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Properties

Label 2-2550-1.1-c1-0-19
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 + 4·7-s − 8-s + 9-s − 2·11-s + 12-s + 6·13-s − 4·14-s + 16-s + 17-s − 18-s + 4·19-s + 4·21-s + 2·22-s − 5·23-s − 24-s − 6·26-s + 27-s + 4·28-s + 10·31-s − 32-s − 2·33-s − 34-s + 36-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s + 1.51·7-s − 0.353·8-s + 1/3·9-s − 0.603·11-s + 0.288·12-s + 1.66·13-s − 1.06·14-s + 1/4·16-s + 0.242·17-s − 0.235·18-s + 0.917·19-s + 0.872·21-s + 0.426·22-s − 1.04·23-s − 0.204·24-s − 1.17·26-s + 0.192·27-s + 0.755·28-s + 1.79·31-s − 0.176·32-s − 0.348·33-s − 0.171·34-s + 1/6·36-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 2.1473512022.147351202
L(12)L(\frac12) ≈\approx 2.1473512022.147351202
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 1−4T+pT2 1 - 4 T + p T^{2}
11 1+2T+pT2 1 + 2 T + p T^{2}
13 1−6T+pT2 1 - 6 T + p T^{2}
19 1−4T+pT2 1 - 4 T + p T^{2}
23 1+5T+pT2 1 + 5 T + p T^{2}
29 1+pT2 1 + p T^{2}
31 1−10T+pT2 1 - 10 T + p T^{2}
37 1+9T+pT2 1 + 9 T + p T^{2}
41 1−11T+pT2 1 - 11 T + p T^{2}
43 1+10T+pT2 1 + 10 T + p T^{2}
47 1+8T+pT2 1 + 8 T + p T^{2}
53 1−11T+pT2 1 - 11 T + p T^{2}
59 1+15T+pT2 1 + 15 T + p T^{2}
61 1+T+pT2 1 + T + p T^{2}
67 1−14T+pT2 1 - 14 T + p T^{2}
71 1−11T+pT2 1 - 11 T + p T^{2}
73 1+8T+pT2 1 + 8 T + p T^{2}
79 1+8T+pT2 1 + 8 T + p T^{2}
83 1−5T+pT2 1 - 5 T + p T^{2}
89 1+6T+pT2 1 + 6 T + p T^{2}
97 1+8T+pT2 1 + 8 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

−8.605637047293088628845868985582, −8.188287204909209328453849372785, −7.80740081157824912500538828494, −6.79463681343707149067236637719, −5.84084108193046638467245630585, −4.99585376514666012367397104697, −3.99225334470857281631445650152, −3.01308414951224896614167330334, −1.88247361842834831005657001491, −1.11251462700235043615266083539, 1.11251462700235043615266083539, 1.88247361842834831005657001491, 3.01308414951224896614167330334, 3.99225334470857281631445650152, 4.99585376514666012367397104697, 5.84084108193046638467245630585, 6.79463681343707149067236637719, 7.80740081157824912500538828494, 8.188287204909209328453849372785, 8.605637047293088628845868985582

Graph of the ZZ-function along the critical line