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Properties

Label 2-2550-1.1-c1-0-45
Degree 22
Conductor 25502550
Sign −1-1
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 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s + 4-s − 6-s − 7-s + 8-s + 9-s − 3·11-s − 12-s + 4·13-s − 14-s + 16-s − 17-s + 18-s − 5·19-s + 21-s − 3·22-s − 8·23-s − 24-s + 4·26-s − 27-s − 28-s − 4·29-s − 3·31-s + 32-s + 3·33-s − 34-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.904·11-s − 0.288·12-s + 1.10·13-s − 0.267·14-s + 1/4·16-s − 0.242·17-s + 0.235·18-s − 1.14·19-s + 0.218·21-s − 0.639·22-s − 1.66·23-s − 0.204·24-s + 0.784·26-s − 0.192·27-s − 0.188·28-s − 0.742·29-s − 0.538·31-s + 0.176·32-s + 0.522·33-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: −1-1
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: 11
Selberg data: (2, 2550, ( :1/2), −1)(2,\ 2550,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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+T+pT2 1 + T + p T^{2}
11 1+3T+pT2 1 + 3 T + p T^{2}
13 1−4T+pT2 1 - 4 T + p T^{2}
19 1+5T+pT2 1 + 5 T + p T^{2}
23 1+8T+pT2 1 + 8 T + p T^{2}
29 1+4T+pT2 1 + 4 T + p T^{2}
31 1+3T+pT2 1 + 3 T + p T^{2}
37 1−7T+pT2 1 - 7 T + p T^{2}
41 1−2T+pT2 1 - 2 T + p T^{2}
43 1−T+pT2 1 - T + p T^{2}
47 1−7T+pT2 1 - 7 T + p T^{2}
53 1+7T+pT2 1 + 7 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 1−2T+pT2 1 - 2 T + p T^{2}
79 1+15T+pT2 1 + 15 T + p T^{2}
83 1+16T+pT2 1 + 16 T + p T^{2}
89 1−2T+pT2 1 - 2 T + p T^{2}
97 1+10T+pT2 1 + 10 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

−8.328679747447743988696308911741, −7.70514363062754001760074818196, −6.70805793765460903024093009589, −6.01105343681128848375965087529, −5.59357704065219692752888750061, −4.42415495348840661004046299825, −3.89086747195596752872647341737, −2.74556657861042306546288376856, −1.69697211485226681614066053159, 0, 1.69697211485226681614066053159, 2.74556657861042306546288376856, 3.89086747195596752872647341737, 4.42415495348840661004046299825, 5.59357704065219692752888750061, 6.01105343681128848375965087529, 6.70805793765460903024093009589, 7.70514363062754001760074818196, 8.328679747447743988696308911741

Graph of the ZZ-function along the critical line