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Properties

Label 2-363-1.1-c3-0-36
Degree 22
Conductor 363363
Sign 11
Analytic cond. 21.417621.4176
Root an. cond. 4.627924.62792
Motivic weight 33
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5·2-s + 3·3-s + 17·4-s − 14·5-s + 15·6-s + 32·7-s + 45·8-s + 9·9-s − 70·10-s + 51·12-s + 38·13-s + 160·14-s − 42·15-s + 89·16-s + 2·17-s + 45·18-s − 72·19-s − 238·20-s + 96·21-s + 68·23-s + 135·24-s + 71·25-s + 190·26-s + 27·27-s + 544·28-s + 54·29-s − 210·30-s + ⋯
L(s)  = 1  + 1.76·2-s + 0.577·3-s + 17/8·4-s − 1.25·5-s + 1.02·6-s + 1.72·7-s + 1.98·8-s + 1/3·9-s − 2.21·10-s + 1.22·12-s + 0.810·13-s + 3.05·14-s − 0.722·15-s + 1.39·16-s + 0.0285·17-s + 0.589·18-s − 0.869·19-s − 2.66·20-s + 0.997·21-s + 0.616·23-s + 1.14·24-s + 0.567·25-s + 1.43·26-s + 0.192·27-s + 3.67·28-s + 0.345·29-s − 1.27·30-s + ⋯

Functional equation

Λ(s)=(363s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(363s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 363363    =    3⋅1123 \cdot 11^{2}
Sign: 11
Analytic conductor: 21.417621.4176
Root analytic conductor: 4.627924.62792
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 363, ( :3/2), 1)(2,\ 363,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 6.4528258456.452825845
L(12)L(\frac12) ≈\approx 6.4528258456.452825845
L(52)L(\frac{5}{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)
bad3 1−pT 1 - p T
11 1 1
good2 1−5T+p3T2 1 - 5 T + p^{3} T^{2}
5 1+14T+p3T2 1 + 14 T + p^{3} T^{2}
7 1−32T+p3T2 1 - 32 T + p^{3} T^{2}
13 1−38T+p3T2 1 - 38 T + p^{3} T^{2}
17 1−2T+p3T2 1 - 2 T + p^{3} T^{2}
19 1+72T+p3T2 1 + 72 T + p^{3} T^{2}
23 1−68T+p3T2 1 - 68 T + p^{3} T^{2}
29 1−54T+p3T2 1 - 54 T + p^{3} T^{2}
31 1+152T+p3T2 1 + 152 T + p^{3} T^{2}
37 1−174T+p3T2 1 - 174 T + p^{3} T^{2}
41 1+94T+p3T2 1 + 94 T + p^{3} T^{2}
43 1−528T+p3T2 1 - 528 T + p^{3} T^{2}
47 1+340T+p3T2 1 + 340 T + p^{3} T^{2}
53 1+438T+p3T2 1 + 438 T + p^{3} T^{2}
59 1−20T+p3T2 1 - 20 T + p^{3} T^{2}
61 1+570T+p3T2 1 + 570 T + p^{3} T^{2}
67 1+460T+p3T2 1 + 460 T + p^{3} T^{2}
71 1+1092T+p3T2 1 + 1092 T + p^{3} T^{2}
73 1+562T+p3T2 1 + 562 T + p^{3} T^{2}
79 1−16T+p3T2 1 - 16 T + p^{3} T^{2}
83 1+372T+p3T2 1 + 372 T + p^{3} T^{2}
89 1+966T+p3T2 1 + 966 T + p^{3} T^{2}
97 1+526T+p3T2 1 + 526 T + p^{3} 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

−11.19006251218915211552277133289, −10.85660797934741720806117522488, −8.772090369735353356673906875150, −7.926372491329169539228361320705, −7.21315452598863345440018093497, −5.86733141680236281432592538378, −4.59284218701746206559809718511, −4.20318469982873972522876409409, −3.05566527665222354951417324263, −1.64665048604234698926158528203, 1.64665048604234698926158528203, 3.05566527665222354951417324263, 4.20318469982873972522876409409, 4.59284218701746206559809718511, 5.86733141680236281432592538378, 7.21315452598863345440018093497, 7.926372491329169539228361320705, 8.772090369735353356673906875150, 10.85660797934741720806117522488, 11.19006251218915211552277133289

Graph of the ZZ-function along the critical line