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Properties

Label 2-1134-9.7-c1-0-1
Degree $2$
Conductor $1134$
Sign $-0.984 - 0.173i$
Analytic cond. $9.05503$
Root an. cond. $3.00915$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (0.866 + 1.5i)5-s + (−0.5 + 0.866i)7-s + 0.999·8-s − 1.73·10-s + (−2.36 + 4.09i)11-s + (0.5 + 0.866i)13-s + (−0.499 − 0.866i)14-s + (−0.5 + 0.866i)16-s + 6.46·17-s − 6.19·19-s + (0.866 − 1.49i)20-s + (−2.36 − 4.09i)22-s + (−0.633 − 1.09i)23-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.387 + 0.670i)5-s + (−0.188 + 0.327i)7-s + 0.353·8-s − 0.547·10-s + (−0.713 + 1.23i)11-s + (0.138 + 0.240i)13-s + (−0.133 − 0.231i)14-s + (−0.125 + 0.216i)16-s + 1.56·17-s − 1.42·19-s + (0.193 − 0.335i)20-s + (−0.504 − 0.873i)22-s + (−0.132 − 0.228i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 - 0.173i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.984 - 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1134\)    =    \(2 \cdot 3^{4} \cdot 7\)
Sign: $-0.984 - 0.173i$
Analytic conductor: \(9.05503\)
Root analytic conductor: \(3.00915\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1134} (379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1134,\ (\ :1/2),\ -0.984 - 0.173i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8928704308\)
\(L(\frac12)\) \(\approx\) \(0.8928704308\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 \)
7 \( 1 + (0.5 - 0.866i)T \)
good5 \( 1 + (-0.866 - 1.5i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.36 - 4.09i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 6.46T + 17T^{2} \)
19 \( 1 + 6.19T + 19T^{2} \)
23 \( 1 + (0.633 + 1.09i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.23 - 5.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.09 + 3.63i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 3.19T + 37T^{2} \)
41 \( 1 + (1.26 + 2.19i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (6.19 - 10.7i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.09 - 1.90i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 9.46T + 53T^{2} \)
59 \( 1 + (-4.09 - 7.09i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.69 - 8.13i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.09 + 3.63i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 4.39T + 71T^{2} \)
73 \( 1 + 9.19T + 73T^{2} \)
79 \( 1 + (-3.09 + 5.36i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.633 - 1.09i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 12.4T + 89T^{2} \)
97 \( 1 + (-8 + 13.8i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.29248741734989760345210250162, −9.401048887452808720549802809807, −8.518284970199330138505648441594, −7.61878888600240022097939413583, −6.94344919418445877063139565266, −6.10914643178145773999733533574, −5.30686008579854975967802549248, −4.26117355390498120010840731020, −2.87940454206394498583765985496, −1.78112553952568648165674512170, 0.43071588991685060656995847123, 1.72197570241629030810240047315, 3.08190958301147078977212879590, 3.89692800302526546201493055570, 5.22987087406631711106183588426, 5.81000277642061675307514354217, 7.07173081957761231899127233653, 8.173200723456803180993107833165, 8.544245034001757784600638634609, 9.526020768549386509974857334740

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