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Properties

Label 2-2550-1.1-c1-0-3
Degree 22
Conductor 25502550
Sign 11
Analytic cond. 20.361820.3618
Root an. cond. 4.512414.51241
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s − 6-s − 3·7-s − 8-s + 9-s − 5·11-s + 12-s + 2·13-s + 3·14-s + 16-s − 17-s − 18-s + 19-s − 3·21-s + 5·22-s − 6·23-s − 24-s − 2·26-s + 27-s − 3·28-s + 10·29-s + 5·31-s − 32-s − 5·33-s + 34-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 1.13·7-s − 0.353·8-s + 1/3·9-s − 1.50·11-s + 0.288·12-s + 0.554·13-s + 0.801·14-s + 1/4·16-s − 0.242·17-s − 0.235·18-s + 0.229·19-s − 0.654·21-s + 1.06·22-s − 1.25·23-s − 0.204·24-s − 0.392·26-s + 0.192·27-s − 0.566·28-s + 1.85·29-s + 0.898·31-s − 0.176·32-s − 0.870·33-s + 0.171·34-s + ⋯

Functional equation

Λ(s)=(2550s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(2550s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 25502550    =    2⋅3⋅52⋅172 \cdot 3 \cdot 5^{2} \cdot 17
Sign: 11
Analytic conductor: 20.361820.3618
Root analytic conductor: 4.512414.51241
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 2550, ( :1/2), 1)(2,\ 2550,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.1457414181.145741418
L(12)L(\frac12) ≈\approx 1.1457414181.145741418
L(32)L(\frac{3}{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+T 1 + T
3 1−T 1 - T
5 1 1
17 1+T 1 + T
good7 1+3T+pT2 1 + 3 T + p T^{2}
11 1+5T+pT2 1 + 5 T + p T^{2}
13 1−2T+pT2 1 - 2 T + p T^{2}
19 1−T+pT2 1 - T + p T^{2}
23 1+6T+pT2 1 + 6 T + p T^{2}
29 1−10T+pT2 1 - 10 T + p T^{2}
31 1−5T+pT2 1 - 5 T + p T^{2}
37 1−3T+pT2 1 - 3 T + p T^{2}
41 1−6T+pT2 1 - 6 T + p T^{2}
43 1−T+pT2 1 - T + p T^{2}
47 1+3T+pT2 1 + 3 T + p T^{2}
53 1−T+pT2 1 - T + p T^{2}
59 1−8T+pT2 1 - 8 T + p T^{2}
61 1+2T+pT2 1 + 2 T + p T^{2}
67 1−11T+pT2 1 - 11 T + p T^{2}
71 1−6T+pT2 1 - 6 T + p T^{2}
73 1−12T+pT2 1 - 12 T + p T^{2}
79 1−5T+pT2 1 - 5 T + p T^{2}
83 1+18T+pT2 1 + 18 T + p T^{2}
89 1−12T+pT2 1 - 12 T + p T^{2}
97 1−14T+pT2 1 - 14 T + p 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

−8.810504122961107276296448912027, −8.166360002861459473388767211292, −7.65413840568452465144147907114, −6.61072040183152185812517360312, −6.11532498490045678119089308595, −4.99734969623358779987712591625, −3.84795020439861036846716122060, −2.90043687921487017392042208715, −2.30560140959508307974254558712, −0.70376376823350907184750688404, 0.70376376823350907184750688404, 2.30560140959508307974254558712, 2.90043687921487017392042208715, 3.84795020439861036846716122060, 4.99734969623358779987712591625, 6.11532498490045678119089308595, 6.61072040183152185812517360312, 7.65413840568452465144147907114, 8.166360002861459473388767211292, 8.810504122961107276296448912027

Graph of the ZZ-function along the critical line