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Properties

Label 2-189-1.1-c1-0-2
Degree 22
Conductor 189189
Sign 11
Analytic cond. 1.509171.50917
Root an. cond. 1.228481.22848
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s + 3·5-s + 7-s + 6·11-s − 4·13-s + 4·16-s + 3·17-s + 2·19-s − 6·20-s − 6·23-s + 4·25-s − 2·28-s − 6·29-s − 4·31-s + 3·35-s − 7·37-s − 3·41-s − 43-s − 12·44-s + 9·47-s + 49-s + 8·52-s − 6·53-s + 18·55-s + 9·59-s − 10·61-s − 8·64-s + ⋯
L(s)  = 1  − 4-s + 1.34·5-s + 0.377·7-s + 1.80·11-s − 1.10·13-s + 16-s + 0.727·17-s + 0.458·19-s − 1.34·20-s − 1.25·23-s + 4/5·25-s − 0.377·28-s − 1.11·29-s − 0.718·31-s + 0.507·35-s − 1.15·37-s − 0.468·41-s − 0.152·43-s − 1.80·44-s + 1.31·47-s + 1/7·49-s + 1.10·52-s − 0.824·53-s + 2.42·55-s + 1.17·59-s − 1.28·61-s − 64-s + ⋯

Functional equation

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

Invariants

Degree: 22
Conductor: 189189    =    33⋅73^{3} \cdot 7
Sign: 11
Analytic conductor: 1.509171.50917
Root analytic conductor: 1.228481.22848
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 189, ( :1/2), 1)(2,\ 189,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.2453440161.245344016
L(12)L(\frac12) ≈\approx 1.2453440161.245344016
L(32)L(\frac{3}{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)Isogeny Class over Fp\mathbf{F}_p
bad3 1 1
7 1−T 1 - T
good2 1+pT2 1 + p T^{2} 1.2.a
5 1−3T+pT2 1 - 3 T + p T^{2} 1.5.ad
11 1−6T+pT2 1 - 6 T + p T^{2} 1.11.ag
13 1+4T+pT2 1 + 4 T + p T^{2} 1.13.e
17 1−3T+pT2 1 - 3 T + p T^{2} 1.17.ad
19 1−2T+pT2 1 - 2 T + p T^{2} 1.19.ac
23 1+6T+pT2 1 + 6 T + p T^{2} 1.23.g
29 1+6T+pT2 1 + 6 T + p T^{2} 1.29.g
31 1+4T+pT2 1 + 4 T + p T^{2} 1.31.e
37 1+7T+pT2 1 + 7 T + p T^{2} 1.37.h
41 1+3T+pT2 1 + 3 T + p T^{2} 1.41.d
43 1+T+pT2 1 + T + p T^{2} 1.43.b
47 1−9T+pT2 1 - 9 T + p T^{2} 1.47.aj
53 1+6T+pT2 1 + 6 T + p T^{2} 1.53.g
59 1−9T+pT2 1 - 9 T + p T^{2} 1.59.aj
61 1+10T+pT2 1 + 10 T + p T^{2} 1.61.k
67 1+4T+pT2 1 + 4 T + p T^{2} 1.67.e
71 1+pT2 1 + p T^{2} 1.71.a
73 1−2T+pT2 1 - 2 T + p T^{2} 1.73.ac
79 1+T+pT2 1 + T + p T^{2} 1.79.b
83 1−3T+pT2 1 - 3 T + p T^{2} 1.83.ad
89 1−6T+pT2 1 - 6 T + p T^{2} 1.89.ag
97 1+10T+pT2 1 + 10 T + p T^{2} 1.97.k
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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

−12.58433731583081615557168762807, −11.80712467067776826965075311823, −10.22213033820361484191010028356, −9.540146555327995507333228878724, −8.905165354580512836683090859524, −7.46477788862935127887709071190, −6.04902466517769458594559190180, −5.13832633779419862585354738180, −3.79566092998735399945677628017, −1.69414023663856774565794959548, 1.69414023663856774565794959548, 3.79566092998735399945677628017, 5.13832633779419862585354738180, 6.04902466517769458594559190180, 7.46477788862935127887709071190, 8.905165354580512836683090859524, 9.540146555327995507333228878724, 10.22213033820361484191010028356, 11.80712467067776826965075311823, 12.58433731583081615557168762807

Graph of the ZZ-function along the critical line