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Properties

Label 2-43560-1.1-c1-0-5
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 − 7-s + 13-s − 17-s + 3·19-s − 6·23-s + 25-s − 3·29-s − 35-s − 5·37-s − 12·41-s − 8·43-s − 12·47-s − 6·49-s − 2·53-s + 65-s + 12·67-s − 71-s + 8·73-s − 8·79-s − 83-s − 85-s − 6·89-s − 91-s + 3·95-s − 8·97-s + 101-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.377·7-s + 0.277·13-s − 0.242·17-s + 0.688·19-s − 1.25·23-s + 1/5·25-s − 0.557·29-s − 0.169·35-s − 0.821·37-s − 1.87·41-s − 1.21·43-s − 1.75·47-s − 6/7·49-s − 0.274·53-s + 0.124·65-s + 1.46·67-s − 0.118·71-s + 0.936·73-s − 0.900·79-s − 0.109·83-s − 0.108·85-s − 0.635·89-s − 0.104·91-s + 0.307·95-s − 0.812·97-s + 0.0995·101-s + ⋯

Functional equation

Λ(s)=(43560s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 43560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(43560s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\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    =    23⋅32⋅5⋅1122^{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 1.4374380291.437438029
L(12)L(\frac12) ≈\approx 1.4374380291.437438029
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
11 1 1
good7 1+T+pT2 1 + T + p T^{2}
13 1−T+pT2 1 - T + p T^{2}
17 1+T+pT2 1 + T + p T^{2}
19 1−3T+pT2 1 - 3 T + p T^{2}
23 1+6T+pT2 1 + 6 T + p T^{2}
29 1+3T+pT2 1 + 3 T + p T^{2}
31 1+pT2 1 + p T^{2}
37 1+5T+pT2 1 + 5 T + p T^{2}
41 1+12T+pT2 1 + 12 T + p T^{2}
43 1+8T+pT2 1 + 8 T + p T^{2}
47 1+12T+pT2 1 + 12 T + p T^{2}
53 1+2T+pT2 1 + 2 T + p T^{2}
59 1+pT2 1 + p T^{2}
61 1+pT2 1 + p T^{2}
67 1−12T+pT2 1 - 12 T + p T^{2}
71 1+T+pT2 1 + T + p T^{2}
73 1−8T+pT2 1 - 8 T + p T^{2}
79 1+8T+pT2 1 + 8 T + p T^{2}
83 1+T+pT2 1 + T + p T^{2}
89 1+6T+pT2 1 + 6 T + p T^{2}
97 1+8T+pT2 1 + 8 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

−14.60409168763892, −14.09511658455942, −13.65489268341840, −13.19791997403017, −12.67820553997972, −12.11873491389468, −11.47519313240256, −11.21138447910445, −10.30869727020044, −9.910290249913593, −9.640419254864521, −8.843686175372679, −8.308898334770781, −7.897453206422812, −6.990511880704977, −6.648745044865900, −6.085178329065698, −5.356516716178182, −4.988599897724440, −4.138169817625749, −3.398888351337852, −3.065401437495274, −1.890586287072609, −1.677739677192176, −0.4119463335781092, 0.4119463335781092, 1.677739677192176, 1.890586287072609, 3.065401437495274, 3.398888351337852, 4.138169817625749, 4.988599897724440, 5.356516716178182, 6.085178329065698, 6.648745044865900, 6.990511880704977, 7.897453206422812, 8.308898334770781, 8.843686175372679, 9.640419254864521, 9.910290249913593, 10.30869727020044, 11.21138447910445, 11.47519313240256, 12.11873491389468, 12.67820553997972, 13.19791997403017, 13.65489268341840, 14.09511658455942, 14.60409168763892

Graph of the ZZ-function along the critical line