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Properties

Label 2-5-1.1-c3-0-0
Degree 22
Conductor 55
Sign 11
Analytic cond. 0.2950090.295009
Root an. cond. 0.5431470.543147
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  − 4·2-s + 2·3-s + 8·4-s − 5·5-s − 8·6-s + 6·7-s − 23·9-s + 20·10-s + 32·11-s + 16·12-s − 38·13-s − 24·14-s − 10·15-s − 64·16-s + 26·17-s + 92·18-s + 100·19-s − 40·20-s + 12·21-s − 128·22-s − 78·23-s + 25·25-s + 152·26-s − 100·27-s + 48·28-s − 50·29-s + 40·30-s + ⋯
L(s)  = 1  − 1.41·2-s + 0.384·3-s + 4-s − 0.447·5-s − 0.544·6-s + 0.323·7-s − 0.851·9-s + 0.632·10-s + 0.877·11-s + 0.384·12-s − 0.810·13-s − 0.458·14-s − 0.172·15-s − 16-s + 0.370·17-s + 1.20·18-s + 1.20·19-s − 0.447·20-s + 0.124·21-s − 1.24·22-s − 0.707·23-s + 1/5·25-s + 1.14·26-s − 0.712·27-s + 0.323·28-s − 0.320·29-s + 0.243·30-s + ⋯

Functional equation

Λ(s)=(5s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(5s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 55
Sign: 11
Analytic conductor: 0.2950090.295009
Root analytic conductor: 0.5431470.543147
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 5, ( :3/2), 1)(2,\ 5,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 0.41186132830.4118613283
L(12)L(\frac12) ≈\approx 0.41186132830.4118613283
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)
bad5 1+pT 1 + p T
good2 1+p2T+p3T2 1 + p^{2} T + p^{3} T^{2}
3 1−2T+p3T2 1 - 2 T + p^{3} T^{2}
7 1−6T+p3T2 1 - 6 T + p^{3} T^{2}
11 1−32T+p3T2 1 - 32 T + p^{3} T^{2}
13 1+38T+p3T2 1 + 38 T + p^{3} T^{2}
17 1−26T+p3T2 1 - 26 T + p^{3} T^{2}
19 1−100T+p3T2 1 - 100 T + p^{3} T^{2}
23 1+78T+p3T2 1 + 78 T + p^{3} T^{2}
29 1+50T+p3T2 1 + 50 T + p^{3} T^{2}
31 1+108T+p3T2 1 + 108 T + p^{3} T^{2}
37 1−266T+p3T2 1 - 266 T + p^{3} T^{2}
41 1−22T+p3T2 1 - 22 T + p^{3} T^{2}
43 1−442T+p3T2 1 - 442 T + p^{3} T^{2}
47 1+514T+p3T2 1 + 514 T + p^{3} T^{2}
53 1−2T+p3T2 1 - 2 T + p^{3} T^{2}
59 1−500T+p3T2 1 - 500 T + p^{3} T^{2}
61 1+518T+p3T2 1 + 518 T + p^{3} T^{2}
67 1−126T+p3T2 1 - 126 T + p^{3} T^{2}
71 1−412T+p3T2 1 - 412 T + p^{3} T^{2}
73 1+878T+p3T2 1 + 878 T + p^{3} T^{2}
79 1−600T+p3T2 1 - 600 T + p^{3} T^{2}
83 1−282T+p3T2 1 - 282 T + p^{3} T^{2}
89 1+150T+p3T2 1 + 150 T + p^{3} T^{2}
97 1−386T+p3T2 1 - 386 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

−24.53719930434053859680587393842, −22.51370164112999450106246009488, −20.24253638460194980453802000495, −19.39406085946034311371495394382, −17.80936002011203375803360468238, −16.49202096766309066044503764833, −14.42700262096538081523197027989, −11.52156568135998034842009582479, −9.415016459828749011981823541780, −7.80368599340441310768273857061, 7.80368599340441310768273857061, 9.415016459828749011981823541780, 11.52156568135998034842009582479, 14.42700262096538081523197027989, 16.49202096766309066044503764833, 17.80936002011203375803360468238, 19.39406085946034311371495394382, 20.24253638460194980453802000495, 22.51370164112999450106246009488, 24.53719930434053859680587393842

Graph of the ZZ-function along the critical line