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Properties

Label 2-1960-1.1-c1-0-15
Degree 22
Conductor 19601960
Sign 11
Analytic cond. 15.650615.6506
Root an. cond. 3.956093.95609
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 − 11-s + 3·13-s + 2·15-s + 2·17-s + 5·19-s + 7·23-s + 25-s − 4·27-s − 6·29-s − 4·31-s − 2·33-s − 5·37-s + 6·39-s + 5·41-s + 6·43-s + 45-s + 9·47-s + 4·51-s + 11·53-s − 55-s + 10·57-s − 8·59-s + 12·61-s + 3·65-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.447·5-s + 1/3·9-s − 0.301·11-s + 0.832·13-s + 0.516·15-s + 0.485·17-s + 1.14·19-s + 1.45·23-s + 1/5·25-s − 0.769·27-s − 1.11·29-s − 0.718·31-s − 0.348·33-s − 0.821·37-s + 0.960·39-s + 0.780·41-s + 0.914·43-s + 0.149·45-s + 1.31·47-s + 0.560·51-s + 1.51·53-s − 0.134·55-s + 1.32·57-s − 1.04·59-s + 1.53·61-s + 0.372·65-s + ⋯

Functional equation

Λ(s)=(1960s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(1960s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 19601960    =    23⋅5⋅722^{3} \cdot 5 \cdot 7^{2}
Sign: 11
Analytic conductor: 15.650615.6506
Root analytic conductor: 3.956093.95609
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1960, ( :1/2), 1)(2,\ 1960,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.9984192412.998419241
L(12)L(\frac12) ≈\approx 2.9984192412.998419241
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−T 1 - T
7 1 1
good3 1−2T+pT2 1 - 2 T + p T^{2}
11 1+T+pT2 1 + T + p T^{2}
13 1−3T+pT2 1 - 3 T + p T^{2}
17 1−2T+pT2 1 - 2 T + p T^{2}
19 1−5T+pT2 1 - 5 T + p T^{2}
23 1−7T+pT2 1 - 7 T + p T^{2}
29 1+6T+pT2 1 + 6 T + p T^{2}
31 1+4T+pT2 1 + 4 T + p T^{2}
37 1+5T+pT2 1 + 5 T + p T^{2}
41 1−5T+pT2 1 - 5 T + p T^{2}
43 1−6T+pT2 1 - 6 T + p T^{2}
47 1−9T+pT2 1 - 9 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 1−12T+pT2 1 - 12 T + p T^{2}
67 1+4T+pT2 1 + 4 T + p T^{2}
71 1+4T+pT2 1 + 4 T + p T^{2}
73 1+12T+pT2 1 + 12 T + p T^{2}
79 1−14T+pT2 1 - 14 T + p T^{2}
83 1−4T+pT2 1 - 4 T + p T^{2}
89 1+6T+pT2 1 + 6 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

−9.091732998138559844786811523090, −8.597255807018327695840385730332, −7.58664983209587469841750189347, −7.15183087845906613930192459977, −5.82239473250670380943876399125, −5.31661172074901202674013179149, −3.95097779424508233039643008599, −3.21701786226494141604504472434, −2.39676310674079664374662908188, −1.20655101631795718573335213936, 1.20655101631795718573335213936, 2.39676310674079664374662908188, 3.21701786226494141604504472434, 3.95097779424508233039643008599, 5.31661172074901202674013179149, 5.82239473250670380943876399125, 7.15183087845906613930192459977, 7.58664983209587469841750189347, 8.597255807018327695840385730332, 9.091732998138559844786811523090

Graph of the ZZ-function along the critical line