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Properties

Label 2-176-1.1-c5-0-14
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  + 29.5·3-s + 41.2·5-s − 77.0·7-s + 632.·9-s + 121·11-s − 637.·13-s + 1.22e3·15-s + 1.42e3·17-s + 1.71e3·19-s − 2.28e3·21-s + 4.19e3·23-s − 1.42e3·25-s + 1.15e4·27-s + 878.·29-s − 6.76e3·31-s + 3.58e3·33-s − 3.17e3·35-s − 4.00e3·37-s − 1.88e4·39-s − 1.36e4·41-s − 1.62e3·43-s + 2.60e4·45-s + 1.71e4·47-s − 1.08e4·49-s + 4.21e4·51-s + 1.51e4·53-s + 4.98e3·55-s + ⋯
L(s)  = 1  + 1.89·3-s + 0.737·5-s − 0.594·7-s + 2.60·9-s + 0.301·11-s − 1.04·13-s + 1.40·15-s + 1.19·17-s + 1.09·19-s − 1.12·21-s + 1.65·23-s − 0.455·25-s + 3.04·27-s + 0.193·29-s − 1.26·31-s + 0.572·33-s − 0.438·35-s − 0.481·37-s − 1.98·39-s − 1.27·41-s − 0.134·43-s + 1.91·45-s + 1.13·47-s − 0.646·49-s + 2.26·51-s + 0.740·53-s + 0.222·55-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 4.3805188854.380518885
L(12)L(\frac12) ≈\approx 4.3805188854.380518885
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 1−29.5T+243T2 1 - 29.5T + 243T^{2}
5 1−41.2T+3.12e3T2 1 - 41.2T + 3.12e3T^{2}
7 1+77.0T+1.68e4T2 1 + 77.0T + 1.68e4T^{2}
13 1+637.T+3.71e5T2 1 + 637.T + 3.71e5T^{2}
17 1−1.42e3T+1.41e6T2 1 - 1.42e3T + 1.41e6T^{2}
19 1−1.71e3T+2.47e6T2 1 - 1.71e3T + 2.47e6T^{2}
23 1−4.19e3T+6.43e6T2 1 - 4.19e3T + 6.43e6T^{2}
29 1−878.T+2.05e7T2 1 - 878.T + 2.05e7T^{2}
31 1+6.76e3T+2.86e7T2 1 + 6.76e3T + 2.86e7T^{2}
37 1+4.00e3T+6.93e7T2 1 + 4.00e3T + 6.93e7T^{2}
41 1+1.36e4T+1.15e8T2 1 + 1.36e4T + 1.15e8T^{2}
43 1+1.62e3T+1.47e8T2 1 + 1.62e3T + 1.47e8T^{2}
47 1−1.71e4T+2.29e8T2 1 - 1.71e4T + 2.29e8T^{2}
53 1−1.51e4T+4.18e8T2 1 - 1.51e4T + 4.18e8T^{2}
59 1+1.22e4T+7.14e8T2 1 + 1.22e4T + 7.14e8T^{2}
61 1+3.52e4T+8.44e8T2 1 + 3.52e4T + 8.44e8T^{2}
67 1−5.84e4T+1.35e9T2 1 - 5.84e4T + 1.35e9T^{2}
71 1−5.56e3T+1.80e9T2 1 - 5.56e3T + 1.80e9T^{2}
73 1+5.96e4T+2.07e9T2 1 + 5.96e4T + 2.07e9T^{2}
79 1+8.07e4T+3.07e9T2 1 + 8.07e4T + 3.07e9T^{2}
83 1−7.29e4T+3.93e9T2 1 - 7.29e4T + 3.93e9T^{2}
89 1−1.07e5T+5.58e9T2 1 - 1.07e5T + 5.58e9T^{2}
97 1−3.44e4T+8.58e9T2 1 - 3.44e4T + 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

−12.16439923588664503455133900016, −10.27104608776563147031566061527, −9.558287216805528934135622477954, −9.018833966861268827972839676658, −7.70152003114787942435264094707, −6.95871997629458097131436093722, −5.21322987011102319725784400200, −3.56514418249012423677702258015, −2.74042401048019756521052565843, −1.45557761468651860679098532501, 1.45557761468651860679098532501, 2.74042401048019756521052565843, 3.56514418249012423677702258015, 5.21322987011102319725784400200, 6.95871997629458097131436093722, 7.70152003114787942435264094707, 9.018833966861268827972839676658, 9.558287216805528934135622477954, 10.27104608776563147031566061527, 12.16439923588664503455133900016

Graph of the ZZ-function along the critical line