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Properties

Label 2-5760-1.1-c1-0-33
Degree 22
Conductor 57605760
Sign 11
Analytic cond. 45.993845.9938
Root an. cond. 6.781876.78187
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 − 4·13-s + 6·17-s + 6·19-s + 6·23-s + 25-s + 2·29-s − 8·31-s + 8·37-s + 4·43-s − 6·47-s − 7·49-s − 10·53-s + 4·55-s − 12·59-s + 6·61-s − 4·65-s − 8·67-s + 4·71-s + 14·73-s − 4·79-s − 4·83-s + 6·85-s + 4·89-s + 6·95-s + 18·97-s + ⋯
L(s)  = 1  + 0.447·5-s + 1.20·11-s − 1.10·13-s + 1.45·17-s + 1.37·19-s + 1.25·23-s + 1/5·25-s + 0.371·29-s − 1.43·31-s + 1.31·37-s + 0.609·43-s − 0.875·47-s − 49-s − 1.37·53-s + 0.539·55-s − 1.56·59-s + 0.768·61-s − 0.496·65-s − 0.977·67-s + 0.474·71-s + 1.63·73-s − 0.450·79-s − 0.439·83-s + 0.650·85-s + 0.423·89-s + 0.615·95-s + 1.82·97-s + ⋯

Functional equation

Λ(s)=(5760s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 5760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(5760s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 5760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 57605760    =    27⋅32⋅52^{7} \cdot 3^{2} \cdot 5
Sign: 11
Analytic conductor: 45.993845.9938
Root analytic conductor: 6.781876.78187
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 5760, ( :1/2), 1)(2,\ 5760,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.5629905142.562990514
L(12)L(\frac12) ≈\approx 2.5629905142.562990514
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+4T+pT2 1 + 4 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 1−6T+pT2 1 - 6 T + p T^{2}
29 1−2T+pT2 1 - 2 T + p T^{2}
31 1+8T+pT2 1 + 8 T + p T^{2}
37 1−8T+pT2 1 - 8 T + p T^{2}
41 1+pT2 1 + p T^{2}
43 1−4T+pT2 1 - 4 T + p T^{2}
47 1+6T+pT2 1 + 6 T + p T^{2}
53 1+10T+pT2 1 + 10 T + p T^{2}
59 1+12T+pT2 1 + 12 T + p T^{2}
61 1−6T+pT2 1 - 6 T + p T^{2}
67 1+8T+pT2 1 + 8 T + p T^{2}
71 1−4T+pT2 1 - 4 T + p T^{2}
73 1−14T+pT2 1 - 14 T + p T^{2}
79 1+4T+pT2 1 + 4 T + p T^{2}
83 1+4T+pT2 1 + 4 T + p T^{2}
89 1−4T+pT2 1 - 4 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

−7.915498693344195473459991934156, −7.45645405407327466794189181251, −6.72715814240596862911710437882, −5.95657367615252029699134653277, −5.21208760644277380449490969485, −4.63080528810134641587934365750, −3.46642982637163098219628506995, −2.96149597344385008296549009236, −1.72613225983972960150385260557, −0.910382245506312098936969232690, 0.910382245506312098936969232690, 1.72613225983972960150385260557, 2.96149597344385008296549009236, 3.46642982637163098219628506995, 4.63080528810134641587934365750, 5.21208760644277380449490969485, 5.95657367615252029699134653277, 6.72715814240596862911710437882, 7.45645405407327466794189181251, 7.915498693344195473459991934156

Graph of the ZZ-function along the critical line