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Properties

Label 2-29988-1.1-c1-0-9
Degree 22
Conductor 2998829988
Sign 11
Analytic cond. 239.455239.455
Root an. cond. 15.474315.4743
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  + 2·5-s − 6·13-s − 17-s + 2·19-s − 25-s − 8·29-s + 2·37-s + 2·41-s + 8·43-s − 8·47-s + 2·53-s + 12·59-s + 4·61-s − 12·65-s + 12·67-s − 8·73-s − 8·79-s − 2·85-s + 10·89-s + 4·95-s − 12·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  + 0.894·5-s − 1.66·13-s − 0.242·17-s + 0.458·19-s − 1/5·25-s − 1.48·29-s + 0.328·37-s + 0.312·41-s + 1.21·43-s − 1.16·47-s + 0.274·53-s + 1.56·59-s + 0.512·61-s − 1.48·65-s + 1.46·67-s − 0.936·73-s − 0.900·79-s − 0.216·85-s + 1.05·89-s + 0.410·95-s − 1.21·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯

Functional equation

Λ(s)=(29988s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 29988 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(29988s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 29988 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 2998829988    =    22⋅32⋅72⋅172^{2} \cdot 3^{2} \cdot 7^{2} \cdot 17
Sign: 11
Analytic conductor: 239.455239.455
Root analytic conductor: 15.474315.4743
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 29988, ( :1/2), 1)(2,\ 29988,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.9448866991.944886699
L(12)L(\frac12) ≈\approx 1.9448866991.944886699
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
7 1 1
17 1+T 1 + T
good5 1−2T+pT2 1 - 2 T + p T^{2}
11 1+pT2 1 + p T^{2}
13 1+6T+pT2 1 + 6 T + p T^{2}
19 1−2T+pT2 1 - 2 T + p T^{2}
23 1+pT2 1 + p T^{2}
29 1+8T+pT2 1 + 8 T + p T^{2}
31 1+pT2 1 + p T^{2}
37 1−2T+pT2 1 - 2 T + p T^{2}
41 1−2T+pT2 1 - 2 T + p T^{2}
43 1−8T+pT2 1 - 8 T + p T^{2}
47 1+8T+pT2 1 + 8 T + p T^{2}
53 1−2T+pT2 1 - 2 T + p T^{2}
59 1−12T+pT2 1 - 12 T + p T^{2}
61 1−4T+pT2 1 - 4 T + p T^{2}
67 1−12T+pT2 1 - 12 T + p T^{2}
71 1+pT2 1 + 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+pT2 1 + p T^{2}
89 1−10T+pT2 1 - 10 T + p T^{2}
97 1+12T+pT2 1 + 12 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

−14.92033386886470, −14.60747275354898, −14.19019192091677, −13.49420818976538, −13.02560512655082, −12.61903082830999, −11.92548308701613, −11.42695364775616, −10.86693941565370, −10.07181652922708, −9.740348274542234, −9.387423472243391, −8.713832390062314, −7.922595582493134, −7.399627663538299, −6.915647770174595, −6.194942068017204, −5.474567590415255, −5.231014158596817, −4.381544210874853, −3.737985486332155, −2.767691867071923, −2.281794252799006, −1.633748226340058, −0.5145497515938038, 0.5145497515938038, 1.633748226340058, 2.281794252799006, 2.767691867071923, 3.737985486332155, 4.381544210874853, 5.231014158596817, 5.474567590415255, 6.194942068017204, 6.915647770174595, 7.399627663538299, 7.922595582493134, 8.713832390062314, 9.387423472243391, 9.740348274542234, 10.07181652922708, 10.86693941565370, 11.42695364775616, 11.92548308701613, 12.61903082830999, 13.02560512655082, 13.49420818976538, 14.19019192091677, 14.60747275354898, 14.92033386886470

Graph of the ZZ-function along the critical line