↑

Properties

Label 2-1134-21.17-c1-0-18
Degree $2$
Conductor $1134$
Sign $-0.540 + 0.841i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + (−1.58 − 2.74i)5-s + (−0.457 + 2.60i)7-s + 0.999i·8-s + (2.74 + 1.58i)10-s + (0.578 + 0.333i)11-s + 0.359i·13-s + (−0.906 − 2.48i)14-s + (−0.5 − 0.866i)16-s + (2.03 − 3.51i)17-s + (0.692 − 0.399i)19-s − 3.16·20-s − 0.667·22-s + (−1.55 + 0.899i)23-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + (−0.708 − 1.22i)5-s + (−0.172 + 0.984i)7-s + 0.353i·8-s + (0.868 + 0.501i)10-s + (0.174 + 0.100i)11-s + 0.0997i·13-s + (−0.242 − 0.664i)14-s + (−0.125 − 0.216i)16-s + (0.492 − 0.852i)17-s + (0.158 − 0.0917i)19-s − 0.708·20-s − 0.142·22-s + (−0.324 + 0.187i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.540 + 0.841i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.540 + 0.841i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1134\)    =    \(2 \cdot 3^{4} \cdot 7\)
Sign: $-0.540 + 0.841i$
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: \((2,\ 1134,\ (\ :1/2),\ -0.540 + 0.841i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4900724346\)
\(L(\frac12)\) \(\approx\) \(0.4900724346\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.457 - 2.60i)T \)
good5 \( 1 + (1.58 + 2.74i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.578 - 0.333i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 0.359iT - 13T^{2} \)
17 \( 1 + (-2.03 + 3.51i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.692 + 0.399i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.55 - 0.899i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 7.25iT - 29T^{2} \)
31 \( 1 + (-5.80 - 3.35i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.94 + 6.83i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 1.92T + 41T^{2} \)
43 \( 1 + 11.3T + 43T^{2} \)
47 \( 1 + (4.08 + 7.08i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (11.5 + 6.67i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (7.00 - 12.1i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (10.4 - 6.06i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.76 + 9.98i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 1.98iT - 71T^{2} \)
73 \( 1 + (-7.99 - 4.61i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.70 + 2.96i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 10.0T + 83T^{2} \)
89 \( 1 + (-1.29 - 2.24i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 18.3iT - 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.431706899545951021711982640878, −8.581404350011109339108385368255, −8.173486913153684706046516228334, −7.22240691920514826590575507165, −6.17527658569705559191119567723, −5.25629644883077334178016202560, −4.55715892191840060797949905239, −3.18023477315552863453065302812, −1.73421500683073069979641685848, −0.27300847630528677524646434273, 1.45778123835550789813856535210, 3.11879041954272881431241592488, 3.53209790080022799190158351296, 4.69263235364764977378113060611, 6.37522697286857289964269499321, 6.80231107142598224629994120701, 7.82512656623052067235127570689, 8.177846442839446108819615173377, 9.514518171443681600165125843738, 10.26142130084113743371488450086

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