↑

Properties

Label 2-3700-1.1-c1-0-14
Degree 22
Conductor 37003700
Sign 11
Analytic cond. 29.544629.5446
Root an. cond. 5.435495.43549
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 2·7-s + 9-s + 6·11-s − 4·13-s − 19-s − 4·21-s + 3·23-s + 4·27-s + 8·31-s − 12·33-s + 37-s + 8·39-s + 3·41-s − 43-s − 6·47-s − 3·49-s − 9·53-s + 2·57-s − 3·59-s + 14·61-s + 2·63-s − 4·67-s − 6·69-s − 6·71-s + 11·73-s + 12·77-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.755·7-s + 1/3·9-s + 1.80·11-s − 1.10·13-s − 0.229·19-s − 0.872·21-s + 0.625·23-s + 0.769·27-s + 1.43·31-s − 2.08·33-s + 0.164·37-s + 1.28·39-s + 0.468·41-s − 0.152·43-s − 0.875·47-s − 3/7·49-s − 1.23·53-s + 0.264·57-s − 0.390·59-s + 1.79·61-s + 0.251·63-s − 0.488·67-s − 0.722·69-s − 0.712·71-s + 1.28·73-s + 1.36·77-s + ⋯

Functional equation

Λ(s)=(3700s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(3700s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 37003700    =    22⋅52⋅372^{2} \cdot 5^{2} \cdot 37
Sign: 11
Analytic conductor: 29.544629.5446
Root analytic conductor: 5.435495.43549
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 3700, ( :1/2), 1)(2,\ 3700,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.3678728191.367872819
L(12)L(\frac12) ≈\approx 1.3678728191.367872819
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 1
5 1 1
37 1−T 1 - T
good3 1+2T+pT2 1 + 2 T + p T^{2}
7 1−2T+pT2 1 - 2 T + p T^{2}
11 1−6T+pT2 1 - 6 T + p T^{2}
13 1+4T+pT2 1 + 4 T + p T^{2}
17 1+pT2 1 + p T^{2}
19 1+T+pT2 1 + T + p T^{2}
23 1−3T+pT2 1 - 3 T + p T^{2}
29 1+pT2 1 + p T^{2}
31 1−8T+pT2 1 - 8 T + p T^{2}
41 1−3T+pT2 1 - 3 T + p T^{2}
43 1+T+pT2 1 + T + p T^{2}
47 1+6T+pT2 1 + 6 T + p T^{2}
53 1+9T+pT2 1 + 9 T + p T^{2}
59 1+3T+pT2 1 + 3 T + p T^{2}
61 1−14T+pT2 1 - 14 T + p T^{2}
67 1+4T+pT2 1 + 4 T + p T^{2}
71 1+6T+pT2 1 + 6 T + p T^{2}
73 1−11T+pT2 1 - 11 T + p T^{2}
79 1+T+pT2 1 + T + p T^{2}
83 1+pT2 1 + p T^{2}
89 1−12T+pT2 1 - 12 T + p T^{2}
97 1−2T+pT2 1 - 2 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

−8.506863405659471229977930285776, −7.74407899803003989638143010044, −6.67804883401462601781504454328, −6.50108919407440481405845477859, −5.46337353224654334009786014845, −4.77402321633752819724827727350, −4.22545534036621367303405162766, −2.99256630433841580776509883235, −1.73344867858359236084414018866, −0.75607631634855246623818726703, 0.75607631634855246623818726703, 1.73344867858359236084414018866, 2.99256630433841580776509883235, 4.22545534036621367303405162766, 4.77402321633752819724827727350, 5.46337353224654334009786014845, 6.50108919407440481405845477859, 6.67804883401462601781504454328, 7.74407899803003989638143010044, 8.506863405659471229977930285776

Graph of the ZZ-function along the critical line