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Properties

Label 2-14-1.1-c3-0-1
Degree 22
Conductor 1414
Sign 11
Analytic cond. 0.8260260.826026
Root an. cond. 0.9088600.908860
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  + 2·2-s − 2·3-s + 4·4-s − 12·5-s − 4·6-s + 7·7-s + 8·8-s − 23·9-s − 24·10-s + 48·11-s − 8·12-s + 56·13-s + 14·14-s + 24·15-s + 16·16-s − 114·17-s − 46·18-s + 2·19-s − 48·20-s − 14·21-s + 96·22-s − 120·23-s − 16·24-s + 19·25-s + 112·26-s + 100·27-s + 28·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.384·3-s + 1/2·4-s − 1.07·5-s − 0.272·6-s + 0.377·7-s + 0.353·8-s − 0.851·9-s − 0.758·10-s + 1.31·11-s − 0.192·12-s + 1.19·13-s + 0.267·14-s + 0.413·15-s + 1/4·16-s − 1.62·17-s − 0.602·18-s + 0.0241·19-s − 0.536·20-s − 0.145·21-s + 0.930·22-s − 1.08·23-s − 0.136·24-s + 0.151·25-s + 0.844·26-s + 0.712·27-s + 0.188·28-s + ⋯

Functional equation

Λ(s)=(14s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(14s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 1414    =    2⋅72 \cdot 7
Sign: 11
Analytic conductor: 0.8260260.826026
Root analytic conductor: 0.9088600.908860
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 14, ( :3/2), 1)(2,\ 14,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 1.1362033851.136203385
L(12)L(\frac12) ≈\approx 1.1362033851.136203385
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)
bad2 1−pT 1 - p T
7 1−pT 1 - p T
good3 1+2T+p3T2 1 + 2 T + p^{3} T^{2}
5 1+12T+p3T2 1 + 12 T + p^{3} T^{2}
11 1−48T+p3T2 1 - 48 T + p^{3} T^{2}
13 1−56T+p3T2 1 - 56 T + p^{3} T^{2}
17 1+114T+p3T2 1 + 114 T + p^{3} T^{2}
19 1−2T+p3T2 1 - 2 T + p^{3} T^{2}
23 1+120T+p3T2 1 + 120 T + p^{3} T^{2}
29 1+54T+p3T2 1 + 54 T + p^{3} T^{2}
31 1−236T+p3T2 1 - 236 T + p^{3} T^{2}
37 1−146T+p3T2 1 - 146 T + p^{3} T^{2}
41 1−126T+p3T2 1 - 126 T + p^{3} T^{2}
43 1+376T+p3T2 1 + 376 T + p^{3} T^{2}
47 1+12T+p3T2 1 + 12 T + p^{3} T^{2}
53 1−174T+p3T2 1 - 174 T + p^{3} T^{2}
59 1−138T+p3T2 1 - 138 T + p^{3} T^{2}
61 1−380T+p3T2 1 - 380 T + p^{3} T^{2}
67 1+484T+p3T2 1 + 484 T + p^{3} T^{2}
71 1−576T+p3T2 1 - 576 T + p^{3} T^{2}
73 1+1150T+p3T2 1 + 1150 T + p^{3} T^{2}
79 1−776T+p3T2 1 - 776 T + p^{3} T^{2}
83 1−378T+p3T2 1 - 378 T + p^{3} T^{2}
89 1+390T+p3T2 1 + 390 T + p^{3} T^{2}
97 1+1330T+p3T2 1 + 1330 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

−19.55131588676090196951499634617, −17.72055148312217158731195318477, −16.28749175378899646413987168836, −15.09686444946375758885149736496, −13.72439967177253770172734555098, −11.83139263343001526257239926146, −11.22780708054999787555304036773, −8.454298897419624859030526746703, −6.36455880219347397070988808395, −4.10728508949545943221697599589, 4.10728508949545943221697599589, 6.36455880219347397070988808395, 8.454298897419624859030526746703, 11.22780708054999787555304036773, 11.83139263343001526257239926146, 13.72439967177253770172734555098, 15.09686444946375758885149736496, 16.28749175378899646413987168836, 17.72055148312217158731195318477, 19.55131588676090196951499634617

Graph of the ZZ-function along the critical line