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Properties

Label 2-6552-1.1-c1-0-48
Degree 22
Conductor 65526552
Sign 11
Analytic cond. 52.317952.3179
Root an. cond. 7.233117.23311
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  + 2·5-s + 7-s + 4·11-s + 13-s + 6·17-s + 4·19-s − 25-s − 6·29-s + 2·35-s + 6·37-s + 6·41-s + 4·43-s + 49-s + 2·53-s + 8·55-s − 4·59-s − 2·61-s + 2·65-s + 4·67-s − 6·73-s + 4·77-s + 8·79-s − 12·83-s + 12·85-s − 10·89-s + 91-s + 8·95-s + ⋯
L(s)  = 1  + 0.894·5-s + 0.377·7-s + 1.20·11-s + 0.277·13-s + 1.45·17-s + 0.917·19-s − 1/5·25-s − 1.11·29-s + 0.338·35-s + 0.986·37-s + 0.937·41-s + 0.609·43-s + 1/7·49-s + 0.274·53-s + 1.07·55-s − 0.520·59-s − 0.256·61-s + 0.248·65-s + 0.488·67-s − 0.702·73-s + 0.455·77-s + 0.900·79-s − 1.31·83-s + 1.30·85-s − 1.05·89-s + 0.104·91-s + 0.820·95-s + ⋯

Functional equation

Λ(s)=(6552s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 6552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(6552s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 6552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 65526552    =    23⋅32⋅7⋅132^{3} \cdot 3^{2} \cdot 7 \cdot 13
Sign: 11
Analytic conductor: 52.317952.3179
Root analytic conductor: 7.233117.23311
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 6552, ( :1/2), 1)(2,\ 6552,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 3.2046925053.204692505
L(12)L(\frac12) ≈\approx 3.2046925053.204692505
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)
bad2 1 1
3 1 1
7 1−T 1 - T
13 1−T 1 - T
good5 1−2T+pT2 1 - 2 T + p T^{2}
11 1−4T+pT2 1 - 4 T + p T^{2}
17 1−6T+pT2 1 - 6 T + p T^{2}
19 1−4T+pT2 1 - 4 T + p T^{2}
23 1+pT2 1 + p T^{2}
29 1+6T+pT2 1 + 6 T + p T^{2}
31 1+pT2 1 + p T^{2}
37 1−6T+pT2 1 - 6 T + p T^{2}
41 1−6T+pT2 1 - 6 T + p T^{2}
43 1−4T+pT2 1 - 4 T + p T^{2}
47 1+pT2 1 + p T^{2}
53 1−2T+pT2 1 - 2 T + p T^{2}
59 1+4T+pT2 1 + 4 T + p T^{2}
61 1+2T+pT2 1 + 2 T + p T^{2}
67 1−4T+pT2 1 - 4 T + p T^{2}
71 1+pT2 1 + p T^{2}
73 1+6T+pT2 1 + 6 T + p T^{2}
79 1−8T+pT2 1 - 8 T + p T^{2}
83 1+12T+pT2 1 + 12 T + p T^{2}
89 1+10T+pT2 1 + 10 T + p T^{2}
97 1+14T+pT2 1 + 14 T + p T^{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

−7.86808902150910022748272179233, −7.40638260288841524859508132736, −6.48964132982911825714556354626, −5.75789024080702081831307579096, −5.44384505756742781933018364260, −4.32328205164357812303218546295, −3.64183458737992114265619645944, −2.72073979031890227895774674961, −1.65031570417369949078072999027, −1.03907702587109094101826337827, 1.03907702587109094101826337827, 1.65031570417369949078072999027, 2.72073979031890227895774674961, 3.64183458737992114265619645944, 4.32328205164357812303218546295, 5.44384505756742781933018364260, 5.75789024080702081831307579096, 6.48964132982911825714556354626, 7.40638260288841524859508132736, 7.86808902150910022748272179233

Graph of the ZZ-function along the critical line