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Properties

Label 2-6552-1.1-c1-0-26
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  + 7-s + 2·11-s − 13-s + 6·17-s − 8·19-s + 4·23-s − 5·25-s + 6·29-s + 4·31-s − 2·37-s + 4·43-s + 6·47-s + 49-s − 6·53-s − 10·59-s + 10·61-s − 4·67-s + 6·71-s + 6·73-s + 2·77-s − 6·83-s + 12·89-s − 91-s − 2·97-s + 14·101-s + 8·103-s − 6·109-s + ⋯
L(s)  = 1  + 0.377·7-s + 0.603·11-s − 0.277·13-s + 1.45·17-s − 1.83·19-s + 0.834·23-s − 25-s + 1.11·29-s + 0.718·31-s − 0.328·37-s + 0.609·43-s + 0.875·47-s + 1/7·49-s − 0.824·53-s − 1.30·59-s + 1.28·61-s − 0.488·67-s + 0.712·71-s + 0.702·73-s + 0.227·77-s − 0.658·83-s + 1.27·89-s − 0.104·91-s − 0.203·97-s + 1.39·101-s + 0.788·103-s − 0.574·109-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 2.1836257962.183625796
L(12)L(\frac12) ≈\approx 2.1836257962.183625796
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+pT2 1 + p T^{2}
11 1−2T+pT2 1 - 2 T + p T^{2}
17 1−6T+pT2 1 - 6 T + p T^{2}
19 1+8T+pT2 1 + 8 T + p T^{2}
23 1−4T+pT2 1 - 4 T + p T^{2}
29 1−6T+pT2 1 - 6 T + p T^{2}
31 1−4T+pT2 1 - 4 T + p T^{2}
37 1+2T+pT2 1 + 2 T + p T^{2}
41 1+pT2 1 + p T^{2}
43 1−4T+pT2 1 - 4 T + p T^{2}
47 1−6T+pT2 1 - 6 T + p T^{2}
53 1+6T+pT2 1 + 6 T + p T^{2}
59 1+10T+pT2 1 + 10 T + p T^{2}
61 1−10T+pT2 1 - 10 T + p T^{2}
67 1+4T+pT2 1 + 4 T + p T^{2}
71 1−6T+pT2 1 - 6 T + p T^{2}
73 1−6T+pT2 1 - 6 T + p T^{2}
79 1+pT2 1 + p T^{2}
83 1+6T+pT2 1 + 6 T + p T^{2}
89 1−12T+pT2 1 - 12 T + p T^{2}
97 1+2T+pT2 1 + 2 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

−8.020826581727519939478446233088, −7.37342652208140642174399731054, −6.51056181261956265586124256402, −6.00209048136519658428399884673, −5.08670990306916741113411991854, −4.41012325330574180793620936540, −3.66615611131011434255405278277, −2.72348096868692540422321530716, −1.80387105369187633275033274184, −0.78342614189904650346580328544, 0.78342614189904650346580328544, 1.80387105369187633275033274184, 2.72348096868692540422321530716, 3.66615611131011434255405278277, 4.41012325330574180793620936540, 5.08670990306916741113411991854, 6.00209048136519658428399884673, 6.51056181261956265586124256402, 7.37342652208140642174399731054, 8.020826581727519939478446233088

Graph of the ZZ-function along the critical line