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Properties

Label 2-416-104.51-c6-0-53
Degree 22
Conductor 416416
Sign 11
Analytic cond. 95.702495.7024
Root an. cond. 9.782769.78276
Motivic weight 66
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 50·3-s − 218·5-s + 614·7-s + 1.77e3·9-s − 2.19e3·13-s − 1.09e4·15-s + 3.17e3·17-s + 3.07e4·21-s + 3.18e4·25-s + 5.21e4·27-s + 2.78e4·31-s − 1.33e5·35-s + 1.38e4·37-s − 1.09e5·39-s + 1.11e5·43-s − 3.86e5·45-s − 1.28e5·47-s + 2.59e5·49-s + 1.58e5·51-s + 1.08e6·63-s + 4.78e5·65-s + 3.17e5·71-s + 1.59e6·75-s + 1.31e6·81-s − 6.91e5·85-s − 1.34e6·91-s + 1.39e6·93-s + ⋯
L(s)  = 1  + 1.85·3-s − 1.74·5-s + 1.79·7-s + 2.42·9-s − 13-s − 3.22·15-s + 0.645·17-s + 3.31·21-s + 2.04·25-s + 2.64·27-s + 0.934·31-s − 3.12·35-s + 0.274·37-s − 1.85·39-s + 1.40·43-s − 4.23·45-s − 1.23·47-s + 2.20·49-s + 1.19·51-s + 4.34·63-s + 1.74·65-s + 0.888·71-s + 3.78·75-s + 2.47·81-s − 1.12·85-s − 1.79·91-s + 1.72·93-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: yes
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 4.2906452384.290645238
L(12)L(\frac12) ≈\approx 4.2906452384.290645238
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+p3T 1 + p^{3} T
good3 1−50T+p6T2 1 - 50 T + p^{6} T^{2}
5 1+218T+p6T2 1 + 218 T + p^{6} T^{2}
7 1−614T+p6T2 1 - 614 T + p^{6} T^{2}
11 (1−p3T)(1+p3T) ( 1 - p^{3} T )( 1 + p^{3} T )
17 1−3170T+p6T2 1 - 3170 T + p^{6} T^{2}
19 (1−p3T)(1+p3T) ( 1 - p^{3} T )( 1 + p^{3} T )
23 (1−p3T)(1+p3T) ( 1 - p^{3} T )( 1 + p^{3} T )
29 (1−p3T)(1+p3T) ( 1 - p^{3} T )( 1 + p^{3} T )
31 1−27830T+p6T2 1 - 27830 T + p^{6} T^{2}
37 1−13894T+p6T2 1 - 13894 T + p^{6} T^{2}
41 (1−p3T)(1+p3T) ( 1 - p^{3} T )( 1 + p^{3} T )
43 1−111490T+p6T2 1 - 111490 T + p^{6} T^{2}
47 1+128554T+p6T2 1 + 128554 T + p^{6} T^{2}
53 (1−p3T)(1+p3T) ( 1 - p^{3} T )( 1 + p^{3} T )
59 (1−p3T)(1+p3T) ( 1 - p^{3} T )( 1 + p^{3} T )
61 (1−p3T)(1+p3T) ( 1 - p^{3} T )( 1 + p^{3} T )
67 (1−p3T)(1+p3T) ( 1 - p^{3} T )( 1 + p^{3} T )
71 1−317990T+p6T2 1 - 317990 T + p^{6} T^{2}
73 (1−p3T)(1+p3T) ( 1 - p^{3} T )( 1 + p^{3} T )
79 (1−p3T)(1+p3T) ( 1 - p^{3} T )( 1 + p^{3} T )
83 (1−p3T)(1+p3T) ( 1 - p^{3} T )( 1 + p^{3} T )
89 (1−p3T)(1+p3T) ( 1 - p^{3} T )( 1 + p^{3} T )
97 (1−p3T)(1+p3T) ( 1 - p^{3} T )( 1 + p^{3} T )
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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

−10.09555616764959933928864337360, −8.929785115982116395420764856246, −8.141887735203654408895953678611, −7.79779067574422377097957391923, −7.19746986405138588844385494713, −4.82768396846223968724242945694, −4.25792522684558799032857330418, −3.25697536663107794959482148636, −2.20136466490286399083650286051, −0.971924218133457298629039723403, 0.971924218133457298629039723403, 2.20136466490286399083650286051, 3.25697536663107794959482148636, 4.25792522684558799032857330418, 4.82768396846223968724242945694, 7.19746986405138588844385494713, 7.79779067574422377097957391923, 8.141887735203654408895953678611, 8.929785115982116395420764856246, 10.09555616764959933928864337360

Graph of the ZZ-function along the critical line