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Properties

Label 2-1134-21.17-c1-0-1
Degree $2$
Conductor $1134$
Sign $-0.998 + 0.0503i$
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 + (1.77 + 3.07i)5-s + (−1.14 − 2.38i)7-s + 0.999i·8-s + (−3.07 − 1.77i)10-s + (−2.61 − 1.51i)11-s − 1.02i·13-s + (2.18 + 1.49i)14-s + (−0.5 − 0.866i)16-s + (−0.809 + 1.40i)17-s + (−7.12 + 4.11i)19-s + 3.55·20-s + 3.02·22-s + (−2.90 + 1.67i)23-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + (0.794 + 1.37i)5-s + (−0.432 − 0.901i)7-s + 0.353i·8-s + (−0.972 − 0.561i)10-s + (−0.789 − 0.455i)11-s − 0.284i·13-s + (0.583 + 0.399i)14-s + (−0.125 − 0.216i)16-s + (−0.196 + 0.339i)17-s + (−1.63 + 0.943i)19-s + 0.794·20-s + 0.644·22-s + (−0.606 + 0.350i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0503i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0503i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1134\)    =    \(2 \cdot 3^{4} \cdot 7\)
Sign: $-0.998 + 0.0503i$
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: \((2,\ 1134,\ (\ :1/2),\ -0.998 + 0.0503i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3966858918\)
\(L(\frac12)\) \(\approx\) \(0.3966858918\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.14 + 2.38i)T \)
good5 \( 1 + (-1.77 - 3.07i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.61 + 1.51i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 1.02iT - 13T^{2} \)
17 \( 1 + (0.809 - 1.40i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (7.12 - 4.11i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.90 - 1.67i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 4.27iT - 29T^{2} \)
31 \( 1 + (5.18 + 2.99i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.92 - 5.06i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 0.0944T + 41T^{2} \)
43 \( 1 + 6.11T + 43T^{2} \)
47 \( 1 + (-2.57 - 4.45i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.76 + 1.59i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.42 + 7.65i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.06 + 2.34i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.187 + 0.325i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 13.9iT - 71T^{2} \)
73 \( 1 + (-1.13 - 0.655i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.462 + 0.800i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 10.8T + 83T^{2} \)
89 \( 1 + (2.35 + 4.07i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 15.3iT - 97T^{2} \)
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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

−10.19516990948932468335586933897, −9.719588503910907003443496033595, −8.452840604143213914262967660511, −7.74050293488283612203010577210, −6.82452312261315417398817546470, −6.29328639818541842838251861851, −5.51638120162324034537428466624, −3.96080557965295906263497187647, −2.92459040515875971150461402911, −1.82208532951240584920025090089, 0.19163130008744003535856727830, 1.91481842182348756396687360780, 2.53059256209267009933299199555, 4.22661345366896300314675545005, 5.11780489332639545735699973240, 5.95915973478480072162862899441, 6.90922106334175040336434190016, 8.156086932526969535476769634524, 8.788044572607981033305118558117, 9.294789940478996689681219118153

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