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Properties

Label 2-336-1.1-c5-0-10
Degree 22
Conductor 336336
Sign 11
Analytic cond. 53.888953.8889
Root an. cond. 7.340917.34091
Motivic weight 55
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 9·3-s − 34·5-s + 49·7-s + 81·9-s + 340·11-s + 454·13-s − 306·15-s − 798·17-s − 892·19-s + 441·21-s + 3.19e3·23-s − 1.96e3·25-s + 729·27-s − 8.24e3·29-s + 2.49e3·31-s + 3.06e3·33-s − 1.66e3·35-s + 9.79e3·37-s + 4.08e3·39-s + 1.98e4·41-s + 1.72e4·43-s − 2.75e3·45-s − 8.92e3·47-s + 2.40e3·49-s − 7.18e3·51-s + 150·53-s − 1.15e4·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.608·5-s + 0.377·7-s + 1/3·9-s + 0.847·11-s + 0.745·13-s − 0.351·15-s − 0.669·17-s − 0.566·19-s + 0.218·21-s + 1.25·23-s − 0.630·25-s + 0.192·27-s − 1.81·29-s + 0.466·31-s + 0.489·33-s − 0.229·35-s + 1.17·37-s + 0.430·39-s + 1.84·41-s + 1.42·43-s − 0.202·45-s − 0.589·47-s + 1/7·49-s − 0.386·51-s + 0.00733·53-s − 0.515·55-s + ⋯

Functional equation

Λ(s)=(336s/2 ΓC(s) L(s)=( Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
Λ(s)=(336s/2 ΓC(s+5/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 336336    =    24⋅3⋅72^{4} \cdot 3 \cdot 7
Sign: 11
Analytic conductor: 53.888953.8889
Root analytic conductor: 7.340917.34091
Motivic weight: 55
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 336, ( :5/2), 1)(2,\ 336,\ (\ :5/2),\ 1)

Particular Values

L(3)L(3) ≈\approx 2.6083113262.608311326
L(12)L(\frac12) ≈\approx 2.6083113262.608311326
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
3 1−p2T 1 - p^{2} T
7 1−p2T 1 - p^{2} T
good5 1+34T+p5T2 1 + 34 T + p^{5} T^{2}
11 1−340T+p5T2 1 - 340 T + p^{5} T^{2}
13 1−454T+p5T2 1 - 454 T + p^{5} T^{2}
17 1+798T+p5T2 1 + 798 T + p^{5} T^{2}
19 1+892T+p5T2 1 + 892 T + p^{5} T^{2}
23 1−3192T+p5T2 1 - 3192 T + p^{5} T^{2}
29 1+8242T+p5T2 1 + 8242 T + p^{5} T^{2}
31 1−2496T+p5T2 1 - 2496 T + p^{5} T^{2}
37 1−9798T+p5T2 1 - 9798 T + p^{5} T^{2}
41 1−19834T+p5T2 1 - 19834 T + p^{5} T^{2}
43 1−17236T+p5T2 1 - 17236 T + p^{5} T^{2}
47 1+8928T+p5T2 1 + 8928 T + p^{5} T^{2}
53 1−150T+p5T2 1 - 150 T + p^{5} T^{2}
59 1−42396T+p5T2 1 - 42396 T + p^{5} T^{2}
61 1−14758T+p5T2 1 - 14758 T + p^{5} T^{2}
67 1−1676T+p5T2 1 - 1676 T + p^{5} T^{2}
71 1+14568T+p5T2 1 + 14568 T + p^{5} T^{2}
73 1−78378T+p5T2 1 - 78378 T + p^{5} T^{2}
79 1−2272T+p5T2 1 - 2272 T + p^{5} T^{2}
83 1−37764T+p5T2 1 - 37764 T + p^{5} T^{2}
89 1+117286T+p5T2 1 + 117286 T + p^{5} T^{2}
97 1−10002T+p5T2 1 - 10002 T + p^{5} T^{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

−11.05703355156774468292632628458, −9.539647215135225232145350182252, −8.852902043513638274273940261398, −7.948196697666891224299296664988, −7.04924647583409911325506229709, −5.91154036232606829397198566739, −4.39375374417510443571817010023, −3.69732301842623012294772726858, −2.25233064069868240762843230113, −0.894635478060079159301213642759, 0.894635478060079159301213642759, 2.25233064069868240762843230113, 3.69732301842623012294772726858, 4.39375374417510443571817010023, 5.91154036232606829397198566739, 7.04924647583409911325506229709, 7.948196697666891224299296664988, 8.852902043513638274273940261398, 9.539647215135225232145350182252, 11.05703355156774468292632628458

Graph of the ZZ-function along the critical line