Properties

Degree $2$
Conductor $1134$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 3.44·5-s − 7-s + 8-s + 3.44·10-s + 2·11-s − 4.89·13-s − 14-s + 16-s + 2·17-s + 7.44·19-s + 3.44·20-s + 2·22-s − 23-s + 6.89·25-s − 4.89·26-s − 28-s + 2.89·29-s + 6·31-s + 32-s + 2·34-s − 3.44·35-s − 7.79·37-s + 7.44·38-s + 3.44·40-s − 9.79·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + 1.54·5-s − 0.377·7-s + 0.353·8-s + 1.09·10-s + 0.603·11-s − 1.35·13-s − 0.267·14-s + 0.250·16-s + 0.485·17-s + 1.70·19-s + 0.771·20-s + 0.426·22-s − 0.208·23-s + 1.37·25-s − 0.960·26-s − 0.188·28-s + 0.538·29-s + 1.07·31-s + 0.176·32-s + 0.342·34-s − 0.583·35-s − 1.28·37-s + 1.20·38-s + 0.545·40-s − 1.53·41-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: $1$
Motivic weight: \(1\)
Character: $\chi_{1134} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1134,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.235769349\)
\(L(\frac12)\) \(\approx\) \(3.235769349\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.44T + 5T^{2} \)
11 \( 1 - 2T + 11T^{2} \)
13 \( 1 + 4.89T + 13T^{2} \)
17 \( 1 - 2T + 17T^{2} \)
19 \( 1 - 7.44T + 19T^{2} \)
23 \( 1 + T + 23T^{2} \)
29 \( 1 - 2.89T + 29T^{2} \)
31 \( 1 - 6T + 31T^{2} \)
37 \( 1 + 7.79T + 37T^{2} \)
41 \( 1 + 9.79T + 41T^{2} \)
43 \( 1 + 2.89T + 43T^{2} \)
47 \( 1 + 9.79T + 47T^{2} \)
53 \( 1 + 1.10T + 53T^{2} \)
59 \( 1 + 2T + 59T^{2} \)
61 \( 1 - 11.4T + 61T^{2} \)
67 \( 1 + 3.10T + 67T^{2} \)
71 \( 1 - 9.89T + 71T^{2} \)
73 \( 1 - 2.89T + 73T^{2} \)
79 \( 1 - 7.89T + 79T^{2} \)
83 \( 1 - 2T + 83T^{2} \)
89 \( 1 + 7.10T + 89T^{2} \)
97 \( 1 + 6.89T + 97T^{2} \)
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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

−9.871789683529587753535228693256, −9.351794020520797996072876028654, −8.097411820302655090278509220683, −6.93646771189539582000617549097, −6.46185032101993303340984275912, −5.34585629002613168293784273338, −5.00531107582065291770556216251, −3.47190081021483249752507638496, −2.57532945852329164575102546344, −1.45426245522405647146436173250, 1.45426245522405647146436173250, 2.57532945852329164575102546344, 3.47190081021483249752507638496, 5.00531107582065291770556216251, 5.34585629002613168293784273338, 6.46185032101993303340984275912, 6.93646771189539582000617549097, 8.097411820302655090278509220683, 9.351794020520797996072876028654, 9.871789683529587753535228693256

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