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Properties

Label 2-277248-1.1-c1-0-5
Degree 22
Conductor 277248277248
Sign 11
Analytic cond. 2213.832213.83
Root an. cond. 47.051447.0514
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  + 3-s + 4·7-s + 9-s + 4·11-s + 4·13-s − 2·17-s + 4·21-s + 8·23-s − 5·25-s + 27-s − 8·29-s − 4·31-s + 4·33-s − 4·37-s + 4·39-s − 6·41-s + 4·43-s + 8·47-s + 9·49-s − 2·51-s − 8·53-s + 12·59-s − 12·61-s + 4·63-s − 12·67-s + 8·69-s + 8·71-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.51·7-s + 1/3·9-s + 1.20·11-s + 1.10·13-s − 0.485·17-s + 0.872·21-s + 1.66·23-s − 25-s + 0.192·27-s − 1.48·29-s − 0.718·31-s + 0.696·33-s − 0.657·37-s + 0.640·39-s − 0.937·41-s + 0.609·43-s + 1.16·47-s + 9/7·49-s − 0.280·51-s − 1.09·53-s + 1.56·59-s − 1.53·61-s + 0.503·63-s − 1.46·67-s + 0.963·69-s + 0.949·71-s + ⋯

Functional equation

Λ(s)=(277248s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 277248 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(277248s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 277248 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 277248277248    =    28⋅3⋅1922^{8} \cdot 3 \cdot 19^{2}
Sign: 11
Analytic conductor: 2213.832213.83
Root analytic conductor: 47.051447.0514
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 277248, ( :1/2), 1)(2,\ 277248,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 5.6656476515.665647651
L(12)L(\frac12) ≈\approx 5.6656476515.665647651
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−T 1 - T
19 1 1
good5 1+pT2 1 + p T^{2}
7 1−4T+pT2 1 - 4 T + p T^{2}
11 1−4T+pT2 1 - 4 T + p T^{2}
13 1−4T+pT2 1 - 4 T + p T^{2}
17 1+2T+pT2 1 + 2 T + p T^{2}
23 1−8T+pT2 1 - 8 T + p T^{2}
29 1+8T+pT2 1 + 8 T + p T^{2}
31 1+4T+pT2 1 + 4 T + p T^{2}
37 1+4T+pT2 1 + 4 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 1−8T+pT2 1 - 8 T + p T^{2}
53 1+8T+pT2 1 + 8 T + p T^{2}
59 1−12T+pT2 1 - 12 T + p T^{2}
61 1+12T+pT2 1 + 12 T + p T^{2}
67 1+12T+pT2 1 + 12 T + p T^{2}
71 1−8T+pT2 1 - 8 T + p T^{2}
73 1+6T+pT2 1 + 6 T + p T^{2}
79 1+4T+pT2 1 + 4 T + p T^{2}
83 1+4T+pT2 1 + 4 T + p T^{2}
89 1−6T+pT2 1 - 6 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

−12.86569416009618, −12.22259232227594, −11.75773340635083, −11.27425119080885, −11.03660395629445, −10.65577179442125, −9.956101694546623, −9.200543279368042, −9.083786760627470, −8.632054177578185, −8.238442599944072, −7.517246841148061, −7.304943317192642, −6.768903324494763, −6.049417435948066, −5.670473391370517, −5.050157362677421, −4.483962749377612, −4.076914378819987, −3.540980235488512, −3.108545442357180, −2.136666160721966, −1.713347407435079, −1.387953271945893, −0.6163724415038104, 0.6163724415038104, 1.387953271945893, 1.713347407435079, 2.136666160721966, 3.108545442357180, 3.540980235488512, 4.076914378819987, 4.483962749377612, 5.050157362677421, 5.670473391370517, 6.049417435948066, 6.768903324494763, 7.304943317192642, 7.517246841148061, 8.238442599944072, 8.632054177578185, 9.083786760627470, 9.200543279368042, 9.956101694546623, 10.65577179442125, 11.03660395629445, 11.27425119080885, 11.75773340635083, 12.22259232227594, 12.86569416009618

Graph of the ZZ-function along the critical line