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Properties

Label 2-1134-21.5-c1-0-1
Degree $2$
Conductor $1134$
Sign $-0.487 - 0.873i$
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 + (−0.450 + 0.779i)5-s + (−2.62 + 0.296i)7-s − 0.999i·8-s + (0.779 − 0.450i)10-s + (2.70 − 1.56i)11-s + 2.29i·13-s + (2.42 + 1.05i)14-s + (−0.5 + 0.866i)16-s + (−2.57 − 4.46i)17-s + (2.38 + 1.37i)19-s − 0.900·20-s − 3.12·22-s + (1.48 + 0.857i)23-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (−0.201 + 0.348i)5-s + (−0.993 + 0.112i)7-s − 0.353i·8-s + (0.246 − 0.142i)10-s + (0.816 − 0.471i)11-s + 0.637i·13-s + (0.648 + 0.282i)14-s + (−0.125 + 0.216i)16-s + (−0.624 − 1.08i)17-s + (0.546 + 0.315i)19-s − 0.201·20-s − 0.666·22-s + (0.309 + 0.178i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.487 - 0.873i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.487 - 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1134\)    =    \(2 \cdot 3^{4} \cdot 7\)
Sign: $-0.487 - 0.873i$
Analytic conductor: \(9.05503\)
Root analytic conductor: \(3.00915\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1134} (971, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1134,\ (\ :1/2),\ -0.487 - 0.873i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4514621164\)
\(L(\frac12)\) \(\approx\) \(0.4514621164\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.62 - 0.296i)T \)
good5 \( 1 + (0.450 - 0.779i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.70 + 1.56i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 2.29iT - 13T^{2} \)
17 \( 1 + (2.57 + 4.46i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.38 - 1.37i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.48 - 0.857i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 2.14iT - 29T^{2} \)
31 \( 1 + (8.66 - 5.00i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.73 - 8.20i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 2.44T + 41T^{2} \)
43 \( 1 - 0.546T + 43T^{2} \)
47 \( 1 + (3.93 - 6.80i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (12.0 - 6.97i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.99 + 6.91i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.28 + 3.62i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.83 + 3.17i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 14.1iT - 71T^{2} \)
73 \( 1 + (10.9 - 6.30i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.27 + 5.67i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 0.368T + 83T^{2} \)
89 \( 1 + (-6.00 + 10.3i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 10.2iT - 97T^{2} \)
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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

−9.914662702914538651919864537865, −9.237508951818385332520091744723, −8.818917966731292105421392801078, −7.52081180920169424372336393636, −6.87153983100977465575511349073, −6.18051610141973259817373668441, −4.85316266270577264667079665877, −3.53970242466922934157482683804, −2.98678618246632249077430015427, −1.47033942331912610880930248871, 0.24540776084408152701218653372, 1.79792457816372044573422941093, 3.27582309885941703446726662618, 4.27379399277192123585824481640, 5.46719620434161461888540132327, 6.37489954718068710718038812227, 7.04565389702336503654575767765, 7.904404451021105557013965197105, 8.955132963420357943023836384549, 9.273291209158385722178918586140

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