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Properties

Label 2-1134-63.4-c1-0-9
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: $\chi_{1134} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1134,\ (\ :1/2),\ -0.478 - 0.878i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.240481141\)
\(L(\frac12)\) \(\approx\) \(1.240481141\)
\(L(\frac{3}{2})\) 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

−9.630138290111861153518709932041, −9.143323536207276450319087811751, −8.703541368349698672545021836357, −7.50585241320656795703460578950, −6.74528207170365043105674933769, −6.12068317659607657474113509753, −4.89317752426159709548107826351, −4.32961349028772210819817524699, −2.67733514348159704630074558096, −1.48181134383721448973533810843, 0.65444610601915169787308640489, 1.89109963040355958048375276337, 3.29448670201366474105107421368, 4.07976776609271030939225186093, 5.07283158930996307884835932048, 6.25057427413966247214036269429, 7.30292483477505406435780637613, 7.940917950934701129877219398840, 8.764304303577574992128508363504, 9.828640915358322711951825141652

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