Properties

Label 2-43560-1.1-c1-0-43
Degree 22
Conductor 4356043560
Sign 11
Analytic cond. 347.828347.828
Root an. cond. 18.650118.6501
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 + 4·7-s + 6·13-s − 2·17-s + 4·19-s + 8·23-s + 25-s + 6·29-s − 4·35-s − 2·37-s + 2·41-s + 8·43-s + 9·49-s + 2·53-s − 4·59-s + 10·61-s − 6·65-s + 12·67-s + 16·71-s − 6·73-s − 8·79-s − 16·83-s + 2·85-s − 10·89-s + 24·91-s − 4·95-s + 10·97-s + ⋯
L(s)  = 1  − 0.447·5-s + 1.51·7-s + 1.66·13-s − 0.485·17-s + 0.917·19-s + 1.66·23-s + 1/5·25-s + 1.11·29-s − 0.676·35-s − 0.328·37-s + 0.312·41-s + 1.21·43-s + 9/7·49-s + 0.274·53-s − 0.520·59-s + 1.28·61-s − 0.744·65-s + 1.46·67-s + 1.89·71-s − 0.702·73-s − 0.900·79-s − 1.75·83-s + 0.216·85-s − 1.05·89-s + 2.51·91-s − 0.410·95-s + 1.01·97-s + ⋯

Functional equation

Λ(s)=(43560s/2ΓC(s)L(s)=(Λ(2s)\begin{aligned}\Lambda(s)=\mathstrut & 43560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(43560s/2ΓC(s+1/2)L(s)=(Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 43560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 4356043560    =    233251122^{3} \cdot 3^{2} \cdot 5 \cdot 11^{2}
Sign: 11
Analytic conductor: 347.828347.828
Root analytic conductor: 18.650118.6501
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 43560, ( :1/2), 1)(2,\ 43560,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) \approx 3.9960969293.996096929
L(12)L(\frac12) \approx 3.9960969293.996096929
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)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
11 1 1
good7 14T+pT2 1 - 4 T + p T^{2}
13 16T+pT2 1 - 6 T + p T^{2}
17 1+2T+pT2 1 + 2 T + p T^{2}
19 14T+pT2 1 - 4 T + p T^{2}
23 18T+pT2 1 - 8 T + p T^{2}
29 16T+pT2 1 - 6 T + p T^{2}
31 1+pT2 1 + p T^{2}
37 1+2T+pT2 1 + 2 T + p T^{2}
41 12T+pT2 1 - 2 T + p T^{2}
43 18T+pT2 1 - 8 T + p T^{2}
47 1+pT2 1 + p T^{2}
53 12T+pT2 1 - 2 T + p T^{2}
59 1+4T+pT2 1 + 4 T + p T^{2}
61 110T+pT2 1 - 10 T + p T^{2}
67 112T+pT2 1 - 12 T + p T^{2}
71 116T+pT2 1 - 16 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+16T+pT2 1 + 16 T + p T^{2}
89 1+10T+pT2 1 + 10 T + p T^{2}
97 110T+pT2 1 - 10 T + p T^{2}
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   L(s)=p j=12(1αj,pps)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

−14.54148997017571, −14.14824706298236, −13.86839141659273, −13.01809629030151, −12.77780535692275, −11.89104653219775, −11.44191504672753, −11.11268676096999, −10.77162965235900, −10.07911576939431, −9.219054282201628, −8.766963751868313, −8.313754245616861, −7.923746829696269, −7.125696265970288, −6.808510780615954, −5.901465016287536, −5.385265244962661, −4.767158574160939, −4.273283151927510, −3.589731041560660, −2.923620460069711, −2.098670165146183, −1.151954464426075, −0.9016669122759157, 0.9016669122759157, 1.151954464426075, 2.098670165146183, 2.923620460069711, 3.589731041560660, 4.273283151927510, 4.767158574160939, 5.385265244962661, 5.901465016287536, 6.808510780615954, 7.125696265970288, 7.923746829696269, 8.313754245616861, 8.766963751868313, 9.219054282201628, 10.07911576939431, 10.77162965235900, 11.11268676096999, 11.44191504672753, 11.89104653219775, 12.77780535692275, 13.01809629030151, 13.86839141659273, 14.14824706298236, 14.54148997017571

Graph of the ZZ-function along the critical line