Properties

Label 2-2112-1.1-c3-0-93
Degree 22
Conductor 21122112
Sign 1-1
Analytic cond. 124.612124.612
Root an. cond. 11.162911.1629
Motivic weight 33
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s + 4·5-s − 26·7-s + 9·9-s − 11·11-s + 32·13-s + 12·15-s + 74·17-s + 60·19-s − 78·21-s − 182·23-s − 109·25-s + 27·27-s + 90·29-s − 8·31-s − 33·33-s − 104·35-s + 66·37-s + 96·39-s + 422·41-s − 408·43-s + 36·45-s − 506·47-s + 333·49-s + 222·51-s − 348·53-s − 44·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.357·5-s − 1.40·7-s + 1/3·9-s − 0.301·11-s + 0.682·13-s + 0.206·15-s + 1.05·17-s + 0.724·19-s − 0.810·21-s − 1.64·23-s − 0.871·25-s + 0.192·27-s + 0.576·29-s − 0.0463·31-s − 0.174·33-s − 0.502·35-s + 0.293·37-s + 0.394·39-s + 1.60·41-s − 1.44·43-s + 0.119·45-s − 1.57·47-s + 0.970·49-s + 0.609·51-s − 0.901·53-s − 0.107·55-s + ⋯

Functional equation

Λ(s)=(2112s/2ΓC(s)L(s)=(Λ(4s)\begin{aligned}\Lambda(s)=\mathstrut & 2112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(2112s/2ΓC(s+3/2)L(s)=(Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 2112 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 21122112    =    263112^{6} \cdot 3 \cdot 11
Sign: 1-1
Analytic conductor: 124.612124.612
Root analytic conductor: 11.162911.1629
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 2112, ( :3/2), 1)(2,\ 2112,\ (\ :3/2),\ -1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
L(52)L(\frac{5}{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 1pT 1 - p T
11 1+pT 1 + p T
good5 14T+p3T2 1 - 4 T + p^{3} T^{2}
7 1+26T+p3T2 1 + 26 T + p^{3} T^{2}
13 132T+p3T2 1 - 32 T + p^{3} T^{2}
17 174T+p3T2 1 - 74 T + p^{3} T^{2}
19 160T+p3T2 1 - 60 T + p^{3} T^{2}
23 1+182T+p3T2 1 + 182 T + p^{3} T^{2}
29 190T+p3T2 1 - 90 T + p^{3} T^{2}
31 1+8T+p3T2 1 + 8 T + p^{3} T^{2}
37 166T+p3T2 1 - 66 T + p^{3} T^{2}
41 1422T+p3T2 1 - 422 T + p^{3} T^{2}
43 1+408T+p3T2 1 + 408 T + p^{3} T^{2}
47 1+506T+p3T2 1 + 506 T + p^{3} T^{2}
53 1+348T+p3T2 1 + 348 T + p^{3} T^{2}
59 1200T+p3T2 1 - 200 T + p^{3} T^{2}
61 1+132T+p3T2 1 + 132 T + p^{3} T^{2}
67 11036T+p3T2 1 - 1036 T + p^{3} T^{2}
71 1762T+p3T2 1 - 762 T + p^{3} T^{2}
73 1+542T+p3T2 1 + 542 T + p^{3} T^{2}
79 1+550T+p3T2 1 + 550 T + p^{3} T^{2}
83 1132T+p3T2 1 - 132 T + p^{3} T^{2}
89 1570T+p3T2 1 - 570 T + p^{3} T^{2}
97 114T+p3T2 1 - 14 T + p^{3} 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

−8.213967367680691300782723569916, −7.79449479861081698286700699396, −6.66022364838920303122687000699, −6.10248769674964699522562224397, −5.29682195345652693109702144393, −3.94927740677310572827003028734, −3.36199407088840525199809496499, −2.50193778729963704125352144638, −1.32505436075178096330294889103, 0, 1.32505436075178096330294889103, 2.50193778729963704125352144638, 3.36199407088840525199809496499, 3.94927740677310572827003028734, 5.29682195345652693109702144393, 6.10248769674964699522562224397, 6.66022364838920303122687000699, 7.79449479861081698286700699396, 8.213967367680691300782723569916

Graph of the ZZ-function along the critical line