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Properties

Label 2-5760-1.1-c1-0-12
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 1.4797432631.479743263
L(12)L(\frac12) ≈\approx 1.4797432631.479743263
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

−8.221784593047618370443600142588, −7.31234115119924924839519158305, −6.59955013140584688329875296949, −6.31922612991577539753934456878, −4.94021366721806789945391007986, −4.53528577234853491767187121078, −3.78396261931862396131954677074, −2.74258038034632227131235442262, −1.94258996435176496943866233694, −0.63273215052695375258695370179, 0.63273215052695375258695370179, 1.94258996435176496943866233694, 2.74258038034632227131235442262, 3.78396261931862396131954677074, 4.53528577234853491767187121078, 4.94021366721806789945391007986, 6.31922612991577539753934456878, 6.59955013140584688329875296949, 7.31234115119924924839519158305, 8.221784593047618370443600142588

Graph of the ZZ-function along the critical line