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Properties

Label 2-1134-63.16-c1-0-23
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 + (2.5 − 0.866i)7-s − 0.999·8-s + 6·11-s + (−2.5 − 4.33i)13-s + (2 + 1.73i)14-s + (−0.5 − 0.866i)16-s + (−3 − 5.19i)17-s + (2 − 3.46i)19-s + (3 + 5.19i)22-s + 6·23-s − 5·25-s + (2.5 − 4.33i)26-s + (−0.499 + 2.59i)28-s + (−3 + 5.19i)29-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.944 − 0.327i)7-s − 0.353·8-s + 1.80·11-s + (−0.693 − 1.20i)13-s + (0.534 + 0.462i)14-s + (−0.125 − 0.216i)16-s + (−0.727 − 1.26i)17-s + (0.458 − 0.794i)19-s + (0.639 + 1.10i)22-s + 1.25·23-s − 25-s + (0.490 − 0.849i)26-s + (−0.0944 + 0.490i)28-s + (−0.557 + 0.964i)29-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: $\chi_{1134} (541, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1134,\ (\ :1/2),\ 0.975 - 0.220i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.206744625\)
\(L(\frac12)\) \(\approx\) \(2.206744625\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.5 + 0.866i)T \)
good5 \( 1 + 5T^{2} \)
11 \( 1 - 6T + 11T^{2} \)
13 \( 1 + (2.5 + 4.33i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (3 + 5.19i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2 + 3.46i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 6T + 23T^{2} \)
29 \( 1 + (3 - 5.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T + (-18.5 - 32.0i)T^{2} \)
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 \( 1 + (-3 - 5.19i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3 + 5.19i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 + (1 + 1.73i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3 - 5.19i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (8.5 - 14.7i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.465332533031813976941574797422, −9.120305119157853814756542819241, −8.009311720512961542835125142076, −7.25040745162203603056926161355, −6.68477463206211802736074348248, −5.43328226005505491157279947663, −4.80342790808219005803640306317, −3.88000364443450822150639819388, −2.67997420454366301469499555766, −1.00651431091087365133455105355, 1.45918747090600352519881053067, 2.18130811670038671856949365784, 3.88168682084895238490402238393, 4.25284980915254941516585475812, 5.42275544283082452023052915588, 6.33861633985820331710232424977, 7.19898118594152866756694412444, 8.382233093431997321324514039656, 9.105797657443044987251634530276, 9.710392721598376877064088641370

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