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Properties

Label 2-704-1.1-c5-0-66
Degree 22
Conductor 704704
Sign 11
Analytic cond. 112.910112.910
Root an. cond. 10.625910.6259
Motivic weight 55
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 25.2·3-s + 55.5·5-s + 200.·7-s + 395.·9-s + 121·11-s − 727.·13-s + 1.40e3·15-s − 1.10e3·17-s − 1.66e3·19-s + 5.07e3·21-s + 2.98e3·23-s − 40.0·25-s + 3.85e3·27-s + 1.55e3·29-s + 9.51e3·31-s + 3.05e3·33-s + 1.11e4·35-s + 9.43e3·37-s − 1.83e4·39-s + 7.37e3·41-s + 8.52e3·43-s + 2.19e4·45-s + 3.00e4·47-s + 2.35e4·49-s − 2.80e4·51-s − 2.39e4·53-s + 6.72e3·55-s + ⋯
L(s)  = 1  + 1.62·3-s + 0.993·5-s + 1.54·7-s + 1.62·9-s + 0.301·11-s − 1.19·13-s + 1.61·15-s − 0.930·17-s − 1.05·19-s + 2.51·21-s + 1.17·23-s − 0.0128·25-s + 1.01·27-s + 0.342·29-s + 1.77·31-s + 0.488·33-s + 1.53·35-s + 1.13·37-s − 1.93·39-s + 0.684·41-s + 0.703·43-s + 1.61·45-s + 1.98·47-s + 1.39·49-s − 1.50·51-s − 1.17·53-s + 0.299·55-s + ⋯

Functional equation

Λ(s)=(704s/2 ΓC(s) L(s)=( Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
Λ(s)=(704s/2 ΓC(s+5/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 704704    =    26⋅112^{6} \cdot 11
Sign: 11
Analytic conductor: 112.910112.910
Root analytic conductor: 10.625910.6259
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 704, ( :5/2), 1)(2,\ 704,\ (\ :5/2),\ 1)

Particular Values

L(3)L(3) ≈\approx 6.3079811626.307981162
L(12)L(\frac12) ≈\approx 6.3079811626.307981162
L(72)L(\frac{7}{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
11 1−121T 1 - 121T
good3 1−25.2T+243T2 1 - 25.2T + 243T^{2}
5 1−55.5T+3.12e3T2 1 - 55.5T + 3.12e3T^{2}
7 1−200.T+1.68e4T2 1 - 200.T + 1.68e4T^{2}
13 1+727.T+3.71e5T2 1 + 727.T + 3.71e5T^{2}
17 1+1.10e3T+1.41e6T2 1 + 1.10e3T + 1.41e6T^{2}
19 1+1.66e3T+2.47e6T2 1 + 1.66e3T + 2.47e6T^{2}
23 1−2.98e3T+6.43e6T2 1 - 2.98e3T + 6.43e6T^{2}
29 1−1.55e3T+2.05e7T2 1 - 1.55e3T + 2.05e7T^{2}
31 1−9.51e3T+2.86e7T2 1 - 9.51e3T + 2.86e7T^{2}
37 1−9.43e3T+6.93e7T2 1 - 9.43e3T + 6.93e7T^{2}
41 1−7.37e3T+1.15e8T2 1 - 7.37e3T + 1.15e8T^{2}
43 1−8.52e3T+1.47e8T2 1 - 8.52e3T + 1.47e8T^{2}
47 1−3.00e4T+2.29e8T2 1 - 3.00e4T + 2.29e8T^{2}
53 1+2.39e4T+4.18e8T2 1 + 2.39e4T + 4.18e8T^{2}
59 1−6.96e3T+7.14e8T2 1 - 6.96e3T + 7.14e8T^{2}
61 1+4.90e4T+8.44e8T2 1 + 4.90e4T + 8.44e8T^{2}
67 1−2.39e4T+1.35e9T2 1 - 2.39e4T + 1.35e9T^{2}
71 1+1.88e3T+1.80e9T2 1 + 1.88e3T + 1.80e9T^{2}
73 1+1.36e4T+2.07e9T2 1 + 1.36e4T + 2.07e9T^{2}
79 1−1.15e4T+3.07e9T2 1 - 1.15e4T + 3.07e9T^{2}
83 1−5.51e4T+3.93e9T2 1 - 5.51e4T + 3.93e9T^{2}
89 1+2.37e3T+5.58e9T2 1 + 2.37e3T + 5.58e9T^{2}
97 1+7.91e3T+8.58e9T2 1 + 7.91e3T + 8.58e9T^{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

−9.389813301140849158726246338354, −8.881217079265874155313461584433, −8.057147655590802014277155121830, −7.36666750851791548502486630243, −6.23172673601397853986420892717, −4.82460565862799976683098065074, −4.27906679631701291060093210354, −2.58249390203976490695454581090, −2.26239687125334516244887884937, −1.19629078210578017810573943444, 1.19629078210578017810573943444, 2.26239687125334516244887884937, 2.58249390203976490695454581090, 4.27906679631701291060093210354, 4.82460565862799976683098065074, 6.23172673601397853986420892717, 7.36666750851791548502486630243, 8.057147655590802014277155121830, 8.881217079265874155313461584433, 9.389813301140849158726246338354

Graph of the ZZ-function along the critical line