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Properties

Label 2-6080-1.1-c1-0-19
Degree 22
Conductor 60806080
Sign 11
Analytic cond. 48.549048.5490
Root an. cond. 6.967716.96771
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  + 2·3-s − 5-s − 4·7-s + 9-s − 4·11-s − 2·15-s + 6·17-s − 19-s − 8·21-s − 8·23-s + 25-s − 4·27-s + 6·29-s + 8·31-s − 8·33-s + 4·35-s + 8·37-s − 2·41-s − 45-s − 12·47-s + 9·49-s + 12·51-s − 4·53-s + 4·55-s − 2·57-s + 8·59-s + 14·61-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.447·5-s − 1.51·7-s + 1/3·9-s − 1.20·11-s − 0.516·15-s + 1.45·17-s − 0.229·19-s − 1.74·21-s − 1.66·23-s + 1/5·25-s − 0.769·27-s + 1.11·29-s + 1.43·31-s − 1.39·33-s + 0.676·35-s + 1.31·37-s − 0.312·41-s − 0.149·45-s − 1.75·47-s + 9/7·49-s + 1.68·51-s − 0.549·53-s + 0.539·55-s − 0.264·57-s + 1.04·59-s + 1.79·61-s + ⋯

Functional equation

Λ(s)=(6080s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(6080s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 60806080    =    26⋅5⋅192^{6} \cdot 5 \cdot 19
Sign: 11
Analytic conductor: 48.549048.5490
Root analytic conductor: 6.967716.96771
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 6080, ( :1/2), 1)(2,\ 6080,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.6734152931.673415293
L(12)L(\frac12) ≈\approx 1.6734152931.673415293
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
5 1+T 1 + T
19 1+T 1 + T
good3 1−2T+pT2 1 - 2 T + p T^{2}
7 1+4T+pT2 1 + 4 T + p T^{2}
11 1+4T+pT2 1 + 4 T + p T^{2}
13 1+pT2 1 + p T^{2}
17 1−6T+pT2 1 - 6 T + p T^{2}
23 1+8T+pT2 1 + 8 T + p T^{2}
29 1−6T+pT2 1 - 6 T + p T^{2}
31 1−8T+pT2 1 - 8 T + p T^{2}
37 1−8T+pT2 1 - 8 T + p T^{2}
41 1+2T+pT2 1 + 2 T + p T^{2}
43 1+pT2 1 + p T^{2}
47 1+12T+pT2 1 + 12 T + p T^{2}
53 1+4T+pT2 1 + 4 T + p T^{2}
59 1−8T+pT2 1 - 8 T + p T^{2}
61 1−14T+pT2 1 - 14 T + p T^{2}
67 1+2T+pT2 1 + 2 T + p T^{2}
71 1−8T+pT2 1 - 8 T + p T^{2}
73 1+2T+pT2 1 + 2 T + p T^{2}
79 1+4T+pT2 1 + 4 T + p T^{2}
83 1−12T+pT2 1 - 12 T + p T^{2}
89 1−6T+pT2 1 - 6 T + p T^{2}
97 1+pT2 1 + 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

−8.002145793349034303386882789385, −7.74206895127479604135859330168, −6.64159139979810876027460977242, −6.09050200317240069522502521420, −5.19910910847363332156501483830, −4.16104953155253670272146072421, −3.37483914573339538706864448662, −2.94207105490551775645458620008, −2.21972199427398646415186339445, −0.60473382774498273174936177378, 0.60473382774498273174936177378, 2.21972199427398646415186339445, 2.94207105490551775645458620008, 3.37483914573339538706864448662, 4.16104953155253670272146072421, 5.19910910847363332156501483830, 6.09050200317240069522502521420, 6.64159139979810876027460977242, 7.74206895127479604135859330168, 8.002145793349034303386882789385

Graph of the ZZ-function along the critical line