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Properties

Label 2-1134-63.47-c1-0-24
Degree $2$
Conductor $1134$
Sign $0.572 + 0.819i$
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 + 4.18·5-s + (−1 − 2.44i)7-s − 0.999i·8-s + (3.62 − 2.09i)10-s + 3i·11-s + (2.12 − 1.22i)13-s + (−2.09 − 1.62i)14-s + (−0.5 − 0.866i)16-s + (0.507 + 0.878i)17-s + (−0.878 − 0.507i)19-s + (2.09 − 3.62i)20-s + (1.5 + 2.59i)22-s − 4.24i·23-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + 1.87·5-s + (−0.377 − 0.925i)7-s − 0.353i·8-s + (1.14 − 0.661i)10-s + 0.904i·11-s + (0.588 − 0.339i)13-s + (−0.558 − 0.433i)14-s + (−0.125 − 0.216i)16-s + (0.123 + 0.213i)17-s + (−0.201 − 0.116i)19-s + (0.467 − 0.809i)20-s + (0.319 + 0.553i)22-s − 0.884i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.572 + 0.819i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.572 + 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1134\)    =    \(2 \cdot 3^{4} \cdot 7\)
Sign: $0.572 + 0.819i$
Analytic conductor: \(9.05503\)
Root analytic conductor: \(3.00915\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1134} (593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1134,\ (\ :1/2),\ 0.572 + 0.819i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.123971986\)
\(L(\frac12)\) \(\approx\) \(3.123971986\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 + 2.44i)T \)
good5 \( 1 - 4.18T + 5T^{2} \)
11 \( 1 - 3iT - 11T^{2} \)
13 \( 1 + (-2.12 + 1.22i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.507 - 0.878i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.878 + 0.507i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 4.24iT - 23T^{2} \)
29 \( 1 + (-1.07 - 0.621i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.86 + 2.80i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.12 - 7.13i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.01 - 1.75i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.12 - 7.13i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.507 - 0.878i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.07 - 0.621i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.76 + 9.98i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.12 - 2.95i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5 + 8.66i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 10.2iT - 71T^{2} \)
73 \( 1 + (-7.24 + 4.18i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.62 - 9.73i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.58 - 2.74i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (5.19 - 9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.25 + 1.88i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.813146413029954933786360318611, −9.293214269978899224122840159997, −8.027720164257493590611496041135, −6.66717881265866255977767644515, −6.43898298842941712698971381757, −5.34548740381154390659051529798, −4.58034372291974166322524170102, −3.36838057950481315889611143220, −2.26105324077548530217071616699, −1.29068565394466565391515196819, 1.70272564066912518535171548070, 2.66541082644325467460553040412, 3.68441117320273958694575189038, 5.32364656981842164589624692428, 5.60691590456987997248887787908, 6.31380665447826333437114675797, 7.11353563438962215634978654644, 8.620335777498976335929305437619, 8.981266179773787573976375798355, 9.851562597015346146542486321944

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