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Properties

Label 2-1134-9.4-c1-0-5
Degree $2$
Conductor $1134$
Sign $-0.939 - 0.342i$
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 + (−0.5 − 0.866i)7-s − 0.999·8-s + (−2.5 + 4.33i)13-s + (0.499 − 0.866i)14-s + (−0.5 − 0.866i)16-s − 3·17-s + 2·19-s + (−4.5 + 7.79i)23-s + (2.5 + 4.33i)25-s − 5·26-s + 0.999·28-s + (−1.5 − 2.59i)29-s + (−2.5 + 4.33i)31-s + (0.499 − 0.866i)32-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.188 − 0.327i)7-s − 0.353·8-s + (−0.693 + 1.20i)13-s + (0.133 − 0.231i)14-s + (−0.125 − 0.216i)16-s − 0.727·17-s + 0.458·19-s + (−0.938 + 1.62i)23-s + (0.5 + 0.866i)25-s − 0.980·26-s + 0.188·28-s + (−0.278 − 0.482i)29-s + (−0.449 + 0.777i)31-s + (0.0883 − 0.153i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1134\)    =    \(2 \cdot 3^{4} \cdot 7\)
Sign: $-0.939 - 0.342i$
Analytic conductor: \(9.05503\)
Root analytic conductor: \(3.00915\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1134} (757, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1134,\ (\ :1/2),\ -0.939 - 0.342i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.081892474\)
\(L(\frac12)\) \(\approx\) \(1.081892474\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.866i)T \)
good5 \( 1 + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.5 - 4.33i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 3T + 17T^{2} \)
19 \( 1 - 2T + 19T^{2} \)
23 \( 1 + (4.5 - 7.79i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.5 + 2.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.5 - 4.33i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + (3 - 5.19i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (3 + 5.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 3T + 53T^{2} \)
59 \( 1 + (1.5 - 2.59i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5 - 8.66i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.5 + 11.2i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 9T + 71T^{2} \)
73 \( 1 - 2T + 73T^{2} \)
79 \( 1 + (-5 - 8.66i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (6 + 10.3i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 15T + 89T^{2} \)
97 \( 1 + (4 + 6.92i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.885134133042836330935516795291, −9.391262284233501507210926728986, −8.477560232599063629792787379997, −7.41594919424181699244440397273, −6.98406452730295600993489160984, −6.00348404784966079694422444278, −5.05883848137187692748835115015, −4.19752837815257050096715567336, −3.25039431342898257412006554155, −1.79398034052069660191843039230, 0.39941090450697708471924845253, 2.18334898427611332614545841639, 2.97517000591496477531489828277, 4.15878802283952430183615936459, 5.04999033406061977695829306295, 5.92166960088236342353435483151, 6.81751324330959583993421152007, 7.951236135915009545633414434809, 8.696429661124892141416874886261, 9.668560844692927382484509656630

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