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Properties

Label 2-1134-63.16-c1-0-31
Degree $2$
Conductor $1134$
Sign $-0.478 + 0.878i$
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 + (1 − 2.44i)7-s − 0.999·8-s − 4.24·11-s + (−3.12 − 5.40i)13-s + (2.62 − 0.358i)14-s + (−0.5 − 0.866i)16-s + (−3.12 + 5.40i)19-s + (−2.12 − 3.67i)22-s − 7.24·23-s − 5·25-s + (3.12 − 5.40i)26-s + (1.62 + 2.09i)28-s + (−2.12 + 3.67i)29-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.377 − 0.925i)7-s − 0.353·8-s − 1.27·11-s + (−0.865 − 1.49i)13-s + (0.700 − 0.0958i)14-s + (−0.125 − 0.216i)16-s + (−0.716 + 1.24i)19-s + (−0.452 − 0.783i)22-s − 1.51·23-s − 25-s + (0.612 − 1.06i)26-s + (0.306 + 0.395i)28-s + (−0.393 + 0.682i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.478 + 0.878i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.478 + 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1134\)    =    \(2 \cdot 3^{4} \cdot 7\)
Sign: $-0.478 + 0.878i$
Analytic conductor: \(9.05503\)
Root analytic conductor: \(3.00915\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1134} (541, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1134,\ (\ :1/2),\ -0.478 + 0.878i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4390222941\)
\(L(\frac12)\) \(\approx\) \(0.4390222941\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 + 2.44i)T \)
good5 \( 1 + 5T^{2} \)
11 \( 1 + 4.24T + 11T^{2} \)
13 \( 1 + (3.12 + 5.40i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.12 - 5.40i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 7.24T + 23T^{2} \)
29 \( 1 + (2.12 - 3.67i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.378 - 0.655i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2 + 3.46i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.74 + 4.75i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.24 + 5.61i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-6.62 - 11.4i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.12 - 3.67i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.12 + 5.40i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.12 + 7.13i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 7.24T + 71T^{2} \)
73 \( 1 + (-3.5 - 6.06i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.62 + 8.00i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.87 + 6.71i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (2.74 - 4.75i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-6.24 + 10.8i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.687815026382836708621725064390, −8.265230992350317654404363148857, −7.80386540469825176395415404218, −7.32032748487739464158813391828, −5.94486942680931660687090481416, −5.41637994831004954911996578516, −4.37829126499696644239218707552, −3.48959090219732976851824148163, −2.18343041074378861882633693891, −0.15300582918808224398793566783, 2.12626020948935320056648264552, 2.46965044179967238599455400720, 4.06367321453607835245713154791, 4.83635718095075239160703070015, 5.66182230502224107988673588182, 6.60674306128165652978239806337, 7.74241213685127558606005256032, 8.535631754227283701317878829664, 9.466958413324097558301955508871, 10.01749103537218178500653444127

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