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Properties

Label 2-612-1.1-c1-0-5
Degree 22
Conductor 612612
Sign −1-1
Analytic cond. 4.886844.88684
Root an. cond. 2.210622.21062
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  − 5-s − 5·11-s − 5·13-s − 17-s + 19-s + 3·23-s − 4·25-s − 2·29-s + 2·31-s − 8·37-s + 5·41-s − 9·43-s − 6·47-s − 7·49-s + 6·53-s + 5·55-s − 6·59-s − 4·61-s + 5·65-s + 12·67-s + 12·71-s − 2·73-s + 10·79-s + 2·83-s + 85-s − 12·89-s − 95-s + ⋯
L(s)  = 1  − 0.447·5-s − 1.50·11-s − 1.38·13-s − 0.242·17-s + 0.229·19-s + 0.625·23-s − 4/5·25-s − 0.371·29-s + 0.359·31-s − 1.31·37-s + 0.780·41-s − 1.37·43-s − 0.875·47-s − 49-s + 0.824·53-s + 0.674·55-s − 0.781·59-s − 0.512·61-s + 0.620·65-s + 1.46·67-s + 1.42·71-s − 0.234·73-s + 1.12·79-s + 0.219·83-s + 0.108·85-s − 1.27·89-s − 0.102·95-s + ⋯

Functional equation

Λ(s)=(612s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 612 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(612s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 612 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 612612    =    22⋅32⋅172^{2} \cdot 3^{2} \cdot 17
Sign: −1-1
Analytic conductor: 4.886844.88684
Root analytic conductor: 2.210622.21062
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 612, ( :1/2), −1)(2,\ 612,\ (\ :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)
bad2 1 1
3 1 1
17 1+T 1 + T
good5 1+T+pT2 1 + T + p T^{2}
7 1+pT2 1 + p T^{2}
11 1+5T+pT2 1 + 5 T + p T^{2}
13 1+5T+pT2 1 + 5 T + p T^{2}
19 1−T+pT2 1 - T + p T^{2}
23 1−3T+pT2 1 - 3 T + p T^{2}
29 1+2T+pT2 1 + 2 T + p T^{2}
31 1−2T+pT2 1 - 2 T + p T^{2}
37 1+8T+pT2 1 + 8 T + p T^{2}
41 1−5T+pT2 1 - 5 T + p T^{2}
43 1+9T+pT2 1 + 9 T + p T^{2}
47 1+6T+pT2 1 + 6 T + p T^{2}
53 1−6T+pT2 1 - 6 T + p T^{2}
59 1+6T+pT2 1 + 6 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 1−12T+pT2 1 - 12 T + p T^{2}
73 1+2T+pT2 1 + 2 T + p T^{2}
79 1−10T+pT2 1 - 10 T + p T^{2}
83 1−2T+pT2 1 - 2 T + p T^{2}
89 1+12T+pT2 1 + 12 T + p T^{2}
97 1−16T+pT2 1 - 16 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

−10.18490963672513276838211915410, −9.469029240875226710024119915125, −8.242710873880411872797331552948, −7.63736087168536861434605466701, −6.76959891375384935472749260030, −5.37641192809255730852979399474, −4.75477574120717256848740322395, −3.34157120180978815093157505643, −2.23127545325450121489119586078, 0, 2.23127545325450121489119586078, 3.34157120180978815093157505643, 4.75477574120717256848740322395, 5.37641192809255730852979399474, 6.76959891375384935472749260030, 7.63736087168536861434605466701, 8.242710873880411872797331552948, 9.469029240875226710024119915125, 10.18490963672513276838211915410

Graph of the ZZ-function along the critical line