↑

Properties

Label 2-336-1.1-c3-0-15
Degree 22
Conductor 336336
Sign −1-1
Analytic cond. 19.824619.8246
Root an. cond. 4.452484.45248
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  + 3·3-s − 10·5-s − 7·7-s + 9·9-s + 12·11-s + 30·13-s − 30·15-s + 34·17-s − 148·19-s − 21·21-s − 152·23-s − 25·25-s + 27·27-s − 106·29-s − 304·31-s + 36·33-s + 70·35-s − 114·37-s + 90·39-s + 202·41-s − 116·43-s − 90·45-s − 224·47-s + 49·49-s + 102·51-s − 274·53-s − 120·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.894·5-s − 0.377·7-s + 1/3·9-s + 0.328·11-s + 0.640·13-s − 0.516·15-s + 0.485·17-s − 1.78·19-s − 0.218·21-s − 1.37·23-s − 1/5·25-s + 0.192·27-s − 0.678·29-s − 1.76·31-s + 0.189·33-s + 0.338·35-s − 0.506·37-s + 0.369·39-s + 0.769·41-s − 0.411·43-s − 0.298·45-s − 0.695·47-s + 1/7·49-s + 0.280·51-s − 0.710·53-s − 0.294·55-s + ⋯

Functional equation

Λ(s)=(336s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(336s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 336336    =    24⋅3⋅72^{4} \cdot 3 \cdot 7
Sign: −1-1
Analytic conductor: 19.824619.8246
Root analytic conductor: 4.452484.45248
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 336, ( :3/2), −1)(2,\ 336,\ (\ :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 1
3 1−pT 1 - p T
7 1+pT 1 + p T
good5 1+2pT+p3T2 1 + 2 p T + p^{3} T^{2}
11 1−12T+p3T2 1 - 12 T + p^{3} T^{2}
13 1−30T+p3T2 1 - 30 T + p^{3} T^{2}
17 1−2pT+p3T2 1 - 2 p T + p^{3} T^{2}
19 1+148T+p3T2 1 + 148 T + p^{3} T^{2}
23 1+152T+p3T2 1 + 152 T + p^{3} T^{2}
29 1+106T+p3T2 1 + 106 T + p^{3} T^{2}
31 1+304T+p3T2 1 + 304 T + p^{3} T^{2}
37 1+114T+p3T2 1 + 114 T + p^{3} T^{2}
41 1−202T+p3T2 1 - 202 T + p^{3} T^{2}
43 1+116T+p3T2 1 + 116 T + p^{3} T^{2}
47 1+224T+p3T2 1 + 224 T + p^{3} T^{2}
53 1+274T+p3T2 1 + 274 T + p^{3} T^{2}
59 1−660T+p3T2 1 - 660 T + p^{3} T^{2}
61 1−382T+p3T2 1 - 382 T + p^{3} T^{2}
67 1+12T+p3T2 1 + 12 T + p^{3} T^{2}
71 1−552T+p3T2 1 - 552 T + p^{3} T^{2}
73 1+614T+p3T2 1 + 614 T + p^{3} T^{2}
79 1+880T+p3T2 1 + 880 T + p^{3} T^{2}
83 1−108T+p3T2 1 - 108 T + p^{3} T^{2}
89 1+86T+p3T2 1 + 86 T + p^{3} T^{2}
97 1−1426T+p3T2 1 - 1426 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

−10.69016277315891233656152665294, −9.657937804193542364900762643509, −8.638885954802964202956600690123, −7.955765069573701785111334525963, −6.92015590059259223638448872427, −5.81917927383298404226699125277, −4.14588315318691583224823160744, −3.57215610325400514085082382261, −1.94413213541946236781267209249, 0, 1.94413213541946236781267209249, 3.57215610325400514085082382261, 4.14588315318691583224823160744, 5.81917927383298404226699125277, 6.92015590059259223638448872427, 7.955765069573701785111334525963, 8.638885954802964202956600690123, 9.657937804193542364900762643509, 10.69016277315891233656152665294

Graph of the ZZ-function along the critical line