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Properties

Label 2-1134-9.7-c1-0-20
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

\(L(1)\) \(\approx\) \(0.8994193585\)
\(L(\frac12)\) \(\approx\) \(0.8994193585\)
\(L(\frac{3}{2})\) 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

−9.197137925833759302187631050129, −8.772468840750712344833490555380, −8.048792714756181613031127715605, −6.45473547018816935600626349402, −6.13583817042647730762954402209, −4.71519917583657309817246387580, −4.19328386521004793904242747645, −3.07627098327108403018735489422, −1.85237491631129760380828367084, −0.33386906735975994085501704648, 2.01298696169479382691030566272, 3.35374773134845549697044851615, 4.25634080333921945733477776879, 4.97344963616542716136697732867, 6.46583651900422204461457243042, 6.77823557383799948722601038240, 7.47554885864647770778863422604, 8.642864117833244198216535756904, 9.172103799665737207237346291179, 10.45829840233189293689984031025

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