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Properties

Label 2-1134-63.4-c1-0-11
Degree $2$
Conductor $1134$
Sign $0.975 + 0.220i$
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 − 3·5-s + (−2 + 1.73i)7-s − 0.999·8-s + (−1.5 + 2.59i)10-s + 3·11-s + (−1 + 1.73i)13-s + (0.499 + 2.59i)14-s + (−0.5 + 0.866i)16-s + (3 − 5.19i)17-s + (−1 − 1.73i)19-s + (1.49 + 2.59i)20-s + (1.5 − 2.59i)22-s + 6·23-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s − 1.34·5-s + (−0.755 + 0.654i)7-s − 0.353·8-s + (−0.474 + 0.821i)10-s + 0.904·11-s + (−0.277 + 0.480i)13-s + (0.133 + 0.694i)14-s + (−0.125 + 0.216i)16-s + (0.727 − 1.26i)17-s + (−0.229 − 0.397i)19-s + (0.335 + 0.580i)20-s + (0.319 − 0.553i)22-s + 1.25·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 + 0.220i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.975 + 0.220i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1134\)    =    \(2 \cdot 3^{4} \cdot 7\)
Sign: $0.975 + 0.220i$
Analytic conductor: \(9.05503\)
Root analytic conductor: \(3.00915\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1134} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1134,\ (\ :1/2),\ 0.975 + 0.220i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.300254646\)
\(L(\frac12)\) \(\approx\) \(1.300254646\)
\(L(\frac{3}{2})\) 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 \( 1 + (2 - 1.73i)T \)
good5 \( 1 + 3T + 5T^{2} \)
11 \( 1 - 3T + 11T^{2} \)
13 \( 1 + (1 - 1.73i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3 + 5.19i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1 + 1.73i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 6T + 23T^{2} \)
29 \( 1 + (-4.5 - 7.79i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.5 - 6.06i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5 - 8.66i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2 - 3.46i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-6 + 10.3i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.5 - 2.59i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.5 + 2.59i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2 + 3.46i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1 + 1.73i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (1 - 1.73i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.5 - 4.33i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.5 - 7.79i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (3 + 5.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6.5 - 11.2i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.726054653558983734764329727628, −9.070680312802161235305342108040, −8.353231050658178692623954606532, −7.06952485579367031972517453728, −6.63261219628726278398161055772, −5.21324057022628905359156653046, −4.49991948914571717453111671184, −3.38669133562200595296484428438, −2.82458963834384751437891636840, −0.975484638579017762636638776897, 0.70598724177145393018985165130, 2.96768067327223288966911017286, 4.02488429265996079437928500239, 4.24487051295829938467574555157, 5.81213056561974991646725969432, 6.48825635455924513510815005127, 7.52661575484205842675603170925, 7.85397161044719272567085924444, 8.851647777639541928662659320889, 9.787139695390997440403493160242

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