Properties

Label 2-275-1.1-c5-0-3
Degree 22
Conductor 275275
Sign 11
Analytic cond. 44.105544.1055
Root an. cond. 6.641206.64120
Motivic weight 55
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 10.3·2-s − 3.25·3-s + 75.7·4-s + 33.7·6-s − 50.5·7-s − 454.·8-s − 232.·9-s − 121·11-s − 246.·12-s − 990.·13-s + 525.·14-s + 2.29e3·16-s − 427.·17-s + 2.41e3·18-s + 2.33e3·19-s + 164.·21-s + 1.25e3·22-s − 4.20e3·23-s + 1.47e3·24-s + 1.02e4·26-s + 1.54e3·27-s − 3.83e3·28-s + 1.54e3·29-s − 6.33e3·31-s − 9.26e3·32-s + 393.·33-s + 4.43e3·34-s + ⋯
L(s)  = 1  − 1.83·2-s − 0.208·3-s + 2.36·4-s + 0.383·6-s − 0.390·7-s − 2.51·8-s − 0.956·9-s − 0.301·11-s − 0.494·12-s − 1.62·13-s + 0.716·14-s + 2.24·16-s − 0.358·17-s + 1.75·18-s + 1.48·19-s + 0.0814·21-s + 0.553·22-s − 1.65·23-s + 0.524·24-s + 2.98·26-s + 0.408·27-s − 0.924·28-s + 0.340·29-s − 1.18·31-s − 1.60·32-s + 0.0629·33-s + 0.658·34-s + ⋯

Functional equation

Λ(s)=(275s/2ΓC(s)L(s)=(Λ(6s)\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
Λ(s)=(275s/2ΓC(s+5/2)L(s)=(Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 275275    =    52115^{2} \cdot 11
Sign: 11
Analytic conductor: 44.105544.1055
Root analytic conductor: 6.641206.64120
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 275, ( :5/2), 1)(2,\ 275,\ (\ :5/2),\ 1)

Particular Values

L(3)L(3) \approx 0.21409899190.2140989919
L(12)L(\frac12) \approx 0.21409899190.2140989919
L(72)L(\frac{7}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad5 1 1
11 1+121T 1 + 121T
good2 1+10.3T+32T2 1 + 10.3T + 32T^{2}
3 1+3.25T+243T2 1 + 3.25T + 243T^{2}
7 1+50.5T+1.68e4T2 1 + 50.5T + 1.68e4T^{2}
13 1+990.T+3.71e5T2 1 + 990.T + 3.71e5T^{2}
17 1+427.T+1.41e6T2 1 + 427.T + 1.41e6T^{2}
19 12.33e3T+2.47e6T2 1 - 2.33e3T + 2.47e6T^{2}
23 1+4.20e3T+6.43e6T2 1 + 4.20e3T + 6.43e6T^{2}
29 11.54e3T+2.05e7T2 1 - 1.54e3T + 2.05e7T^{2}
31 1+6.33e3T+2.86e7T2 1 + 6.33e3T + 2.86e7T^{2}
37 11.21e3T+6.93e7T2 1 - 1.21e3T + 6.93e7T^{2}
41 1+1.49e4T+1.15e8T2 1 + 1.49e4T + 1.15e8T^{2}
43 11.79e3T+1.47e8T2 1 - 1.79e3T + 1.47e8T^{2}
47 11.66e3T+2.29e8T2 1 - 1.66e3T + 2.29e8T^{2}
53 1+1.19e4T+4.18e8T2 1 + 1.19e4T + 4.18e8T^{2}
59 1+1.63e4T+7.14e8T2 1 + 1.63e4T + 7.14e8T^{2}
61 1+6.93e3T+8.44e8T2 1 + 6.93e3T + 8.44e8T^{2}
67 1+4.86e3T+1.35e9T2 1 + 4.86e3T + 1.35e9T^{2}
71 1+1.88e4T+1.80e9T2 1 + 1.88e4T + 1.80e9T^{2}
73 16.61e4T+2.07e9T2 1 - 6.61e4T + 2.07e9T^{2}
79 18.30e4T+3.07e9T2 1 - 8.30e4T + 3.07e9T^{2}
83 1+5.98e3T+3.93e9T2 1 + 5.98e3T + 3.93e9T^{2}
89 14.84e4T+5.58e9T2 1 - 4.84e4T + 5.58e9T^{2}
97 1+1.09e5T+8.58e9T2 1 + 1.09e5T + 8.58e9T^{2}
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   L(s)=p j=12(1αj,pps)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

−10.76640571532332951063037904829, −9.837214606078333535755446728915, −9.330687308231510067369370362003, −8.162992714569501733990685766102, −7.47485158610776787015991343584, −6.44884762413589452063930017343, −5.27945466485864042904645586997, −3.06434062027717690144530819571, −1.99392290195506979509547184689, −0.33062775347875548383636817176, 0.33062775347875548383636817176, 1.99392290195506979509547184689, 3.06434062027717690144530819571, 5.27945466485864042904645586997, 6.44884762413589452063930017343, 7.47485158610776787015991343584, 8.162992714569501733990685766102, 9.330687308231510067369370362003, 9.837214606078333535755446728915, 10.76640571532332951063037904829

Graph of the ZZ-function along the critical line