↑

Properties

Label 2-45e2-1.1-c1-0-46
Degree 22
Conductor 20252025
Sign −1-1
Analytic cond. 16.169716.1697
Root an. cond. 4.021154.02115
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s − 2·7-s + 3·11-s + 4·13-s + 4·16-s − 6·17-s − 19-s − 6·23-s + 4·28-s + 9·29-s − 31-s − 8·37-s − 3·41-s + 4·43-s − 6·44-s + 12·47-s − 3·49-s − 8·52-s + 6·53-s − 3·59-s − 10·61-s − 8·64-s − 14·67-s + 12·68-s + 3·71-s − 2·73-s + 2·76-s + ⋯
L(s)  = 1  − 4-s − 0.755·7-s + 0.904·11-s + 1.10·13-s + 16-s − 1.45·17-s − 0.229·19-s − 1.25·23-s + 0.755·28-s + 1.67·29-s − 0.179·31-s − 1.31·37-s − 0.468·41-s + 0.609·43-s − 0.904·44-s + 1.75·47-s − 3/7·49-s − 1.10·52-s + 0.824·53-s − 0.390·59-s − 1.28·61-s − 64-s − 1.71·67-s + 1.45·68-s + 0.356·71-s − 0.234·73-s + 0.229·76-s + ⋯

Functional equation

Λ(s)=(2025s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(2025s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 20252025    =    34⋅523^{4} \cdot 5^{2}
Sign: −1-1
Analytic conductor: 16.169716.1697
Root analytic conductor: 4.021154.02115
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 2025, ( :1/2), −1)(2,\ 2025,\ (\ :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
good2 1+pT2 1 + p T^{2}
7 1+2T+pT2 1 + 2 T + p T^{2}
11 1−3T+pT2 1 - 3 T + p T^{2}
13 1−4T+pT2 1 - 4 T + p T^{2}
17 1+6T+pT2 1 + 6 T + p T^{2}
19 1+T+pT2 1 + T + p T^{2}
23 1+6T+pT2 1 + 6 T + p T^{2}
29 1−9T+pT2 1 - 9 T + p T^{2}
31 1+T+pT2 1 + T + p T^{2}
37 1+8T+pT2 1 + 8 T + p T^{2}
41 1+3T+pT2 1 + 3 T + p T^{2}
43 1−4T+pT2 1 - 4 T + p T^{2}
47 1−12T+pT2 1 - 12 T + p T^{2}
53 1−6T+pT2 1 - 6 T + p T^{2}
59 1+3T+pT2 1 + 3 T + p T^{2}
61 1+10T+pT2 1 + 10 T + p T^{2}
67 1+14T+pT2 1 + 14 T + p T^{2}
71 1−3T+pT2 1 - 3 T + p T^{2}
73 1+2T+pT2 1 + 2 T + p T^{2}
79 1+16T+pT2 1 + 16 T + p T^{2}
83 1+12T+pT2 1 + 12 T + p T^{2}
89 1+15T+pT2 1 + 15 T + p T^{2}
97 1−4T+pT2 1 - 4 T + p T^{2}
show more
show less
   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

−8.767747284753786098117223574742, −8.335548836313514290240539244878, −7.05773573312300104815129757004, −6.32468375473848937056457133649, −5.68073732624305661536135613556, −4.35117009626133411095198923203, −4.03163071651258240001742541940, −2.94203711656515483067854528639, −1.43817397608101885315913429842, 0, 1.43817397608101885315913429842, 2.94203711656515483067854528639, 4.03163071651258240001742541940, 4.35117009626133411095198923203, 5.68073732624305661536135613556, 6.32468375473848937056457133649, 7.05773573312300104815129757004, 8.335548836313514290240539244878, 8.767747284753786098117223574742

Graph of the ZZ-function along the critical line