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Properties

Label 2-3700-1.1-c1-0-38
Degree 22
Conductor 37003700
Sign −1-1
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 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s + 3·7-s + 6·9-s + 5·11-s − 2·13-s − 4·17-s − 4·19-s − 9·21-s − 6·23-s − 9·27-s + 6·29-s − 4·31-s − 15·33-s + 37-s + 6·39-s − 9·41-s − 10·43-s + 11·47-s + 2·49-s + 12·51-s + 11·53-s + 12·57-s − 8·59-s − 8·61-s + 18·63-s + 8·67-s + 18·69-s + ⋯
L(s)  = 1  − 1.73·3-s + 1.13·7-s + 2·9-s + 1.50·11-s − 0.554·13-s − 0.970·17-s − 0.917·19-s − 1.96·21-s − 1.25·23-s − 1.73·27-s + 1.11·29-s − 0.718·31-s − 2.61·33-s + 0.164·37-s + 0.960·39-s − 1.40·41-s − 1.52·43-s + 1.60·47-s + 2/7·49-s + 1.68·51-s + 1.51·53-s + 1.58·57-s − 1.04·59-s − 1.02·61-s + 2.26·63-s + 0.977·67-s + 2.16·69-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: −1-1
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: 11
Selberg data: (2, 3700, ( :1/2), −1)(2,\ 3700,\ (\ :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 1
5 1 1
37 1−T 1 - T
good3 1+pT+pT2 1 + p T + p T^{2}
7 1−3T+pT2 1 - 3 T + p T^{2}
11 1−5T+pT2 1 - 5 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+4T+pT2 1 + 4 T + p T^{2}
23 1+6T+pT2 1 + 6 T + p T^{2}
29 1−6T+pT2 1 - 6 T + p T^{2}
31 1+4T+pT2 1 + 4 T + p T^{2}
41 1+9T+pT2 1 + 9 T + p T^{2}
43 1+10T+pT2 1 + 10 T + p T^{2}
47 1−11T+pT2 1 - 11 T + p T^{2}
53 1−11T+pT2 1 - 11 T + p T^{2}
59 1+8T+pT2 1 + 8 T + p T^{2}
61 1+8T+pT2 1 + 8 T + p T^{2}
67 1−8T+pT2 1 - 8 T + p T^{2}
71 1−3T+pT2 1 - 3 T + p T^{2}
73 1+7T+pT2 1 + 7 T + p T^{2}
79 1−8T+pT2 1 - 8 T + p T^{2}
83 1−9T+pT2 1 - 9 T + p T^{2}
89 1+16T+pT2 1 + 16 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

−8.154290748669424388477852300389, −7.07577695780277543069908710611, −6.60981396730997064078221589991, −5.96334860425801266051119721220, −5.09888884088066200767920008704, −4.48295490912051991181364428829, −3.93322311807360794055015553666, −2.09146483456977231198455242250, −1.30728896437273467280345954384, 0, 1.30728896437273467280345954384, 2.09146483456977231198455242250, 3.93322311807360794055015553666, 4.48295490912051991181364428829, 5.09888884088066200767920008704, 5.96334860425801266051119721220, 6.60981396730997064078221589991, 7.07577695780277543069908710611, 8.154290748669424388477852300389

Graph of the ZZ-function along the critical line