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Properties

Label 2-1920-1.1-c1-0-11
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 + 4·7-s + 9-s − 2·11-s − 15-s + 4·21-s + 8·23-s + 25-s + 27-s − 2·29-s + 2·31-s − 2·33-s − 4·35-s + 8·37-s − 2·41-s + 4·43-s − 45-s + 9·49-s + 6·53-s + 2·55-s − 14·59-s + 14·61-s + 4·63-s − 4·67-s + 8·69-s − 8·71-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 1.51·7-s + 1/3·9-s − 0.603·11-s − 0.258·15-s + 0.872·21-s + 1.66·23-s + 1/5·25-s + 0.192·27-s − 0.371·29-s + 0.359·31-s − 0.348·33-s − 0.676·35-s + 1.31·37-s − 0.312·41-s + 0.609·43-s − 0.149·45-s + 9/7·49-s + 0.824·53-s + 0.269·55-s − 1.82·59-s + 1.79·61-s + 0.503·63-s − 0.488·67-s + 0.963·69-s − 0.949·71-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 2.4081411602.408141160
L(12)L(\frac12) ≈\approx 2.4081411602.408141160
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 1−4T+pT2 1 - 4 T + p T^{2}
11 1+2T+pT2 1 + 2 T + p T^{2}
13 1+pT2 1 + p T^{2}
17 1+pT2 1 + p T^{2}
19 1+pT2 1 + p T^{2}
23 1−8T+pT2 1 - 8 T + p T^{2}
29 1+2T+pT2 1 + 2 T + p T^{2}
31 1−2T+pT2 1 - 2 T + p T^{2}
37 1−8T+pT2 1 - 8 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+pT2 1 + p T^{2}
53 1−6T+pT2 1 - 6 T + p T^{2}
59 1+14T+pT2 1 + 14 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+8T+pT2 1 + 8 T + p T^{2}
73 1+10T+pT2 1 + 10 T + p T^{2}
79 1+6T+pT2 1 + 6 T + p T^{2}
83 1+12T+pT2 1 + 12 T + p T^{2}
89 1−14T+pT2 1 - 14 T + p T^{2}
97 1−18T+pT2 1 - 18 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

−8.905958225250015730932642240476, −8.475697246379034923748800257665, −7.58903760041826735478913178216, −7.27919207522379862388962403738, −5.92704687742133064020277025243, −4.90827321544753146649955163789, −4.42748833343524287397809510890, −3.23500564434491538144898168837, −2.27720663910374506444794308813, −1.09505201742716397244922543371, 1.09505201742716397244922543371, 2.27720663910374506444794308813, 3.23500564434491538144898168837, 4.42748833343524287397809510890, 4.90827321544753146649955163789, 5.92704687742133064020277025243, 7.27919207522379862388962403738, 7.58903760041826735478913178216, 8.475697246379034923748800257665, 8.905958225250015730932642240476

Graph of the ZZ-function along the critical line