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Properties

Label 2-1575-1.1-c1-0-41
Degree 22
Conductor 15751575
Sign −1-1
Analytic cond. 12.576412.5764
Root an. cond. 3.546323.54632
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 − 4-s − 7-s − 3·8-s + 6·11-s − 2·13-s − 14-s − 16-s − 4·17-s − 6·19-s + 6·22-s − 2·26-s + 28-s + 2·29-s − 10·31-s + 5·32-s − 4·34-s − 4·37-s − 6·38-s − 2·41-s − 4·43-s − 6·44-s + 49-s + 2·52-s − 6·53-s + 3·56-s + 2·58-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s − 0.377·7-s − 1.06·8-s + 1.80·11-s − 0.554·13-s − 0.267·14-s − 1/4·16-s − 0.970·17-s − 1.37·19-s + 1.27·22-s − 0.392·26-s + 0.188·28-s + 0.371·29-s − 1.79·31-s + 0.883·32-s − 0.685·34-s − 0.657·37-s − 0.973·38-s − 0.312·41-s − 0.609·43-s − 0.904·44-s + 1/7·49-s + 0.277·52-s − 0.824·53-s + 0.400·56-s + 0.262·58-s + ⋯

Functional equation

Λ(s)=(1575s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(1575s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 15751575    =    32⋅52⋅73^{2} \cdot 5^{2} \cdot 7
Sign: −1-1
Analytic conductor: 12.576412.5764
Root analytic conductor: 3.546323.54632
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 1575, ( :1/2), −1)(2,\ 1575,\ (\ :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)
bad3 1 1
5 1 1
7 1+T 1 + T
good2 1−T+pT2 1 - T + p T^{2}
11 1−6T+pT2 1 - 6 T + p T^{2}
13 1+2T+pT2 1 + 2 T + p T^{2}
17 1+4T+pT2 1 + 4 T + p T^{2}
19 1+6T+pT2 1 + 6 T + p T^{2}
23 1+pT2 1 + p T^{2}
29 1−2T+pT2 1 - 2 T + p T^{2}
31 1+10T+pT2 1 + 10 T + p T^{2}
37 1+4T+pT2 1 + 4 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+pT2 1 + p T^{2}
53 1+6T+pT2 1 + 6 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+16T+pT2 1 + 16 T + p T^{2}
71 1+10T+pT2 1 + 10 T + p T^{2}
73 1+6T+pT2 1 + 6 T + p T^{2}
79 1−4T+pT2 1 - 4 T + p T^{2}
83 1+8T+pT2 1 + 8 T + p T^{2}
89 1+6T+pT2 1 + 6 T + p T^{2}
97 1+2T+pT2 1 + 2 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

−8.971552311057573237641723762458, −8.564726035632039332608333026566, −7.11999628519012507821966096100, −6.49227489268027016194575261734, −5.75408159665580195145733679557, −4.60578903619236206845761227834, −4.08242632050905699099504942411, −3.20269986936965107647955994590, −1.82947662480420584992219247222, 0, 1.82947662480420584992219247222, 3.20269986936965107647955994590, 4.08242632050905699099504942411, 4.60578903619236206845761227834, 5.75408159665580195145733679557, 6.49227489268027016194575261734, 7.11999628519012507821966096100, 8.564726035632039332608333026566, 8.971552311057573237641723762458

Graph of the ZZ-function along the critical line