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Properties

Label 2-1080-1.1-c1-0-4
Degree 22
Conductor 10801080
Sign 11
Analytic cond. 8.623848.62384
Root an. cond. 2.936632.93663
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  − 5-s + 2·7-s − 11-s + 13-s + 17-s + 4·19-s − 23-s + 25-s + 5·29-s + 31-s − 2·35-s + 6·37-s + 7·43-s − 7·47-s − 3·49-s + 12·53-s + 55-s + 4·59-s + 10·61-s − 65-s − 4·67-s − 12·71-s + 6·73-s − 2·77-s + 15·79-s − 2·83-s − 85-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.755·7-s − 0.301·11-s + 0.277·13-s + 0.242·17-s + 0.917·19-s − 0.208·23-s + 1/5·25-s + 0.928·29-s + 0.179·31-s − 0.338·35-s + 0.986·37-s + 1.06·43-s − 1.02·47-s − 3/7·49-s + 1.64·53-s + 0.134·55-s + 0.520·59-s + 1.28·61-s − 0.124·65-s − 0.488·67-s − 1.42·71-s + 0.702·73-s − 0.227·77-s + 1.68·79-s − 0.219·83-s − 0.108·85-s + ⋯

Functional equation

Λ(s)=(1080s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(1080s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 10801080    =    23⋅33⋅52^{3} \cdot 3^{3} \cdot 5
Sign: 11
Analytic conductor: 8.623848.62384
Root analytic conductor: 2.936632.93663
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1080, ( :1/2), 1)(2,\ 1080,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.6556762921.655676292
L(12)L(\frac12) ≈\approx 1.6556762921.655676292
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 1
5 1+T 1 + T
good7 1−2T+pT2 1 - 2 T + p T^{2}
11 1+T+pT2 1 + T + p T^{2}
13 1−T+pT2 1 - T + p T^{2}
17 1−T+pT2 1 - T + p T^{2}
19 1−4T+pT2 1 - 4 T + p T^{2}
23 1+T+pT2 1 + T + p T^{2}
29 1−5T+pT2 1 - 5 T + p T^{2}
31 1−T+pT2 1 - T + p T^{2}
37 1−6T+pT2 1 - 6 T + p T^{2}
41 1+pT2 1 + p T^{2}
43 1−7T+pT2 1 - 7 T + p T^{2}
47 1+7T+pT2 1 + 7 T + p T^{2}
53 1−12T+pT2 1 - 12 T + p T^{2}
59 1−4T+pT2 1 - 4 T + p T^{2}
61 1−10T+pT2 1 - 10 T + p T^{2}
67 1+4T+pT2 1 + 4 T + p T^{2}
71 1+12T+pT2 1 + 12 T + p T^{2}
73 1−6T+pT2 1 - 6 T + p T^{2}
79 1−15T+pT2 1 - 15 T + p T^{2}
83 1+2T+pT2 1 + 2 T + p T^{2}
89 1−12T+pT2 1 - 12 T + p T^{2}
97 1−10T+pT2 1 - 10 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

−9.938016913173052801027619458454, −8.971544127814454878311880442588, −8.098215835619989410546011808466, −7.59623005890538420604075910258, −6.55466152525426843819318782499, −5.49385970565338845590944925790, −4.68426941255478221849145341689, −3.68561025556385408354635008380, −2.51888632026023556389235975025, −1.04466524673073804361911141068, 1.04466524673073804361911141068, 2.51888632026023556389235975025, 3.68561025556385408354635008380, 4.68426941255478221849145341689, 5.49385970565338845590944925790, 6.55466152525426843819318782499, 7.59623005890538420604075910258, 8.098215835619989410546011808466, 8.971544127814454878311880442588, 9.938016913173052801027619458454

Graph of the ZZ-function along the critical line