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Properties

Label 2-60840-1.1-c1-0-33
Degree 22
Conductor 6084060840
Sign 11
Analytic cond. 485.809485.809
Root an. cond. 22.041022.0410
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 + 6·11-s + 2·17-s − 2·19-s + 6·23-s + 25-s + 6·29-s + 8·31-s − 10·37-s + 4·41-s + 8·47-s − 7·49-s + 4·53-s + 6·55-s − 10·59-s + 10·61-s − 8·67-s + 4·71-s − 2·73-s + 16·79-s + 2·85-s − 2·95-s + 2·97-s + 101-s + 103-s + 107-s + 109-s + ⋯
L(s)  = 1  + 0.447·5-s + 1.80·11-s + 0.485·17-s − 0.458·19-s + 1.25·23-s + 1/5·25-s + 1.11·29-s + 1.43·31-s − 1.64·37-s + 0.624·41-s + 1.16·47-s − 49-s + 0.549·53-s + 0.809·55-s − 1.30·59-s + 1.28·61-s − 0.977·67-s + 0.474·71-s − 0.234·73-s + 1.80·79-s + 0.216·85-s − 0.205·95-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯

Functional equation

Λ(s)=(60840s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 60840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(60840s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 60840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 6084060840    =    23⋅32⋅5⋅1322^{3} \cdot 3^{2} \cdot 5 \cdot 13^{2}
Sign: 11
Analytic conductor: 485.809485.809
Root analytic conductor: 22.041022.0410
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 60840, ( :1/2), 1)(2,\ 60840,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 3.9284099503.928409950
L(12)L(\frac12) ≈\approx 3.9284099503.928409950
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
13 1 1
good7 1+pT2 1 + p T^{2}
11 1−6T+pT2 1 - 6 T + p T^{2}
17 1−2T+pT2 1 - 2 T + p T^{2}
19 1+2T+pT2 1 + 2 T + p T^{2}
23 1−6T+pT2 1 - 6 T + p T^{2}
29 1−6T+pT2 1 - 6 T + p T^{2}
31 1−8T+pT2 1 - 8 T + p T^{2}
37 1+10T+pT2 1 + 10 T + p T^{2}
41 1−4T+pT2 1 - 4 T + p T^{2}
43 1+pT2 1 + p T^{2}
47 1−8T+pT2 1 - 8 T + p T^{2}
53 1−4T+pT2 1 - 4 T + p T^{2}
59 1+10T+pT2 1 + 10 T + p T^{2}
61 1−10T+pT2 1 - 10 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+2T+pT2 1 + 2 T + p T^{2}
79 1−16T+pT2 1 - 16 T + p T^{2}
83 1+pT2 1 + p T^{2}
89 1+pT2 1 + p T^{2}
97 1−2T+pT2 1 - 2 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

−14.29339437370987, −13.71599525100302, −13.55989444940581, −12.63657846483549, −12.20143569792429, −11.94682620016307, −11.24782630463265, −10.68885868327080, −10.27001364086844, −9.484586141076773, −9.330666668010182, −8.534875643138117, −8.363334643961922, −7.369509685458457, −6.867280671343378, −6.450046501849809, −5.982928132453553, −5.218401660012913, −4.644039761377877, −4.095850070272240, −3.386891995082693, −2.837273536640425, −1.996732602711521, −1.258627211488363, −0.7610565762242350, 0.7610565762242350, 1.258627211488363, 1.996732602711521, 2.837273536640425, 3.386891995082693, 4.095850070272240, 4.644039761377877, 5.218401660012913, 5.982928132453553, 6.450046501849809, 6.867280671343378, 7.369509685458457, 8.363334643961922, 8.534875643138117, 9.330666668010182, 9.484586141076773, 10.27001364086844, 10.68885868327080, 11.24782630463265, 11.94682620016307, 12.20143569792429, 12.63657846483549, 13.55989444940581, 13.71599525100302, 14.29339437370987

Graph of the ZZ-function along the critical line