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Properties

Label 2-550-1.1-c1-0-2
Degree 22
Conductor 550550
Sign 11
Analytic cond. 4.391774.39177
Root an. cond. 2.095652.09565
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 − 2·9-s + 11-s + 12-s + 6·13-s + 3·14-s + 16-s + 7·17-s + 2·18-s + 5·19-s − 3·21-s − 22-s + 6·23-s − 24-s − 6·26-s − 5·27-s − 3·28-s + 5·29-s − 3·31-s − 32-s + 33-s − 7·34-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 − 2/3·9-s + 0.301·11-s + 0.288·12-s + 1.66·13-s + 0.801·14-s + 1/4·16-s + 1.69·17-s + 0.471·18-s + 1.14·19-s − 0.654·21-s − 0.213·22-s + 1.25·23-s − 0.204·24-s − 1.17·26-s − 0.962·27-s − 0.566·28-s + 0.928·29-s − 0.538·31-s − 0.176·32-s + 0.174·33-s − 1.20·34-s + ⋯

Functional equation

Λ(s)=(550s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(550s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 550550    =    2⋅52⋅112 \cdot 5^{2} \cdot 11
Sign: 11
Analytic conductor: 4.391774.39177
Root analytic conductor: 2.095652.09565
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 550, ( :1/2), 1)(2,\ 550,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.2160111661.216011166
L(12)L(\frac12) ≈\approx 1.2160111661.216011166
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
5 1 1
11 1−T 1 - T
good3 1−T+pT2 1 - T + p T^{2}
7 1+3T+pT2 1 + 3 T + p T^{2}
13 1−6T+pT2 1 - 6 T + p T^{2}
17 1−7T+pT2 1 - 7 T + p T^{2}
19 1−5T+pT2 1 - 5 T + p T^{2}
23 1−6T+pT2 1 - 6 T + p T^{2}
29 1−5T+pT2 1 - 5 T + p T^{2}
31 1+3T+pT2 1 + 3 T + p T^{2}
37 1+3T+pT2 1 + 3 T + p T^{2}
41 1−2T+pT2 1 - 2 T + p T^{2}
43 1+4T+pT2 1 + 4 T + p T^{2}
47 1−2T+pT2 1 - 2 T + p T^{2}
53 1−T+pT2 1 - T + p T^{2}
59 1+10T+pT2 1 + 10 T + p T^{2}
61 1−7T+pT2 1 - 7 T + p T^{2}
67 1+8T+pT2 1 + 8 T + p T^{2}
71 1−7T+pT2 1 - 7 T + p T^{2}
73 1+14T+pT2 1 + 14 T + p T^{2}
79 1−10T+pT2 1 - 10 T + p T^{2}
83 1−6T+pT2 1 - 6 T + p T^{2}
89 1+15T+pT2 1 + 15 T + p T^{2}
97 1−12T+pT2 1 - 12 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

−10.62761614296171945475675158355, −9.696869758195084295021589593937, −9.051313158030191365780373658814, −8.319735201663290337491520314526, −7.35838560113530362823348866724, −6.32452856749704006618306134382, −5.51294812234663797155432190910, −3.46996281594193181675719351221, −3.07694177217990984214438626883, −1.14442070453994046878747912184, 1.14442070453994046878747912184, 3.07694177217990984214438626883, 3.46996281594193181675719351221, 5.51294812234663797155432190910, 6.32452856749704006618306134382, 7.35838560113530362823348866724, 8.319735201663290337491520314526, 9.051313158030191365780373658814, 9.696869758195084295021589593937, 10.62761614296171945475675158355

Graph of the ZZ-function along the critical line