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Properties

Label 2-1134-7.2-c1-0-3
Degree $2$
Conductor $1134$
Sign $-0.0996 - 0.995i$
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 + (−0.794 + 1.37i)5-s + (1.40 + 2.24i)7-s − 0.999·8-s + (0.794 + 1.37i)10-s + (0.794 + 1.37i)11-s − 4.81·13-s + (2.64 − 0.0963i)14-s + (−0.5 + 0.866i)16-s + (−2.69 − 4.67i)17-s + (−3.54 + 6.14i)19-s + 1.58·20-s + 1.58·22-s + (−0.150 + 0.260i)23-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.355 + 0.615i)5-s + (0.531 + 0.847i)7-s − 0.353·8-s + (0.251 + 0.434i)10-s + (0.239 + 0.414i)11-s − 1.33·13-s + (0.706 − 0.0257i)14-s + (−0.125 + 0.216i)16-s + (−0.654 − 1.13i)17-s + (−0.814 + 1.41i)19-s + 0.355·20-s + 0.338·22-s + (−0.0313 + 0.0542i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0996 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0996 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1134\)    =    \(2 \cdot 3^{4} \cdot 7\)
Sign: $-0.0996 - 0.995i$
Analytic conductor: \(9.05503\)
Root analytic conductor: \(3.00915\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1134} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1134,\ (\ :1/2),\ -0.0996 - 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9571699882\)
\(L(\frac12)\) \(\approx\) \(0.9571699882\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.40 - 2.24i)T \)
good5 \( 1 + (0.794 - 1.37i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.794 - 1.37i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 4.81T + 13T^{2} \)
17 \( 1 + (2.69 + 4.67i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.54 - 6.14i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.150 - 0.260i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 8.27T + 29T^{2} \)
31 \( 1 + (-1.35 - 2.34i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 5.87T + 41T^{2} \)
43 \( 1 - 1.66T + 43T^{2} \)
47 \( 1 + (1.33 - 2.30i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.44 - 4.23i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.23 + 5.60i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.23 + 3.87i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.02 - 8.70i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 12.7T + 71T^{2} \)
73 \( 1 + (-8.02 - 13.9i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.19 - 7.26i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 2.36T + 83T^{2} \)
89 \( 1 + (-1.60 + 2.78i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 1.42T + 97T^{2} \)
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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

−10.00483079172165342574242178178, −9.444454625980450251061520200518, −8.479432527918774528431181485604, −7.50225287768629732968826552213, −6.73551465815983117512114274982, −5.55703976800695274111517555236, −4.83783695518978570808572298696, −3.82517997703292050499498052373, −2.67209960011429352830630890602, −1.88715868979233250334510859128, 0.34951405689446561372440713971, 2.15375555088186542980703910146, 3.72093124740181347776476123985, 4.52159226329993061107549219855, 5.08944172263743685607848463060, 6.35183166633152964357663755909, 7.09342651361578960602277079298, 7.918756039392911359242498570301, 8.591571170538832276275867015657, 9.383182665533292263337540379516

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