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Properties

Label 2-399e2-1.1-c1-0-41
Degree 22
Conductor 159201159201
Sign 11
Analytic cond. 1271.221271.22
Root an. cond. 35.654235.6542
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 22

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 4-s − 2·5-s + 3·8-s + 2·10-s − 4·11-s − 2·13-s − 16-s − 6·17-s + 2·20-s + 4·22-s − 25-s + 2·26-s − 2·29-s − 5·32-s + 6·34-s − 6·37-s − 6·40-s − 2·41-s − 4·43-s + 4·44-s + 50-s + 2·52-s + 6·53-s + 8·55-s + 2·58-s − 12·59-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s − 0.894·5-s + 1.06·8-s + 0.632·10-s − 1.20·11-s − 0.554·13-s − 1/4·16-s − 1.45·17-s + 0.447·20-s + 0.852·22-s − 1/5·25-s + 0.392·26-s − 0.371·29-s − 0.883·32-s + 1.02·34-s − 0.986·37-s − 0.948·40-s − 0.312·41-s − 0.609·43-s + 0.603·44-s + 0.141·50-s + 0.277·52-s + 0.824·53-s + 1.07·55-s + 0.262·58-s − 1.56·59-s + ⋯

Functional equation

Λ(s)=(159201s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 159201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(159201s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 159201 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 159201159201    =    32⋅72⋅1923^{2} \cdot 7^{2} \cdot 19^{2}
Sign: 11
Analytic conductor: 1271.221271.22
Root analytic conductor: 35.654235.6542
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 22
Selberg data: (2, 159201, ( :1/2), 1)(2,\ 159201,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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)
bad3 1 1
7 1 1
19 1 1
good2 1+T+pT2 1 + T + p T^{2}
5 1+2T+pT2 1 + 2 T + p T^{2}
11 1+4T+pT2 1 + 4 T + p T^{2}
13 1+2T+pT2 1 + 2 T + p T^{2}
17 1+6T+pT2 1 + 6 T + p T^{2}
23 1+pT2 1 + p T^{2}
29 1+2T+pT2 1 + 2 T + p T^{2}
31 1+pT2 1 + p T^{2}
37 1+6T+pT2 1 + 6 T + p T^{2}
41 1+2T+pT2 1 + 2 T + p T^{2}
43 1+4T+pT2 1 + 4 T + p T^{2}
47 1+pT2 1 + p T^{2}
53 1−6T+pT2 1 - 6 T + p T^{2}
59 1+12T+pT2 1 + 12 T + p T^{2}
61 1−2T+pT2 1 - 2 T + p T^{2}
67 1+4T+pT2 1 + 4 T + p T^{2}
71 1+pT2 1 + p T^{2}
73 1−6T+pT2 1 - 6 T + p T^{2}
79 1−16T+pT2 1 - 16 T + p T^{2}
83 1+12T+pT2 1 + 12 T + p T^{2}
89 1−14T+pT2 1 - 14 T + p T^{2}
97 1−18T+pT2 1 - 18 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

−13.58479926748091, −13.32956705146113, −12.85196651487021, −12.33256530370328, −11.79895401413914, −11.28265369958407, −10.81967205866097, −10.34222096137701, −10.04619574278562, −9.312028814032149, −8.935336808339231, −8.468144798181291, −8.014654432941583, −7.487715485647086, −7.291883505892065, −6.563914002998695, −5.914206367633884, −5.107716161782147, −4.836878954265275, −4.376586694003391, −3.655949753886024, −3.244902852346174, −2.230109395067092, −1.989583292537695, −0.9027801776158123, 0, 0, 0.9027801776158123, 1.989583292537695, 2.230109395067092, 3.244902852346174, 3.655949753886024, 4.376586694003391, 4.836878954265275, 5.107716161782147, 5.914206367633884, 6.563914002998695, 7.291883505892065, 7.487715485647086, 8.014654432941583, 8.468144798181291, 8.935336808339231, 9.312028814032149, 10.04619574278562, 10.34222096137701, 10.81967205866097, 11.28265369958407, 11.79895401413914, 12.33256530370328, 12.85196651487021, 13.32956705146113, 13.58479926748091

Graph of the ZZ-function along the critical line