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Properties

Label 2-332010-1.1-c1-0-2
Degree 22
Conductor 332010332010
Sign 11
Analytic cond. 2651.112651.11
Root an. cond. 51.488951.4889
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  + 2-s + 4-s + 5-s + 7-s + 8-s + 10-s + 4·11-s − 2·13-s + 14-s + 16-s + 17-s − 4·19-s + 20-s + 4·22-s − 8·23-s + 25-s − 2·26-s + 28-s − 6·29-s − 31-s + 32-s + 34-s + 35-s − 2·37-s − 4·38-s + 40-s − 6·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.377·7-s + 0.353·8-s + 0.316·10-s + 1.20·11-s − 0.554·13-s + 0.267·14-s + 1/4·16-s + 0.242·17-s − 0.917·19-s + 0.223·20-s + 0.852·22-s − 1.66·23-s + 1/5·25-s − 0.392·26-s + 0.188·28-s − 1.11·29-s − 0.179·31-s + 0.176·32-s + 0.171·34-s + 0.169·35-s − 0.328·37-s − 0.648·38-s + 0.158·40-s − 0.937·41-s + ⋯

Functional equation

Λ(s)=(332010s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 332010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(332010s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 332010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 332010332010    =    2⋅32⋅5⋅7⋅17⋅312 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 17 \cdot 31
Sign: 11
Analytic conductor: 2651.112651.11
Root analytic conductor: 51.488951.4889
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 332010, ( :1/2), 1)(2,\ 332010,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 3.3563549523.356354952
L(12)L(\frac12) ≈\approx 3.3563549523.356354952
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−T 1 - T
3 1 1
5 1−T 1 - T
7 1−T 1 - T
17 1−T 1 - T
31 1+T 1 + T
good11 1−4T+pT2 1 - 4 T + p T^{2}
13 1+2T+pT2 1 + 2 T + p T^{2}
19 1+4T+pT2 1 + 4 T + p T^{2}
23 1+8T+pT2 1 + 8 T + p T^{2}
29 1+6T+pT2 1 + 6 T + p T^{2}
37 1+2T+pT2 1 + 2 T + p T^{2}
41 1+6T+pT2 1 + 6 T + p T^{2}
43 1−4T+pT2 1 - 4 T + p T^{2}
47 1+pT2 1 + p T^{2}
53 1−2T+pT2 1 - 2 T + p T^{2}
59 1+8T+pT2 1 + 8 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 1−8T+pT2 1 - 8 T + p T^{2}
73 1+10T+pT2 1 + 10 T + p T^{2}
79 1+pT2 1 + p T^{2}
83 1+16T+pT2 1 + 16 T + p T^{2}
89 1+6T+pT2 1 + 6 T + p T^{2}
97 1−10T+pT2 1 - 10 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

−12.59002593026896, −12.02556036373247, −11.95370313763771, −11.28569099874544, −10.88718151287492, −10.32287792898240, −9.955481084237297, −9.425875076222978, −8.962440102587565, −8.473858963749353, −7.932054671660936, −7.354816646822239, −7.043506694345519, −6.340519893037189, −6.017787864980794, −5.655974264925660, −5.002806119512825, −4.400864243242397, −4.146777290521653, −3.556251072572242, −2.983923471229090, −2.267599116425896, −1.717219234791073, −1.508135304923061, −0.3849710781798285, 0.3849710781798285, 1.508135304923061, 1.717219234791073, 2.267599116425896, 2.983923471229090, 3.556251072572242, 4.146777290521653, 4.400864243242397, 5.002806119512825, 5.655974264925660, 6.017787864980794, 6.340519893037189, 7.043506694345519, 7.354816646822239, 7.932054671660936, 8.473858963749353, 8.962440102587565, 9.425875076222978, 9.955481084237297, 10.32287792898240, 10.88718151287492, 11.28569099874544, 11.95370313763771, 12.02556036373247, 12.59002593026896

Graph of the ZZ-function along the critical line