↑

Properties

Label 2-1134-63.16-c1-0-4
Degree $2$
Conductor $1134$
Sign $-0.272 - 0.962i$
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  + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (1 + 2.44i)7-s + 0.999·8-s − 4.24·11-s + (1.12 + 1.94i)13-s + (1.62 − 2.09i)14-s + (−0.5 − 0.866i)16-s + (1.12 − 1.94i)19-s + (2.12 + 3.67i)22-s − 1.24·23-s − 5·25-s + (1.12 − 1.94i)26-s + (−2.62 − 0.358i)28-s + (−2.12 + 3.67i)29-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.377 + 0.925i)7-s + 0.353·8-s − 1.27·11-s + (0.310 + 0.538i)13-s + (0.433 − 0.558i)14-s + (−0.125 − 0.216i)16-s + (0.257 − 0.445i)19-s + (0.452 + 0.783i)22-s − 0.259·23-s − 25-s + (0.219 − 0.380i)26-s + (−0.495 − 0.0677i)28-s + (−0.393 + 0.682i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.272 - 0.962i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.272 - 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1134\)    =    \(2 \cdot 3^{4} \cdot 7\)
Sign: $-0.272 - 0.962i$
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: \((2,\ 1134,\ (\ :1/2),\ -0.272 - 0.962i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5986501720\)
\(L(\frac12)\) \(\approx\) \(0.5986501720\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 - 2.44i)T \)
good5 \( 1 + 5T^{2} \)
11 \( 1 + 4.24T + 11T^{2} \)
13 \( 1 + (-1.12 - 1.94i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.12 + 1.94i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 1.24T + 23T^{2} \)
29 \( 1 + (2.12 - 3.67i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.62 - 8.00i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2 + 3.46i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (5.74 + 9.94i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.24 - 9.08i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.37 + 4.11i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.12 - 3.67i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.12 - 1.94i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.121 - 0.210i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 1.24T + 71T^{2} \)
73 \( 1 + (-3.5 - 6.06i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.378 + 0.655i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (8.12 - 14.0i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (5.74 - 9.94i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.24 - 3.88i)T + (-48.5 - 84.0i)T^{2} \)
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

−10.08996194822123519337035733947, −9.207921705888677276065889635374, −8.564021270245922945802324161932, −7.81634703191489328730060482672, −6.87708769315606026957582710087, −5.58627063716124900968811832027, −4.99426523144062054147704524639, −3.70265121400582511755735012562, −2.62728265712285782351786773730, −1.70588502117923769650824382184, 0.28285688505091177593490776333, 1.87590327727779244153885706450, 3.41653563218559081533233980822, 4.48816177799705843246656208368, 5.43678604752902672511834879266, 6.18985479641183182138583620257, 7.38176049022753632150441278065, 7.84914065510461504422038844665, 8.445517574699154157201712507166, 9.821259954927868971610520879266

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