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Properties

Label 2-1134-63.47-c1-0-16
Degree $2$
Conductor $1134$
Sign $0.993 + 0.110i$
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 + 1.73·5-s + (0.5 + 2.59i)7-s − 0.999i·8-s + (1.49 − 0.866i)10-s + (1.73 + 2i)14-s + (−0.5 − 0.866i)16-s + (1.73 + 3i)17-s + (6 + 3.46i)19-s + (0.866 − 1.49i)20-s − 6i·23-s − 2.00·25-s + (2.5 + 0.866i)28-s + (7.79 + 4.5i)29-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + 0.774·5-s + (0.188 + 0.981i)7-s − 0.353i·8-s + (0.474 − 0.273i)10-s + (0.462 + 0.534i)14-s + (−0.125 − 0.216i)16-s + (0.420 + 0.727i)17-s + (1.37 + 0.794i)19-s + (0.193 − 0.335i)20-s − 1.25i·23-s − 0.400·25-s + (0.472 + 0.163i)28-s + (1.44 + 0.835i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.110i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 + 0.110i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1134\)    =    \(2 \cdot 3^{4} \cdot 7\)
Sign: $0.993 + 0.110i$
Analytic conductor: \(9.05503\)
Root analytic conductor: \(3.00915\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1134} (593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1134,\ (\ :1/2),\ 0.993 + 0.110i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.795533612\)
\(L(\frac12)\) \(\approx\) \(2.795533612\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 2.59i)T \)
good5 \( 1 - 1.73T + 5T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.73 - 3i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-6 - 3.46i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 6iT - 23T^{2} \)
29 \( 1 + (-7.79 - 4.5i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (3 + 1.73i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (2 - 3.46i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.73 + 3i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4 + 6.92i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.73 - 3i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.59 + 1.5i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6.06 + 10.5i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3 - 1.73i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (7 - 12.1i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 6iT - 71T^{2} \)
73 \( 1 + (10.5 - 6.06i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.5 + 9.52i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-8.66 + 15i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-5.19 + 9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-6 - 3.46i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.05537507804210476656021406734, −9.035041131724654924214461402758, −8.323848127501314230610520633866, −7.16179633061442411212625238381, −6.05093445852663352246468487451, −5.62489608776219843822928289588, −4.72256886851633731956872472458, −3.46626130918610480432785189005, −2.47879456273766357897552518537, −1.46923222089454465193784103114, 1.20161706674668295012847074045, 2.69575563735575496139580642784, 3.70296973349955313367848209947, 4.80682614768122444897990905927, 5.48816446748388282156307967318, 6.43833051108593193571865909884, 7.35176069053409261721157146276, 7.82031477277873499258537374449, 9.165489438207780886426561409362, 9.780379990646521993993058757304

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