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Properties

Label 2-3724-1.1-c1-0-19
Degree 22
Conductor 37243724
Sign 11
Analytic cond. 29.736229.7362
Root an. cond. 5.453095.45309
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 5-s + 9-s + 5·11-s + 4·13-s − 2·15-s + 3·17-s + 19-s + 8·23-s − 4·25-s + 4·27-s − 2·29-s − 4·31-s − 10·33-s + 10·37-s − 8·39-s − 10·41-s + 43-s + 45-s + 47-s − 6·51-s − 4·53-s + 5·55-s − 2·57-s − 6·59-s + 13·61-s + 4·65-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.447·5-s + 1/3·9-s + 1.50·11-s + 1.10·13-s − 0.516·15-s + 0.727·17-s + 0.229·19-s + 1.66·23-s − 4/5·25-s + 0.769·27-s − 0.371·29-s − 0.718·31-s − 1.74·33-s + 1.64·37-s − 1.28·39-s − 1.56·41-s + 0.152·43-s + 0.149·45-s + 0.145·47-s − 0.840·51-s − 0.549·53-s + 0.674·55-s − 0.264·57-s − 0.781·59-s + 1.66·61-s + 0.496·65-s + ⋯

Functional equation

Λ(s)=(3724s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(3724s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 37243724    =    22⋅72⋅192^{2} \cdot 7^{2} \cdot 19
Sign: 11
Analytic conductor: 29.736229.7362
Root analytic conductor: 5.453095.45309
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 3724, ( :1/2), 1)(2,\ 3724,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.6443013091.644301309
L(12)L(\frac12) ≈\approx 1.6443013091.644301309
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
7 1 1
19 1−T 1 - T
good3 1+2T+pT2 1 + 2 T + p T^{2}
5 1−T+pT2 1 - T + p T^{2}
11 1−5T+pT2 1 - 5 T + p T^{2}
13 1−4T+pT2 1 - 4 T + p T^{2}
17 1−3T+pT2 1 - 3 T + p T^{2}
23 1−8T+pT2 1 - 8 T + p T^{2}
29 1+2T+pT2 1 + 2 T + p T^{2}
31 1+4T+pT2 1 + 4 T + p T^{2}
37 1−10T+pT2 1 - 10 T + p T^{2}
41 1+10T+pT2 1 + 10 T + p T^{2}
43 1−T+pT2 1 - T + p T^{2}
47 1−T+pT2 1 - T + p T^{2}
53 1+4T+pT2 1 + 4 T + p T^{2}
59 1+6T+pT2 1 + 6 T + p T^{2}
61 1−13T+pT2 1 - 13 T + p T^{2}
67 1+12T+pT2 1 + 12 T + p T^{2}
71 1−2T+pT2 1 - 2 T + p T^{2}
73 1+9T+pT2 1 + 9 T + p T^{2}
79 1−8T+pT2 1 - 8 T + p T^{2}
83 1−12T+pT2 1 - 12 T + p T^{2}
89 1+12T+pT2 1 + 12 T + p T^{2}
97 1−8T+pT2 1 - 8 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

−8.709339720534257240327886562953, −7.64552271968190118551531459773, −6.76474142604661213574836140046, −6.21156913138879588632301451179, −5.67409965982210077643645681036, −4.91419379168876950061294361671, −3.93593184874987401028815060034, −3.14091634253431942337825536555, −1.62016711898696823707962362202, −0.869557064463155631256543449078, 0.869557064463155631256543449078, 1.62016711898696823707962362202, 3.14091634253431942337825536555, 3.93593184874987401028815060034, 4.91419379168876950061294361671, 5.67409965982210077643645681036, 6.21156913138879588632301451179, 6.76474142604661213574836140046, 7.64552271968190118551531459773, 8.709339720534257240327886562953

Graph of the ZZ-function along the critical line