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Properties

Label 2-189-1.1-c3-0-7
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  + 1.26·2-s − 6.39·4-s − 3.92·5-s + 7·7-s − 18.2·8-s − 4.98·10-s + 51.7·11-s + 67.5·13-s + 8.87·14-s + 27.9·16-s + 63.4·17-s − 71.9·19-s + 25.1·20-s + 65.5·22-s + 147.·23-s − 109.·25-s + 85.6·26-s − 44.7·28-s + 117.·29-s + 54.7·31-s + 181.·32-s + 80.5·34-s − 27.4·35-s + 9.70·37-s − 91.1·38-s + 71.6·40-s − 236.·41-s + ⋯
L(s)  = 1  + 0.448·2-s − 0.799·4-s − 0.351·5-s + 0.377·7-s − 0.806·8-s − 0.157·10-s + 1.41·11-s + 1.44·13-s + 0.169·14-s + 0.437·16-s + 0.905·17-s − 0.868·19-s + 0.280·20-s + 0.635·22-s + 1.33·23-s − 0.876·25-s + 0.646·26-s − 0.302·28-s + 0.753·29-s + 0.316·31-s + 1.00·32-s + 0.406·34-s − 0.132·35-s + 0.0431·37-s − 0.389·38-s + 0.283·40-s − 0.900·41-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 1.8900092731.890009273
L(12)L(\frac12) ≈\approx 1.8900092731.890009273
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 1−1.26T+8T2 1 - 1.26T + 8T^{2}
5 1+3.92T+125T2 1 + 3.92T + 125T^{2}
11 1−51.7T+1.33e3T2 1 - 51.7T + 1.33e3T^{2}
13 1−67.5T+2.19e3T2 1 - 67.5T + 2.19e3T^{2}
17 1−63.4T+4.91e3T2 1 - 63.4T + 4.91e3T^{2}
19 1+71.9T+6.85e3T2 1 + 71.9T + 6.85e3T^{2}
23 1−147.T+1.21e4T2 1 - 147.T + 1.21e4T^{2}
29 1−117.T+2.43e4T2 1 - 117.T + 2.43e4T^{2}
31 1−54.7T+2.97e4T2 1 - 54.7T + 2.97e4T^{2}
37 1−9.70T+5.06e4T2 1 - 9.70T + 5.06e4T^{2}
41 1+236.T+6.89e4T2 1 + 236.T + 6.89e4T^{2}
43 1+489.T+7.95e4T2 1 + 489.T + 7.95e4T^{2}
47 1−613.T+1.03e5T2 1 - 613.T + 1.03e5T^{2}
53 1−316.T+1.48e5T2 1 - 316.T + 1.48e5T^{2}
59 1−2.43T+2.05e5T2 1 - 2.43T + 2.05e5T^{2}
61 1−482.T+2.26e5T2 1 - 482.T + 2.26e5T^{2}
67 1−646.T+3.00e5T2 1 - 646.T + 3.00e5T^{2}
71 1+459.T+3.57e5T2 1 + 459.T + 3.57e5T^{2}
73 1−137.T+3.89e5T2 1 - 137.T + 3.89e5T^{2}
79 1+816.T+4.93e5T2 1 + 816.T + 4.93e5T^{2}
83 1+1.00e3T+5.71e5T2 1 + 1.00e3T + 5.71e5T^{2}
89 1+255.T+7.04e5T2 1 + 255.T + 7.04e5T^{2}
97 1+62.9T+9.12e5T2 1 + 62.9T + 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

−12.11050418120095162927090243564, −11.38877843987926079005848866597, −10.11273323659505284049996330443, −8.883230783940692852695745409819, −8.342258761951479821104636953239, −6.73961172359149170084649142515, −5.62749816620686311188033767704, −4.30097411355325161304113297813, −3.48890578831053268065585151651, −1.10424273914690404018298632997, 1.10424273914690404018298632997, 3.48890578831053268065585151651, 4.30097411355325161304113297813, 5.62749816620686311188033767704, 6.73961172359149170084649142515, 8.342258761951479821104636953239, 8.883230783940692852695745409819, 10.11273323659505284049996330443, 11.38877843987926079005848866597, 12.11050418120095162927090243564

Graph of the ZZ-function along the critical line