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Properties

Label 2-1134-7.4-c1-0-10
Degree $2$
Conductor $1134$
Sign $-0.944 - 0.329i$
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 + (1.59 + 2.75i)5-s + (1.85 + 1.88i)7-s − 0.999·8-s + (−1.59 + 2.75i)10-s + (−1.59 + 2.75i)11-s − 5.70·13-s + (−0.710 + 2.54i)14-s + (−0.5 − 0.866i)16-s + (−0.760 + 1.31i)17-s + (−0.641 − 1.11i)19-s − 3.18·20-s − 3.18·22-s + (−1.11 − 1.93i)23-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.711 + 1.23i)5-s + (0.699 + 0.714i)7-s − 0.353·8-s + (−0.503 + 0.871i)10-s + (−0.479 + 0.830i)11-s − 1.58·13-s + (−0.189 + 0.681i)14-s + (−0.125 − 0.216i)16-s + (−0.184 + 0.319i)17-s + (−0.147 − 0.254i)19-s − 0.711·20-s − 0.678·22-s + (−0.233 − 0.404i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.944 - 0.329i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.944 - 0.329i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1134\)    =    \(2 \cdot 3^{4} \cdot 7\)
Sign: $-0.944 - 0.329i$
Analytic conductor: \(9.05503\)
Root analytic conductor: \(3.00915\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1134} (487, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1134,\ (\ :1/2),\ -0.944 - 0.329i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.869085575\)
\(L(\frac12)\) \(\approx\) \(1.869085575\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.85 - 1.88i)T \)
good5 \( 1 + (-1.59 - 2.75i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.59 - 2.75i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 5.70T + 13T^{2} \)
17 \( 1 + (0.760 - 1.31i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.641 + 1.11i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.11 + 1.93i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 7.08T + 29T^{2} \)
31 \( 1 + (-4.71 + 8.15i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 5.60T + 41T^{2} \)
43 \( 1 + 6.82T + 43T^{2} \)
47 \( 1 + (-2.91 - 5.04i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.02 + 1.78i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.562 + 0.974i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.56 + 2.70i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.48 - 9.49i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 8.69T + 71T^{2} \)
73 \( 1 + (2.48 - 4.30i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.06 - 3.58i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 8.06T + 83T^{2} \)
89 \( 1 + (-0.112 - 0.195i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 14.8T + 97T^{2} \)
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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

−10.04296635566507981929615311391, −9.520069106591257353197710169073, −8.298160480872879912079814577957, −7.58447873116808102106397832796, −6.77885162443105384243562850611, −6.06970804619443227687339552934, −5.09481085628054121325194699005, −4.39083704206111320791882918663, −2.65467428457696803382658650197, −2.32356884060463520394219276037, 0.70669708902964603690244121896, 1.84202883338058939313569372162, 3.01526161670410340705017863544, 4.47957589547385222976034971406, 4.93908990427110358206093554071, 5.67938817851363839070376442607, 6.91795353534151415967185404003, 8.030090939479523499743233913424, 8.688861890710764420257839663302, 9.632725903994225556716707708161

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