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Properties

Label 2-184-1.1-c1-0-1
Degree 22
Conductor 184184
Sign 11
Analytic cond. 1.469241.46924
Root an. cond. 1.212121.21212
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  + 4·7-s − 3·9-s + 6·11-s − 2·13-s + 6·17-s − 6·19-s + 23-s − 5·25-s − 6·29-s − 8·37-s + 6·41-s − 2·43-s − 8·47-s + 9·49-s − 8·53-s + 4·59-s − 4·61-s − 12·63-s + 2·67-s − 8·71-s + 6·73-s + 24·77-s + 12·79-s + 9·81-s + 10·83-s + 10·89-s − 8·91-s + ⋯
L(s)  = 1  + 1.51·7-s − 9-s + 1.80·11-s − 0.554·13-s + 1.45·17-s − 1.37·19-s + 0.208·23-s − 25-s − 1.11·29-s − 1.31·37-s + 0.937·41-s − 0.304·43-s − 1.16·47-s + 9/7·49-s − 1.09·53-s + 0.520·59-s − 0.512·61-s − 1.51·63-s + 0.244·67-s − 0.949·71-s + 0.702·73-s + 2.73·77-s + 1.35·79-s + 81-s + 1.09·83-s + 1.05·89-s − 0.838·91-s + ⋯

Functional equation

Λ(s)=(184s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(184s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 184184    =    23⋅232^{3} \cdot 23
Sign: 11
Analytic conductor: 1.469241.46924
Root analytic conductor: 1.212121.21212
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 184, ( :1/2), 1)(2,\ 184,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.2989747961.298974796
L(12)L(\frac12) ≈\approx 1.2989747961.298974796
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
bad2 1 1
23 1−T 1 - T
good3 1+pT2 1 + p T^{2} 1.3.a
5 1+pT2 1 + p T^{2} 1.5.a
7 1−4T+pT2 1 - 4 T + p T^{2} 1.7.ae
11 1−6T+pT2 1 - 6 T + p T^{2} 1.11.ag
13 1+2T+pT2 1 + 2 T + p T^{2} 1.13.c
17 1−6T+pT2 1 - 6 T + p T^{2} 1.17.ag
19 1+6T+pT2 1 + 6 T + p T^{2} 1.19.g
29 1+6T+pT2 1 + 6 T + p T^{2} 1.29.g
31 1+pT2 1 + p T^{2} 1.31.a
37 1+8T+pT2 1 + 8 T + p T^{2} 1.37.i
41 1−6T+pT2 1 - 6 T + p T^{2} 1.41.ag
43 1+2T+pT2 1 + 2 T + p T^{2} 1.43.c
47 1+8T+pT2 1 + 8 T + p T^{2} 1.47.i
53 1+8T+pT2 1 + 8 T + p T^{2} 1.53.i
59 1−4T+pT2 1 - 4 T + p T^{2} 1.59.ae
61 1+4T+pT2 1 + 4 T + p T^{2} 1.61.e
67 1−2T+pT2 1 - 2 T + p T^{2} 1.67.ac
71 1+8T+pT2 1 + 8 T + p T^{2} 1.71.i
73 1−6T+pT2 1 - 6 T + p T^{2} 1.73.ag
79 1−12T+pT2 1 - 12 T + p T^{2} 1.79.am
83 1−10T+pT2 1 - 10 T + p T^{2} 1.83.ak
89 1−10T+pT2 1 - 10 T + p T^{2} 1.89.ak
97 1+18T+pT2 1 + 18 T + p T^{2} 1.97.s
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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

−12.30242903509008952465429621319, −11.65285962273059824600304475216, −10.90027717989387157107373838421, −9.505020586152698277492779622129, −8.532719042369230860273239674478, −7.65153069290109703677672880782, −6.23214576271461606194158210231, −5.07527273844982488833791664872, −3.76029121491255918011647406142, −1.77632184622518044349130059263, 1.77632184622518044349130059263, 3.76029121491255918011647406142, 5.07527273844982488833791664872, 6.23214576271461606194158210231, 7.65153069290109703677672880782, 8.532719042369230860273239674478, 9.505020586152698277492779622129, 10.90027717989387157107373838421, 11.65285962273059824600304475216, 12.30242903509008952465429621319

Graph of the ZZ-function along the critical line