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Properties

Label 2-360-1.1-c1-0-0
Degree 22
Conductor 360360
Sign 11
Analytic cond. 2.874612.87461
Root an. cond. 1.695461.69546
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 + 4·11-s + 6·13-s + 6·17-s − 4·19-s + 25-s + 2·29-s − 8·31-s − 2·37-s + 6·41-s + 12·43-s − 8·47-s − 7·49-s − 6·53-s − 4·55-s − 12·59-s + 14·61-s − 6·65-s + 4·67-s − 8·71-s − 6·73-s − 8·79-s + 12·83-s − 6·85-s − 10·89-s + 4·95-s + 2·97-s + ⋯
L(s)  = 1  − 0.447·5-s + 1.20·11-s + 1.66·13-s + 1.45·17-s − 0.917·19-s + 1/5·25-s + 0.371·29-s − 1.43·31-s − 0.328·37-s + 0.937·41-s + 1.82·43-s − 1.16·47-s − 49-s − 0.824·53-s − 0.539·55-s − 1.56·59-s + 1.79·61-s − 0.744·65-s + 0.488·67-s − 0.949·71-s − 0.702·73-s − 0.900·79-s + 1.31·83-s − 0.650·85-s − 1.05·89-s + 0.410·95-s + 0.203·97-s + ⋯

Functional equation

Λ(s)=(360s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(360s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 360360    =    23⋅32⋅52^{3} \cdot 3^{2} \cdot 5
Sign: 11
Analytic conductor: 2.874612.87461
Root analytic conductor: 1.695461.69546
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 360, ( :1/2), 1)(2,\ 360,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.3930826871.393082687
L(12)L(\frac12) ≈\approx 1.3930826871.393082687
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+pT2 1 + p T^{2}
11 1−4T+pT2 1 - 4 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+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+8T+pT2 1 + 8 T + p T^{2}
37 1+2T+pT2 1 + 2 T + p T^{2}
41 1−6T+pT2 1 - 6 T + p T^{2}
43 1−12T+pT2 1 - 12 T + p T^{2}
47 1+8T+pT2 1 + 8 T + p T^{2}
53 1+6T+pT2 1 + 6 T + p T^{2}
59 1+12T+pT2 1 + 12 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+6T+pT2 1 + 6 T + p T^{2}
79 1+8T+pT2 1 + 8 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 1−2T+pT2 1 - 2 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

−11.35483361723706372877757083384, −10.73084990800958988992236291099, −9.487642964123562756489741043280, −8.663099716937980035576099714984, −7.76560916456884806442341251241, −6.58285993178476791834968599347, −5.74168392413099570963868387782, −4.19803603229033482796144552370, −3.38984665882479408513314051846, −1.36054514292875137606341288502, 1.36054514292875137606341288502, 3.38984665882479408513314051846, 4.19803603229033482796144552370, 5.74168392413099570963868387782, 6.58285993178476791834968599347, 7.76560916456884806442341251241, 8.663099716937980035576099714984, 9.487642964123562756489741043280, 10.73084990800958988992236291099, 11.35483361723706372877757083384

Graph of the ZZ-function along the critical line