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Properties

Label 2-2400-1.1-c1-0-16
Degree 22
Conductor 24002400
Sign 11
Analytic cond. 19.164019.1640
Root an. cond. 4.377684.37768
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  − 3-s + 4·7-s + 9-s + 4·11-s + 2·13-s + 6·17-s − 4·19-s − 4·21-s − 27-s + 2·29-s + 4·31-s − 4·33-s + 2·37-s − 2·39-s + 2·41-s − 4·43-s − 8·47-s + 9·49-s − 6·51-s − 10·53-s + 4·57-s − 4·59-s + 6·61-s + 4·63-s − 4·67-s − 16·71-s + 6·73-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.51·7-s + 1/3·9-s + 1.20·11-s + 0.554·13-s + 1.45·17-s − 0.917·19-s − 0.872·21-s − 0.192·27-s + 0.371·29-s + 0.718·31-s − 0.696·33-s + 0.328·37-s − 0.320·39-s + 0.312·41-s − 0.609·43-s − 1.16·47-s + 9/7·49-s − 0.840·51-s − 1.37·53-s + 0.529·57-s − 0.520·59-s + 0.768·61-s + 0.503·63-s − 0.488·67-s − 1.89·71-s + 0.702·73-s + ⋯

Functional equation

Λ(s)=(2400s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(2400s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 24002400    =    25⋅3⋅522^{5} \cdot 3 \cdot 5^{2}
Sign: 11
Analytic conductor: 19.164019.1640
Root analytic conductor: 4.377684.37768
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 2400, ( :1/2), 1)(2,\ 2400,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.0954999502.095499950
L(12)L(\frac12) ≈\approx 2.0954999502.095499950
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
3 1+T 1 + T
5 1 1
good7 1−4T+pT2 1 - 4 T + p T^{2}
11 1−4T+pT2 1 - 4 T + p T^{2}
13 1−2T+pT2 1 - 2 T + p T^{2}
17 1−6T+pT2 1 - 6 T + p T^{2}
19 1+4T+pT2 1 + 4 T + p T^{2}
23 1+pT2 1 + p T^{2}
29 1−2T+pT2 1 - 2 T + p T^{2}
31 1−4T+pT2 1 - 4 T + p T^{2}
37 1−2T+pT2 1 - 2 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+8T+pT2 1 + 8 T + p T^{2}
53 1+10T+pT2 1 + 10 T + p T^{2}
59 1+4T+pT2 1 + 4 T + p T^{2}
61 1−6T+pT2 1 - 6 T + p T^{2}
67 1+4T+pT2 1 + 4 T + p T^{2}
71 1+16T+pT2 1 + 16 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+12T+pT2 1 + 12 T + p T^{2}
89 1−10T+pT2 1 - 10 T + p T^{2}
97 1−14T+pT2 1 - 14 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

−8.824046048556980758736864363820, −8.187690885601832957542462060811, −7.51599841504707801181092960196, −6.48884074088343388628451242584, −5.91022112565038900100369808945, −4.90911980616858985140087544886, −4.34933392140332560567015192416, −3.33595145158661942591347021628, −1.79509095105059034627127171893, −1.07872352480771079736145349824, 1.07872352480771079736145349824, 1.79509095105059034627127171893, 3.33595145158661942591347021628, 4.34933392140332560567015192416, 4.90911980616858985140087544886, 5.91022112565038900100369808945, 6.48884074088343388628451242584, 7.51599841504707801181092960196, 8.187690885601832957542462060811, 8.824046048556980758736864363820

Graph of the ZZ-function along the critical line