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Properties

Label 2-416-104.51-c6-0-37
Degree 22
Conductor 416416
Sign 11
Analytic cond. 95.702495.7024
Root an. cond. 9.782769.78276
Motivic weight 66
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 7.33·3-s + 214.·5-s − 571.·7-s − 675.·9-s − 2.19e3·13-s − 1.57e3·15-s − 9.63e3·17-s + 4.19e3·21-s + 3.05e4·25-s + 1.03e4·27-s + 2.78e4·31-s − 1.22e5·35-s + 7.99e4·37-s + 1.61e4·39-s + 4.24e4·43-s − 1.45e5·45-s + 2.05e5·47-s + 2.09e5·49-s + 7.07e4·51-s + 3.86e5·63-s − 4.72e5·65-s − 7.14e5·71-s − 2.24e5·75-s + 4.16e5·81-s − 2.07e6·85-s + 1.25e6·91-s − 2.04e5·93-s + ⋯
L(s)  = 1  − 0.271·3-s + 1.71·5-s − 1.66·7-s − 0.926·9-s − 13-s − 0.467·15-s − 1.96·17-s + 0.453·21-s + 1.95·25-s + 0.523·27-s + 0.934·31-s − 2.86·35-s + 1.57·37-s + 0.271·39-s + 0.533·43-s − 1.59·45-s + 1.97·47-s + 1.78·49-s + 0.533·51-s + 1.54·63-s − 1.71·65-s − 1.99·71-s − 0.531·75-s + 0.783·81-s − 3.37·85-s + 1.66·91-s − 0.253·93-s + ⋯

Functional equation

Λ(s)=(416s/2 ΓC(s) L(s)=( Λ(7−s)\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}
Λ(s)=(416s/2 ΓC(s+3) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 416416    =    25⋅132^{5} \cdot 13
Sign: 11
Analytic conductor: 95.702495.7024
Root analytic conductor: 9.782769.78276
Motivic weight: 66
Rational: no
Arithmetic: yes
Character: χ416(207,⋅)\chi_{416} (207, \cdot )
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 416, ( :3), 1)(2,\ 416,\ (\ :3),\ 1)

Particular Values

L(72)L(\frac{7}{2}) ≈\approx 1.3881304551.388130455
L(12)L(\frac12) ≈\approx 1.3881304551.388130455
L(4)L(4) 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
13 1+2.19e3T 1 + 2.19e3T
good3 1+7.33T+729T2 1 + 7.33T + 729T^{2}
5 1−214.T+1.56e4T2 1 - 214.T + 1.56e4T^{2}
7 1+571.T+1.17e5T2 1 + 571.T + 1.17e5T^{2}
11 1−1.77e6T2 1 - 1.77e6T^{2}
17 1+9.63e3T+2.41e7T2 1 + 9.63e3T + 2.41e7T^{2}
19 1−4.70e7T2 1 - 4.70e7T^{2}
23 1−1.48e8T2 1 - 1.48e8T^{2}
29 1−5.94e8T2 1 - 5.94e8T^{2}
31 1−2.78e4T+8.87e8T2 1 - 2.78e4T + 8.87e8T^{2}
37 1−7.99e4T+2.56e9T2 1 - 7.99e4T + 2.56e9T^{2}
41 1−4.75e9T2 1 - 4.75e9T^{2}
43 1−4.24e4T+6.32e9T2 1 - 4.24e4T + 6.32e9T^{2}
47 1−2.05e5T+1.07e10T2 1 - 2.05e5T + 1.07e10T^{2}
53 1−2.21e10T2 1 - 2.21e10T^{2}
59 1−4.21e10T2 1 - 4.21e10T^{2}
61 1−5.15e10T2 1 - 5.15e10T^{2}
67 1−9.04e10T2 1 - 9.04e10T^{2}
71 1+7.14e5T+1.28e11T2 1 + 7.14e5T + 1.28e11T^{2}
73 1−1.51e11T2 1 - 1.51e11T^{2}
79 1−2.43e11T2 1 - 2.43e11T^{2}
83 1−3.26e11T2 1 - 3.26e11T^{2}
89 1−4.96e11T2 1 - 4.96e11T^{2}
97 1−8.32e11T2 1 - 8.32e11T^{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

−10.07992138620381170639806027579, −9.377494799437826969123263268867, −8.819526307071400331354765961406, −7.06160332394129968315041982068, −6.23686324897468646551939786324, −5.80498711493180292741602862185, −4.53841345467309530569612245774, −2.79555366710034490608897012609, −2.33681847047763002359006457511, −0.54402052549489841218121144042, 0.54402052549489841218121144042, 2.33681847047763002359006457511, 2.79555366710034490608897012609, 4.53841345467309530569612245774, 5.80498711493180292741602862185, 6.23686324897468646551939786324, 7.06160332394129968315041982068, 8.819526307071400331354765961406, 9.377494799437826969123263268867, 10.07992138620381170639806027579

Graph of the ZZ-function along the critical line