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Properties

Label 2-320892-1.1-c1-0-11
Degree 22
Conductor 320892320892
Sign −1-1
Analytic cond. 2562.332562.33
Root an. cond. 50.619550.6195
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  + 3-s − 5·7-s + 9-s + 13-s − 17-s − 6·19-s − 5·21-s − 4·23-s − 5·25-s + 27-s + 3·29-s − 9·31-s + 37-s + 39-s + 6·41-s − 3·43-s + 47-s + 18·49-s − 51-s + 2·53-s − 6·57-s − 7·59-s + 4·61-s − 5·63-s − 2·67-s − 4·69-s − 12·71-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.88·7-s + 1/3·9-s + 0.277·13-s − 0.242·17-s − 1.37·19-s − 1.09·21-s − 0.834·23-s − 25-s + 0.192·27-s + 0.557·29-s − 1.61·31-s + 0.164·37-s + 0.160·39-s + 0.937·41-s − 0.457·43-s + 0.145·47-s + 18/7·49-s − 0.140·51-s + 0.274·53-s − 0.794·57-s − 0.911·59-s + 0.512·61-s − 0.629·63-s − 0.244·67-s − 0.481·69-s − 1.42·71-s + ⋯

Functional equation

Λ(s)=(320892s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 320892 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(320892s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 320892 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 320892320892    =    22⋅3⋅112⋅13⋅172^{2} \cdot 3 \cdot 11^{2} \cdot 13 \cdot 17
Sign: −1-1
Analytic conductor: 2562.332562.33
Root analytic conductor: 50.619550.6195
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 320892, ( :1/2), −1)(2,\ 320892,\ (\ :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−T 1 - T
11 1 1
13 1−T 1 - T
17 1+T 1 + T
good5 1+pT2 1 + p T^{2}
7 1+5T+pT2 1 + 5 T + p T^{2}
19 1+6T+pT2 1 + 6 T + p T^{2}
23 1+4T+pT2 1 + 4 T + p T^{2}
29 1−3T+pT2 1 - 3 T + p T^{2}
31 1+9T+pT2 1 + 9 T + p T^{2}
37 1−T+pT2 1 - T + p T^{2}
41 1−6T+pT2 1 - 6 T + p T^{2}
43 1+3T+pT2 1 + 3 T + p T^{2}
47 1−T+pT2 1 - T + p T^{2}
53 1−2T+pT2 1 - 2 T + p T^{2}
59 1+7T+pT2 1 + 7 T + p T^{2}
61 1−4T+pT2 1 - 4 T + p T^{2}
67 1+2T+pT2 1 + 2 T + p T^{2}
71 1+12T+pT2 1 + 12 T + p T^{2}
73 1−9T+pT2 1 - 9 T + p T^{2}
79 1+10T+pT2 1 + 10 T + p T^{2}
83 1−4T+pT2 1 - 4 T + p T^{2}
89 1−6T+pT2 1 - 6 T + p T^{2}
97 1−14T+pT2 1 - 14 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

−12.93989844302772, −12.66982007031465, −11.95980567515224, −11.64500808325094, −10.85608267700391, −10.42196329959175, −10.16354533166047, −9.593329857301495, −9.086331482421966, −8.971192865592366, −8.293924825985034, −7.737641131791404, −7.327962840175278, −6.704515992215826, −6.346551425771763, −5.962531491465874, −5.496197652137974, −4.617729206421582, −4.024696685908197, −3.789816734337057, −3.237112158027357, −2.640012594037948, −2.173237994423510, −1.595205953006060, −0.5610007280523541, 0, 0.5610007280523541, 1.595205953006060, 2.173237994423510, 2.640012594037948, 3.237112158027357, 3.789816734337057, 4.024696685908197, 4.617729206421582, 5.496197652137974, 5.962531491465874, 6.346551425771763, 6.704515992215826, 7.327962840175278, 7.737641131791404, 8.293924825985034, 8.971192865592366, 9.086331482421966, 9.593329857301495, 10.16354533166047, 10.42196329959175, 10.85608267700391, 11.64500808325094, 11.95980567515224, 12.66982007031465, 12.93989844302772

Graph of the ZZ-function along the critical line