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Properties

Label 2-189-1.1-c3-0-17
Degree 22
Conductor 189189
Sign 11
Analytic cond. 11.151311.1513
Root an. cond. 3.339363.33936
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.73·2-s + 14.3·4-s + 9.92·5-s + 7·7-s + 30.2·8-s + 46.9·10-s − 3.71·11-s − 15.5·13-s + 33.1·14-s + 28.0·16-s − 33.4·17-s + 135.·19-s + 142.·20-s − 17.5·22-s − 87.7·23-s − 26.4·25-s − 73.6·26-s + 100.·28-s + 242.·29-s − 194.·31-s − 109.·32-s − 158.·34-s + 69.4·35-s − 239.·37-s + 643.·38-s + 300.·40-s + 470.·41-s + ⋯
L(s)  = 1  + 1.67·2-s + 1.79·4-s + 0.888·5-s + 0.377·7-s + 1.33·8-s + 1.48·10-s − 0.101·11-s − 0.332·13-s + 0.632·14-s + 0.437·16-s − 0.477·17-s + 1.64·19-s + 1.59·20-s − 0.170·22-s − 0.795·23-s − 0.211·25-s − 0.555·26-s + 0.679·28-s + 1.55·29-s − 1.12·31-s − 0.604·32-s − 0.799·34-s + 0.335·35-s − 1.06·37-s + 2.74·38-s + 1.18·40-s + 1.79·41-s + ⋯

Functional equation

Λ(s)=(189s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(189s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 189189    =    33⋅73^{3} \cdot 7
Sign: 11
Analytic conductor: 11.151311.1513
Root analytic conductor: 3.339363.33936
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 189, ( :3/2), 1)(2,\ 189,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 4.9612052304.961205230
L(12)L(\frac12) ≈\approx 4.9612052304.961205230
L(52)L(\frac{5}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad3 1 1
7 1−7T 1 - 7T
good2 1−4.73T+8T2 1 - 4.73T + 8T^{2}
5 1−9.92T+125T2 1 - 9.92T + 125T^{2}
11 1+3.71T+1.33e3T2 1 + 3.71T + 1.33e3T^{2}
13 1+15.5T+2.19e3T2 1 + 15.5T + 2.19e3T^{2}
17 1+33.4T+4.91e3T2 1 + 33.4T + 4.91e3T^{2}
19 1−135.T+6.85e3T2 1 - 135.T + 6.85e3T^{2}
23 1+87.7T+1.21e4T2 1 + 87.7T + 1.21e4T^{2}
29 1−242.T+2.43e4T2 1 - 242.T + 2.43e4T^{2}
31 1+194.T+2.97e4T2 1 + 194.T + 2.97e4T^{2}
37 1+239.T+5.06e4T2 1 + 239.T + 5.06e4T^{2}
41 1−470.T+6.89e4T2 1 - 470.T + 6.89e4T^{2}
43 1+448.T+7.95e4T2 1 + 448.T + 7.95e4T^{2}
47 1−4.15T+1.03e5T2 1 - 4.15T + 1.03e5T^{2}
53 1+736.T+1.48e5T2 1 + 736.T + 1.48e5T^{2}
59 1−279.T+2.05e5T2 1 - 279.T + 2.05e5T^{2}
61 1+514.T+2.26e5T2 1 + 514.T + 2.26e5T^{2}
67 1+102.T+3.00e5T2 1 + 102.T + 3.00e5T^{2}
71 1+44.1T+3.57e5T2 1 + 44.1T + 3.57e5T^{2}
73 1+901.T+3.89e5T2 1 + 901.T + 3.89e5T^{2}
79 1−1.05e3T+4.93e5T2 1 - 1.05e3T + 4.93e5T^{2}
83 1−487.T+5.71e5T2 1 - 487.T + 5.71e5T^{2}
89 1−963.T+7.04e5T2 1 - 963.T + 7.04e5T^{2}
97 1−726.T+9.12e5T2 1 - 726.T + 9.12e5T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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

−12.27632987659299655646781178799, −11.53596273464521670832837473914, −10.39527046502980608642461348633, −9.265470538002553362706371216565, −7.64659685835112797103145769696, −6.44846109697459173396428217355, −5.51343900559437254962157215947, −4.66444943384070729710683771265, −3.22541706504688024702033440526, −1.92515513340998566052728157358, 1.92515513340998566052728157358, 3.22541706504688024702033440526, 4.66444943384070729710683771265, 5.51343900559437254962157215947, 6.44846109697459173396428217355, 7.64659685835112797103145769696, 9.265470538002553362706371216565, 10.39527046502980608642461348633, 11.53596273464521670832837473914, 12.27632987659299655646781178799

Graph of the ZZ-function along the critical line