↑

Properties

Label 2-378-1.1-c3-0-21
Degree 22
Conductor 378378
Sign −1-1
Analytic cond. 22.302722.3027
Root an. cond. 4.722574.72257
Motivic weight 33
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 4·4-s − 7·5-s − 7·7-s + 8·8-s − 14·10-s − 17·11-s + 12·13-s − 14·14-s + 16·16-s − 38·17-s − 43·19-s − 28·20-s − 34·22-s − 131·23-s − 76·25-s + 24·26-s − 28·28-s − 160·29-s + 45·31-s + 32·32-s − 76·34-s + 49·35-s − 331·37-s − 86·38-s − 56·40-s + 111·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.626·5-s − 0.377·7-s + 0.353·8-s − 0.442·10-s − 0.465·11-s + 0.256·13-s − 0.267·14-s + 1/4·16-s − 0.542·17-s − 0.519·19-s − 0.313·20-s − 0.329·22-s − 1.18·23-s − 0.607·25-s + 0.181·26-s − 0.188·28-s − 1.02·29-s + 0.260·31-s + 0.176·32-s − 0.383·34-s + 0.236·35-s − 1.47·37-s − 0.367·38-s − 0.221·40-s + 0.422·41-s + ⋯

Functional equation

Λ(s)=(378s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(378s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 378378    =    2⋅33⋅72 \cdot 3^{3} \cdot 7
Sign: −1-1
Analytic conductor: 22.302722.3027
Root analytic conductor: 4.722574.72257
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 378, ( :3/2), −1)(2,\ 378,\ (\ :3/2),\ -1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
L(52)L(\frac{5}{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−pT 1 - p T
3 1 1
7 1+pT 1 + p T
good5 1+7T+p3T2 1 + 7 T + p^{3} T^{2}
11 1+17T+p3T2 1 + 17 T + p^{3} T^{2}
13 1−12T+p3T2 1 - 12 T + p^{3} T^{2}
17 1+38T+p3T2 1 + 38 T + p^{3} T^{2}
19 1+43T+p3T2 1 + 43 T + p^{3} T^{2}
23 1+131T+p3T2 1 + 131 T + p^{3} T^{2}
29 1+160T+p3T2 1 + 160 T + p^{3} T^{2}
31 1−45T+p3T2 1 - 45 T + p^{3} T^{2}
37 1+331T+p3T2 1 + 331 T + p^{3} T^{2}
41 1−111T+p3T2 1 - 111 T + p^{3} T^{2}
43 1−230T+p3T2 1 - 230 T + p^{3} T^{2}
47 1+6pT+p3T2 1 + 6 p T + p^{3} T^{2}
53 1+396T+p3T2 1 + 396 T + p^{3} T^{2}
59 1+214T+p3T2 1 + 214 T + p^{3} T^{2}
61 1−768T+p3T2 1 - 768 T + p^{3} T^{2}
67 1−388T+p3T2 1 - 388 T + p^{3} T^{2}
71 1+551T+p3T2 1 + 551 T + p^{3} T^{2}
73 1−274T+p3T2 1 - 274 T + p^{3} T^{2}
79 1−390T+p3T2 1 - 390 T + p^{3} T^{2}
83 1+440T+p3T2 1 + 440 T + p^{3} T^{2}
89 1+105T+p3T2 1 + 105 T + p^{3} T^{2}
97 1−304T+p3T2 1 - 304 T + p^{3} T^{2}
show more
show less
   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

−10.70333363817675177931319688289, −9.691943594082112265838131776750, −8.447193006420253300816536773743, −7.57860600489704063044246266631, −6.53854650826415694786622362939, −5.57323611747284592479121334112, −4.33334355544235076741880926114, −3.47778021263476618635090420264, −2.08218217058645274011285682155, 0, 2.08218217058645274011285682155, 3.47778021263476618635090420264, 4.33334355544235076741880926114, 5.57323611747284592479121334112, 6.53854650826415694786622362939, 7.57860600489704063044246266631, 8.447193006420253300816536773743, 9.691943594082112265838131776750, 10.70333363817675177931319688289

Graph of the ZZ-function along the critical line