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Properties

Label 2-1134-21.17-c1-0-21
Degree $2$
Conductor $1134$
Sign $0.939 - 0.341i$
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  + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + (2.08 + 3.61i)5-s + (1.94 − 1.79i)7-s + 0.999i·8-s + (−3.61 − 2.08i)10-s + (0.0814 + 0.0470i)11-s − 3.79i·13-s + (−0.786 + 2.52i)14-s + (−0.5 − 0.866i)16-s + (3.82 − 6.62i)17-s + (6.77 − 3.91i)19-s + 4.17·20-s − 0.0940·22-s + (3.31 − 1.91i)23-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + (0.932 + 1.61i)5-s + (0.734 − 0.678i)7-s + 0.353i·8-s + (−1.14 − 0.659i)10-s + (0.0245 + 0.0141i)11-s − 1.05i·13-s + (−0.210 + 0.675i)14-s + (−0.125 − 0.216i)16-s + (0.928 − 1.60i)17-s + (1.55 − 0.897i)19-s + 0.932·20-s − 0.0200·22-s + (0.691 − 0.399i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.341i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.939 - 0.341i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1134\)    =    \(2 \cdot 3^{4} \cdot 7\)
Sign: $0.939 - 0.341i$
Analytic conductor: \(9.05503\)
Root analytic conductor: \(3.00915\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1134} (647, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1134,\ (\ :1/2),\ 0.939 - 0.341i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.650298421\)
\(L(\frac12)\) \(\approx\) \(1.650298421\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
3 \( 1 \)
7 \( 1 + (-1.94 + 1.79i)T \)
good5 \( 1 + (-2.08 - 3.61i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.0814 - 0.0470i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 3.79iT - 13T^{2} \)
17 \( 1 + (-3.82 + 6.62i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-6.77 + 3.91i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.31 + 1.91i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 2.03iT - 29T^{2} \)
31 \( 1 + (1.45 + 0.840i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.18 + 3.78i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 3.07T + 41T^{2} \)
43 \( 1 + 5.12T + 43T^{2} \)
47 \( 1 + (-2.54 - 4.41i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (6.05 + 3.49i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.21 - 9.02i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.69 + 1.55i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.76 - 6.52i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 8.61iT - 71T^{2} \)
73 \( 1 + (0.129 + 0.0750i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.48 - 11.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 2.58T + 83T^{2} \)
89 \( 1 + (-1.37 - 2.37i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 4.06iT - 97T^{2} \)
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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

−9.861180365676616202309277077855, −9.310293406586615226062504975924, −7.949788380483102397665924929050, −7.25328471576681120392707540177, −6.90190776865642713013894094221, −5.65554572276798262706671974743, −5.08141279182572396526637195280, −3.27143188047518975523673082930, −2.57640672821323983001539520944, −1.04248762407145452981947744609, 1.50371500428630189031356078500, 1.66406004004760020350547973429, 3.45489365333909983716146771451, 4.75600667441920953642114542842, 5.44114931494427170288026891710, 6.24368163561575498805585068488, 7.68480251861846463209387317737, 8.384075207101802073291112992965, 9.003197630448842726834441240798, 9.602000838490566521458571011877

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