Properties

Label 2-1134-63.58-c1-0-3
Degree $2$
Conductor $1134$
Sign $-0.0477 - 0.998i$
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  + 2-s + 4-s + (−0.5 − 0.866i)5-s + (−2.5 + 0.866i)7-s + 8-s + (−0.5 − 0.866i)10-s + (−2.5 + 4.33i)11-s + (−2.5 + 0.866i)14-s + 16-s + (2 + 3.46i)17-s + (−4 + 6.92i)19-s + (−0.5 − 0.866i)20-s + (−2.5 + 4.33i)22-s + (2 + 3.46i)23-s + (2 − 3.46i)25-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + (−0.223 − 0.387i)5-s + (−0.944 + 0.327i)7-s + 0.353·8-s + (−0.158 − 0.273i)10-s + (−0.753 + 1.30i)11-s + (−0.668 + 0.231i)14-s + 0.250·16-s + (0.485 + 0.840i)17-s + (−0.917 + 1.58i)19-s + (−0.111 − 0.193i)20-s + (−0.533 + 0.923i)22-s + (0.417 + 0.722i)23-s + (0.400 − 0.692i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0477 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0477 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1134\)    =    \(2 \cdot 3^{4} \cdot 7\)
Sign: $-0.0477 - 0.998i$
Analytic conductor: \(9.05503\)
Root analytic conductor: \(3.00915\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1134} (919, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1134,\ (\ :1/2),\ -0.0477 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.612144843\)
\(L(\frac12)\) \(\approx\) \(1.612144843\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
7 \( 1 + (2.5 - 0.866i)T \)
good5 \( 1 + (0.5 + 0.866i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.5 - 4.33i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2 - 3.46i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4 - 6.92i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2 - 3.46i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.5 - 4.33i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 3T + 31T^{2} \)
37 \( 1 + (-2 + 3.46i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1 + 1.73i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 6T + 47T^{2} \)
53 \( 1 + (-4.5 - 7.79i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 11T + 59T^{2} \)
61 \( 1 + 6T + 61T^{2} \)
67 \( 1 + 2T + 67T^{2} \)
71 \( 1 - 2T + 71T^{2} \)
73 \( 1 + (5 + 8.66i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 - 3T + 79T^{2} \)
83 \( 1 + (-3.5 - 6.06i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-3 + 5.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.5 + 6.06i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.24325568620545621264617678335, −9.290587224480628138235668515716, −8.246954869797783614569292344123, −7.50963196714324918913973012470, −6.50085396884058343081308401086, −5.78913350808345790580688620004, −4.81077483172443124844892896363, −3.94943249384637026847873623693, −2.93240083776323606761493869618, −1.73792528971302509004567208480, 0.53574952050694922372747049499, 2.79876466069493092936522099232, 3.08468524151635009491501527226, 4.37094120031666981405274753454, 5.27967609763227623772526903188, 6.35268468604198654364906778511, 6.82963748780639103006076793117, 7.82696881808736457327883071137, 8.748594456877857504775649499084, 9.722815032513935836087037178044

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