 7.58938048544217216140004445826, 8.567860918077030616389573513678, 9.207135692349532420199214553518

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