Properties

Label 2-1134-63.4-c1-0-2
Degree $2$
Conductor $1134$
Sign $-0.975 - 0.220i$
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 + (2.5 + 0.866i)7-s + 0.999·8-s − 6·11-s + (−2.5 + 4.33i)13-s + (−2 + 1.73i)14-s + (−0.5 + 0.866i)16-s + (3 − 5.19i)17-s + (2 + 3.46i)19-s + (3 − 5.19i)22-s − 6·23-s − 5·25-s + (−2.5 − 4.33i)26-s + (−0.499 − 2.59i)28-s + (3 + 5.19i)29-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.944 + 0.327i)7-s + 0.353·8-s − 1.80·11-s + (−0.693 + 1.20i)13-s + (−0.534 + 0.462i)14-s + (−0.125 + 0.216i)16-s + (0.727 − 1.26i)17-s + (0.458 + 0.794i)19-s + (0.639 − 1.10i)22-s − 1.25·23-s − 25-s + (−0.490 − 0.849i)26-s + (−0.0944 − 0.490i)28-s + (0.557 + 0.964i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.975 - 0.220i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.975 - 0.220i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1134\)    =    \(2 \cdot 3^{4} \cdot 7\)
Sign: $-0.975 - 0.220i$
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: \((2,\ 1134,\ (\ :1/2),\ -0.975 - 0.220i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6533077557\)
\(L(\frac12)\) \(\approx\) \(0.6533077557\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.5 - 0.866i)T \)
good5 \( 1 + 5T^{2} \)
11 \( 1 + 6T + 11T^{2} \)
13 \( 1 + (2.5 - 4.33i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3 + 5.19i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2 - 3.46i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 6T + 23T^{2} \)
29 \( 1 + (-3 - 5.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (3 - 5.19i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (3 - 5.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3 - 5.19i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3 + 5.19i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 + (1 - 1.73i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3 - 5.19i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (8.5 + 14.7i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.958551526223595855206332710371, −9.460546799018121678264758689516, −8.242833893349697339424528780741, −7.83662499241586706486499338570, −7.13317487854057583238249128023, −5.88254054680190165166433383901, −5.16058731088957680612920971133, −4.48867585967093749663461055931, −2.84055454104471899862961679408, −1.68819024478826595079496178551, 0.30682151304072432538144532853, 1.93404512240286852963362138812, 2.87934063903172108645609724631, 4.07328144093397480858179774440, 5.12998615067236608923242002615, 5.76931195722910655549769074273, 7.40282711257185787720197550572, 8.008131415707988281563912985988, 8.302615108749417880550665931833, 9.800126945367223941535730406455

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