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Properties

Label 2-1083-1.1-c1-0-42
Degree 22
Conductor 10831083
Sign −1-1
Analytic cond. 8.647798.64779
Root an. cond. 2.940712.94071
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s − 4-s − 6-s + 7-s + 3·8-s + 9-s − 2·11-s − 12-s − 5·13-s − 14-s − 16-s − 4·17-s − 18-s + 21-s + 2·22-s − 4·23-s + 3·24-s − 5·25-s + 5·26-s + 27-s − 28-s + 8·29-s + 3·31-s − 5·32-s − 2·33-s + 4·34-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.408·6-s + 0.377·7-s + 1.06·8-s + 1/3·9-s − 0.603·11-s − 0.288·12-s − 1.38·13-s − 0.267·14-s − 1/4·16-s − 0.970·17-s − 0.235·18-s + 0.218·21-s + 0.426·22-s − 0.834·23-s + 0.612·24-s − 25-s + 0.980·26-s + 0.192·27-s − 0.188·28-s + 1.48·29-s + 0.538·31-s − 0.883·32-s − 0.348·33-s + 0.685·34-s + ⋯

Functional equation

Λ(s)=(1083s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1083 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(1083s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1083 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 10831083    =    3⋅1923 \cdot 19^{2}
Sign: −1-1
Analytic conductor: 8.647798.64779
Root analytic conductor: 2.940712.94071
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 1083, ( :1/2), −1)(2,\ 1083,\ (\ :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−T 1 - T
19 1 1
good2 1+T+pT2 1 + T + p T^{2}
5 1+pT2 1 + p T^{2}
7 1−T+pT2 1 - T + p T^{2}
11 1+2T+pT2 1 + 2 T + p T^{2}
13 1+5T+pT2 1 + 5 T + p T^{2}
17 1+4T+pT2 1 + 4 T + p T^{2}
23 1+4T+pT2 1 + 4 T + p T^{2}
29 1−8T+pT2 1 - 8 T + p T^{2}
31 1−3T+pT2 1 - 3 T + p T^{2}
37 1+3T+pT2 1 + 3 T + p T^{2}
41 1−12T+pT2 1 - 12 T + p T^{2}
43 1+T+pT2 1 + T + p T^{2}
47 1+6T+pT2 1 + 6 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+13T+pT2 1 + 13 T + p T^{2}
67 1+11T+pT2 1 + 11 T + p T^{2}
71 1+6T+pT2 1 + 6 T + p T^{2}
73 1+11T+pT2 1 + 11 T + p T^{2}
79 1+T+pT2 1 + T + p T^{2}
83 1+pT2 1 + p T^{2}
89 1−6T+pT2 1 - 6 T + 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

−9.440765134945835622026803358490, −8.660508077875060824408929882116, −7.85631248963382588612315598005, −7.46143359867823390422868408984, −6.16896727785284343819689789988, −4.78782908451205976462170472642, −4.40390657634204558343229968296, −2.85897993273824516683580972698, −1.77743967134967625738915448392, 0, 1.77743967134967625738915448392, 2.85897993273824516683580972698, 4.40390657634204558343229968296, 4.78782908451205976462170472642, 6.16896727785284343819689789988, 7.46143359867823390422868408984, 7.85631248963382588612315598005, 8.660508077875060824408929882116, 9.440765134945835622026803358490

Graph of the ZZ-function along the critical line