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Properties

Label 2-1134-63.59-c1-0-23
Degree $2$
Conductor $1134$
Sign $0.592 + 0.805i$
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 + 3.46·5-s + (0.5 − 2.59i)7-s − 0.999i·8-s + (−2.99 − 1.73i)10-s + (4.5 + 2.59i)13-s + (−1.73 + 2i)14-s + (−0.5 + 0.866i)16-s + (3.46 − 6i)17-s + (−3 + 1.73i)19-s + (1.73 + 2.99i)20-s − 6i·23-s + 6.99·25-s + (−2.59 − 4.5i)26-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + 1.54·5-s + (0.188 − 0.981i)7-s − 0.353i·8-s + (−0.948 − 0.547i)10-s + (1.24 + 0.720i)13-s + (−0.462 + 0.534i)14-s + (−0.125 + 0.216i)16-s + (0.840 − 1.45i)17-s + (−0.688 + 0.397i)19-s + (0.387 + 0.670i)20-s − 1.25i·23-s + 1.39·25-s + (−0.509 − 0.882i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.592 + 0.805i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.592 + 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1134\)    =    \(2 \cdot 3^{4} \cdot 7\)
Sign: $0.592 + 0.805i$
Analytic conductor: \(9.05503\)
Root analytic conductor: \(3.00915\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1134} (1025, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1134,\ (\ :1/2),\ 0.592 + 0.805i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.729096041\)
\(L(\frac12)\) \(\approx\) \(1.729096041\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.46T + 5T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 + (-4.5 - 2.59i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3.46 + 6i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3 - 1.73i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 6iT - 23T^{2} \)
29 \( 1 + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (7.5 - 4.33i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.5 - 4.33i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.46 - 6i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.46 + 6i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5.19 - 3i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.46 + 6i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.5 - 0.866i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.5 - 11.2i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 6iT - 71T^{2} \)
73 \( 1 + (6 + 3.46i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.5 + 6.06i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.73 - 3i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (5.19 + 9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.5 + 0.866i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.804988358516987340047160300069, −9.008015514258715032469789186747, −8.297847055229863104247472385351, −7.09903971888349281669683948799, −6.51955314544245385921659048027, −5.53677288571378977613681156721, −4.41552384747140738778809795764, −3.23016112147725485361010732927, −2.00089476286532776473425344788, −1.06365222997550164639941068247, 1.44812717591421672896527115259, 2.24218084898688641667300516939, 3.62117535414110725256865446361, 5.36135529618011675339660585359, 5.82382758513697999875299612455, 6.28899952274190684794586431328, 7.61155411884312763931735772365, 8.493169866876115126537891369337, 9.097600050843224233605405908377, 9.766920868729669500507721518402

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