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Properties

Label 2-1134-7.2-c1-0-11
Degree $2$
Conductor $1134$
Sign $0.605 - 0.795i$
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 + 3.46i)5-s + (0.5 − 2.59i)7-s + 0.999·8-s + (−1.99 − 3.46i)10-s + (−1 − 1.73i)11-s + 6·13-s + (2 + 1.73i)14-s + (−0.5 + 0.866i)16-s + (−1 − 1.73i)17-s + (2 − 3.46i)19-s + 3.99·20-s + 1.99·22-s + (0.5 − 0.866i)23-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.894 + 1.54i)5-s + (0.188 − 0.981i)7-s + 0.353·8-s + (−0.632 − 1.09i)10-s + (−0.301 − 0.522i)11-s + 1.66·13-s + (0.534 + 0.462i)14-s + (−0.125 + 0.216i)16-s + (−0.242 − 0.420i)17-s + (0.458 − 0.794i)19-s + 0.894·20-s + 0.426·22-s + (0.104 − 0.180i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1134\)    =    \(2 \cdot 3^{4} \cdot 7\)
Sign: $0.605 - 0.795i$
Analytic conductor: \(9.05503\)
Root analytic conductor: \(3.00915\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1134} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1134,\ (\ :1/2),\ 0.605 - 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.159154444\)
\(L(\frac12)\) \(\approx\) \(1.159154444\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 + 2.59i)T \)
good5 \( 1 + (2 - 3.46i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1 + 1.73i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 6T + 13T^{2} \)
17 \( 1 + (1 + 1.73i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2 + 3.46i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 4T + 29T^{2} \)
31 \( 1 + (-4.5 - 7.79i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4 - 6.92i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 3T + 41T^{2} \)
43 \( 1 - 2T + 43T^{2} \)
47 \( 1 + (-4.5 + 7.79i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6 - 10.3i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (2 + 3.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3 - 5.19i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7 - 12.1i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + T + 71T^{2} \)
73 \( 1 + (3.5 + 6.06i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.5 + 2.59i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 14T + 83T^{2} \)
89 \( 1 + (-1.5 + 2.59i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 10T + 97T^{2} \)
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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

−10.25737184758959382265477222857, −8.820794616823318769592956929817, −8.202532682907544812943221800991, −7.30154233033532838532509207918, −6.82105643504306638255227336053, −6.09185949045987581664774918540, −4.71096349616497830434552604123, −3.69568954276267644779935056547, −2.92101296086047482613191160163, −0.876661825813929335953256138876, 0.902563278206519705593890782152, 2.02650303208289799089492690495, 3.56859320293782943933975143561, 4.32458217783164232559932871444, 5.27086118095282339132565153868, 6.15213765450410592060512003283, 7.77924231238583808540504852395, 8.181077312189295711908508927066, 8.916149983533698404039515493193, 9.420141844727443610376752719788

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