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Properties

Label 2-176-1.1-c5-0-15
Degree 22
Conductor 176176
Sign −1-1
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 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 19.2·3-s + 33.5·5-s − 67.1·7-s + 128.·9-s + 121·11-s + 504.·13-s − 646.·15-s + 984.·17-s − 281.·19-s + 1.29e3·21-s − 359.·23-s − 1.99e3·25-s + 2.20e3·27-s − 5.02e3·29-s + 7.01e3·31-s − 2.33e3·33-s − 2.25e3·35-s − 5.24e3·37-s − 9.72e3·39-s − 1.38e4·41-s − 2.01e4·43-s + 4.30e3·45-s − 6.78e3·47-s − 1.22e4·49-s − 1.89e4·51-s − 2.72e4·53-s + 4.05e3·55-s + ⋯
L(s)  = 1  − 1.23·3-s + 0.600·5-s − 0.518·7-s + 0.528·9-s + 0.301·11-s + 0.828·13-s − 0.741·15-s + 0.826·17-s − 0.178·19-s + 0.640·21-s − 0.141·23-s − 0.639·25-s + 0.583·27-s − 1.10·29-s + 1.31·31-s − 0.372·33-s − 0.310·35-s − 0.629·37-s − 1.02·39-s − 1.28·41-s − 1.66·43-s + 0.316·45-s − 0.447·47-s − 0.731·49-s − 1.02·51-s − 1.33·53-s + 0.180·55-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: −1-1
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: 11
Selberg data: (2, 176, ( :5/2), −1)(2,\ 176,\ (\ :5/2),\ -1)

Particular Values

L(3)L(3) == 00
L(12)L(\frac12) == 00
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+19.2T+243T2 1 + 19.2T + 243T^{2}
5 1−33.5T+3.12e3T2 1 - 33.5T + 3.12e3T^{2}
7 1+67.1T+1.68e4T2 1 + 67.1T + 1.68e4T^{2}
13 1−504.T+3.71e5T2 1 - 504.T + 3.71e5T^{2}
17 1−984.T+1.41e6T2 1 - 984.T + 1.41e6T^{2}
19 1+281.T+2.47e6T2 1 + 281.T + 2.47e6T^{2}
23 1+359.T+6.43e6T2 1 + 359.T + 6.43e6T^{2}
29 1+5.02e3T+2.05e7T2 1 + 5.02e3T + 2.05e7T^{2}
31 1−7.01e3T+2.86e7T2 1 - 7.01e3T + 2.86e7T^{2}
37 1+5.24e3T+6.93e7T2 1 + 5.24e3T + 6.93e7T^{2}
41 1+1.38e4T+1.15e8T2 1 + 1.38e4T + 1.15e8T^{2}
43 1+2.01e4T+1.47e8T2 1 + 2.01e4T + 1.47e8T^{2}
47 1+6.78e3T+2.29e8T2 1 + 6.78e3T + 2.29e8T^{2}
53 1+2.72e4T+4.18e8T2 1 + 2.72e4T + 4.18e8T^{2}
59 1+1.90e4T+7.14e8T2 1 + 1.90e4T + 7.14e8T^{2}
61 1−2.40e4T+8.44e8T2 1 - 2.40e4T + 8.44e8T^{2}
67 1+5.32e4T+1.35e9T2 1 + 5.32e4T + 1.35e9T^{2}
71 1−4.42e4T+1.80e9T2 1 - 4.42e4T + 1.80e9T^{2}
73 1−2.19e4T+2.07e9T2 1 - 2.19e4T + 2.07e9T^{2}
79 1−2.63e4T+3.07e9T2 1 - 2.63e4T + 3.07e9T^{2}
83 1−1.94e4T+3.93e9T2 1 - 1.94e4T + 3.93e9T^{2}
89 1+3.13e4T+5.58e9T2 1 + 3.13e4T + 5.58e9T^{2}
97 1−1.34e5T+8.58e9T2 1 - 1.34e5T + 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

−11.39029789818719813212124866064, −10.36018330384490362369781241231, −9.579703294523501185611750644654, −8.228228237735083129416469802353, −6.64901142587139128158836994933, −6.02433672669980097549679474663, −5.03796223483521583507132212908, −3.43760760367188613810706518846, −1.49458103574551766913901550212, 0, 1.49458103574551766913901550212, 3.43760760367188613810706518846, 5.03796223483521583507132212908, 6.02433672669980097549679474663, 6.64901142587139128158836994933, 8.228228237735083129416469802353, 9.579703294523501185611750644654, 10.36018330384490362369781241231, 11.39029789818719813212124866064

Graph of the ZZ-function along the critical line