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Properties

Label 2-1134-63.4-c1-0-29
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

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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: $\chi_{1134} (109, \cdot )$
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 \( 1 + (2.24 + 3.88i)T + (-48.5 + 84.0i)T^{2} \)
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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

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

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