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Properties

Label 2-176-1.1-c5-0-16
Degree 22
Conductor 176176
Sign 11
Analytic cond. 28.227528.2275
Root an. cond. 5.312965.31296
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  + 19.4·3-s + 82.2·5-s + 200.·7-s + 135.·9-s + 121·11-s + 418.·13-s + 1.60e3·15-s + 292.·17-s − 2.60e3·19-s + 3.90e3·21-s − 3.64e3·23-s + 3.64e3·25-s − 2.08e3·27-s − 13.9·29-s − 692.·31-s + 2.35e3·33-s + 1.64e4·35-s + 4.78e3·37-s + 8.14e3·39-s − 1.81e4·41-s − 1.59e4·43-s + 1.11e4·45-s − 1.05e4·47-s + 2.33e4·49-s + 5.68e3·51-s + 2.35e4·53-s + 9.95e3·55-s + ⋯
L(s)  = 1  + 1.24·3-s + 1.47·5-s + 1.54·7-s + 0.559·9-s + 0.301·11-s + 0.686·13-s + 1.83·15-s + 0.245·17-s − 1.65·19-s + 1.93·21-s − 1.43·23-s + 1.16·25-s − 0.550·27-s − 0.00307·29-s − 0.129·31-s + 0.376·33-s + 2.27·35-s + 0.575·37-s + 0.857·39-s − 1.68·41-s − 1.31·43-s + 0.823·45-s − 0.696·47-s + 1.39·49-s + 0.306·51-s + 1.15·53-s + 0.443·55-s + ⋯

Functional equation

Λ(s)=(176s/2 ΓC(s) L(s)=( Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
Λ(s)=(176s/2 ΓC(s+5/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 176176    =    24⋅112^{4} \cdot 11
Sign: 11
Analytic conductor: 28.227528.2275
Root analytic conductor: 5.312965.31296
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 176, ( :5/2), 1)(2,\ 176,\ (\ :5/2),\ 1)

Particular Values

L(3)L(3) ≈\approx 4.4879225284.487922528
L(12)L(\frac12) ≈\approx 4.4879225284.487922528
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 1−19.4T+243T2 1 - 19.4T + 243T^{2}
5 1−82.2T+3.12e3T2 1 - 82.2T + 3.12e3T^{2}
7 1−200.T+1.68e4T2 1 - 200.T + 1.68e4T^{2}
13 1−418.T+3.71e5T2 1 - 418.T + 3.71e5T^{2}
17 1−292.T+1.41e6T2 1 - 292.T + 1.41e6T^{2}
19 1+2.60e3T+2.47e6T2 1 + 2.60e3T + 2.47e6T^{2}
23 1+3.64e3T+6.43e6T2 1 + 3.64e3T + 6.43e6T^{2}
29 1+13.9T+2.05e7T2 1 + 13.9T + 2.05e7T^{2}
31 1+692.T+2.86e7T2 1 + 692.T + 2.86e7T^{2}
37 1−4.78e3T+6.93e7T2 1 - 4.78e3T + 6.93e7T^{2}
41 1+1.81e4T+1.15e8T2 1 + 1.81e4T + 1.15e8T^{2}
43 1+1.59e4T+1.47e8T2 1 + 1.59e4T + 1.47e8T^{2}
47 1+1.05e4T+2.29e8T2 1 + 1.05e4T + 2.29e8T^{2}
53 1−2.35e4T+4.18e8T2 1 - 2.35e4T + 4.18e8T^{2}
59 1−4.52e4T+7.14e8T2 1 - 4.52e4T + 7.14e8T^{2}
61 1−1.11e4T+8.44e8T2 1 - 1.11e4T + 8.44e8T^{2}
67 1−3.45e4T+1.35e9T2 1 - 3.45e4T + 1.35e9T^{2}
71 1+5.91e4T+1.80e9T2 1 + 5.91e4T + 1.80e9T^{2}
73 1−5.32e4T+2.07e9T2 1 - 5.32e4T + 2.07e9T^{2}
79 1−7.75e3T+3.07e9T2 1 - 7.75e3T + 3.07e9T^{2}
83 1−9.13e4T+3.93e9T2 1 - 9.13e4T + 3.93e9T^{2}
89 1−1.17e5T+5.58e9T2 1 - 1.17e5T + 5.58e9T^{2}
97 1+1.11e5T+8.58e9T2 1 + 1.11e5T + 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

−11.78374263580951575150924057629, −10.60546710592988308651516979325, −9.693882061216968535584651939428, −8.541865127790941975086532263303, −8.184167921799683615760428183299, −6.52148973862538518303442003983, −5.33816573820448082853794309052, −3.93498995180439610569072104704, −2.22518636473064861338222809968, −1.65864343952665049043025395389, 1.65864343952665049043025395389, 2.22518636473064861338222809968, 3.93498995180439610569072104704, 5.33816573820448082853794309052, 6.52148973862538518303442003983, 8.184167921799683615760428183299, 8.541865127790941975086532263303, 9.693882061216968535584651939428, 10.60546710592988308651516979325, 11.78374263580951575150924057629

Graph of the ZZ-function along the critical line