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Properties

Label 2-1920-1.1-c1-0-9
Degree 22
Conductor 19201920
Sign 11
Analytic cond. 15.331215.3312
Root an. cond. 3.915513.91551
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 − 5-s + 2·7-s + 9-s − 2·11-s + 6·13-s − 15-s − 6·17-s + 6·19-s + 2·21-s − 2·23-s + 25-s + 27-s + 2·29-s + 4·31-s − 2·33-s − 2·35-s + 10·37-s + 6·39-s − 2·41-s − 8·43-s − 45-s + 6·47-s − 3·49-s − 6·51-s + 6·53-s + 2·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 0.755·7-s + 1/3·9-s − 0.603·11-s + 1.66·13-s − 0.258·15-s − 1.45·17-s + 1.37·19-s + 0.436·21-s − 0.417·23-s + 1/5·25-s + 0.192·27-s + 0.371·29-s + 0.718·31-s − 0.348·33-s − 0.338·35-s + 1.64·37-s + 0.960·39-s − 0.312·41-s − 1.21·43-s − 0.149·45-s + 0.875·47-s − 3/7·49-s − 0.840·51-s + 0.824·53-s + 0.269·55-s + ⋯

Functional equation

Λ(s)=(1920s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(1920s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 19201920    =    27⋅3⋅52^{7} \cdot 3 \cdot 5
Sign: 11
Analytic conductor: 15.331215.3312
Root analytic conductor: 3.915513.91551
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1920, ( :1/2), 1)(2,\ 1920,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.2869675112.286967511
L(12)L(\frac12) ≈\approx 2.2869675112.286967511
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+T 1 + T
good7 1−2T+pT2 1 - 2 T + p T^{2}
11 1+2T+pT2 1 + 2 T + p T^{2}
13 1−6T+pT2 1 - 6 T + p T^{2}
17 1+6T+pT2 1 + 6 T + p T^{2}
19 1−6T+pT2 1 - 6 T + p T^{2}
23 1+2T+pT2 1 + 2 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+2T+pT2 1 + 2 T + p T^{2}
43 1+8T+pT2 1 + 8 T + p T^{2}
47 1−6T+pT2 1 - 6 T + p T^{2}
53 1−6T+pT2 1 - 6 T + p T^{2}
59 1−10T+pT2 1 - 10 T + p T^{2}
61 1+14T+pT2 1 + 14 T + p T^{2}
67 1−8T+pT2 1 - 8 T + p T^{2}
71 1−8T+pT2 1 - 8 T + p T^{2}
73 1−2T+pT2 1 - 2 T + p T^{2}
79 1+pT2 1 + 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+6T+pT2 1 + 6 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

−9.009782191031173971561105640243, −8.300454644250546298995728078650, −7.921959866288433993862715383143, −6.94048247529211023854157343618, −6.07380284801590675671363852873, −5.00646292439135152348096305314, −4.21182094038261900638983310748, −3.34183055256449834784989322034, −2.28231008886779719005891093166, −1.05573399756661793291429489696, 1.05573399756661793291429489696, 2.28231008886779719005891093166, 3.34183055256449834784989322034, 4.21182094038261900638983310748, 5.00646292439135152348096305314, 6.07380284801590675671363852873, 6.94048247529211023854157343618, 7.921959866288433993862715383143, 8.300454644250546298995728078650, 9.009782191031173971561105640243

Graph of the ZZ-function along the critical line