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Properties

Label 2-1134-21.20-c1-0-18
Degree $2$
Conductor $1134$
Sign $0.202 - 0.979i$
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  + i·2-s − 4-s + 3.79·5-s + (2.59 + 0.536i)7-s − i·8-s + 3.79i·10-s + 1.61i·11-s + 5.11i·13-s + (−0.536 + 2.59i)14-s + 16-s − 2.33·17-s − 4.24i·19-s − 3.79·20-s − 1.61·22-s + 2.62i·23-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + 1.69·5-s + (0.979 + 0.202i)7-s − 0.353i·8-s + 1.20i·10-s + 0.486i·11-s + 1.41i·13-s + (−0.143 + 0.692i)14-s + 0.250·16-s − 0.565·17-s − 0.974i·19-s − 0.849·20-s − 0.344·22-s + 0.548i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.202 - 0.979i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.202 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1134\)    =    \(2 \cdot 3^{4} \cdot 7\)
Sign: $0.202 - 0.979i$
Analytic conductor: \(9.05503\)
Root analytic conductor: \(3.00915\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1134} (1133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1134,\ (\ :1/2),\ 0.202 - 0.979i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.285726046\)
\(L(\frac12)\) \(\approx\) \(2.285726046\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 \)
7 \( 1 + (-2.59 - 0.536i)T \)
good5 \( 1 - 3.79T + 5T^{2} \)
11 \( 1 - 1.61iT - 11T^{2} \)
13 \( 1 - 5.11iT - 13T^{2} \)
17 \( 1 + 2.33T + 17T^{2} \)
19 \( 1 + 4.24iT - 19T^{2} \)
23 \( 1 - 2.62iT - 23T^{2} \)
29 \( 1 - 4.01iT - 29T^{2} \)
31 \( 1 + 7.98iT - 31T^{2} \)
37 \( 1 + 5.74T + 37T^{2} \)
41 \( 1 + 8.51T + 41T^{2} \)
43 \( 1 - 9.25T + 43T^{2} \)
47 \( 1 - 9.69T + 47T^{2} \)
53 \( 1 + 2.19iT - 53T^{2} \)
59 \( 1 - 1.46T + 59T^{2} \)
61 \( 1 + 14.4iT - 61T^{2} \)
67 \( 1 + 5.78T + 67T^{2} \)
71 \( 1 - 15.6iT - 71T^{2} \)
73 \( 1 + 7.65iT - 73T^{2} \)
79 \( 1 + 1.64T + 79T^{2} \)
83 \( 1 + 3.02T + 83T^{2} \)
89 \( 1 + 17.5T + 89T^{2} \)
97 \( 1 - 10.3iT - 97T^{2} \)
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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

−9.659989781981640366307619398208, −9.180381923300522742430104476379, −8.541709304947299867814589257591, −7.25718429473666848403334843505, −6.69262357452328827579018909153, −5.74505715828615385439523129054, −5.04941637572660581454144179757, −4.24085299878464603119886898691, −2.38023550537410588774104072499, −1.61075660822856267618384849614, 1.13766801243606081862597922595, 2.11390592013887237831910781401, 3.08740704980418101166653291575, 4.44999840249090634897792575276, 5.48889904848136657924777766800, 5.86309962023662655472964239705, 7.16757433626379697418469969729, 8.359392422414851646690919068566, 8.817012548612519631717257454294, 9.916162863987691167298752469265

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