↑

Properties

Label 2-209-1.1-c1-0-14
Degree 22
Conductor 209209
Sign −1-1
Analytic cond. 1.668871.66887
Root an. cond. 1.291841.29184
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 2·4-s − 3·5-s − 4·7-s − 2·9-s + 11-s − 2·12-s + 2·13-s − 3·15-s + 4·16-s + 19-s + 6·20-s − 4·21-s + 3·23-s + 4·25-s − 5·27-s + 8·28-s − 6·29-s − 7·31-s + 33-s + 12·35-s + 4·36-s − 7·37-s + 2·39-s − 10·43-s − 2·44-s + 6·45-s + ⋯
L(s)  = 1  + 0.577·3-s − 4-s − 1.34·5-s − 1.51·7-s − 2/3·9-s + 0.301·11-s − 0.577·12-s + 0.554·13-s − 0.774·15-s + 16-s + 0.229·19-s + 1.34·20-s − 0.872·21-s + 0.625·23-s + 4/5·25-s − 0.962·27-s + 1.51·28-s − 1.11·29-s − 1.25·31-s + 0.174·33-s + 2.02·35-s + 2/3·36-s − 1.15·37-s + 0.320·39-s − 1.52·43-s − 0.301·44-s + 0.894·45-s + ⋯

Functional equation

Λ(s)=(209s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(209s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 209209    =    11⋅1911 \cdot 19
Sign: −1-1
Analytic conductor: 1.668871.66887
Root analytic conductor: 1.291841.29184
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 209, ( :1/2), −1)(2,\ 209,\ (\ :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)
bad11 1−T 1 - T
19 1−T 1 - T
good2 1+pT2 1 + p T^{2}
3 1−T+pT2 1 - T + p T^{2}
5 1+3T+pT2 1 + 3 T + p T^{2}
7 1+4T+pT2 1 + 4 T + p T^{2}
13 1−2T+pT2 1 - 2 T + p T^{2}
17 1+pT2 1 + p T^{2}
23 1−3T+pT2 1 - 3 T + p T^{2}
29 1+6T+pT2 1 + 6 T + p T^{2}
31 1+7T+pT2 1 + 7 T + p T^{2}
37 1+7T+pT2 1 + 7 T + p T^{2}
41 1+pT2 1 + p T^{2}
43 1+10T+pT2 1 + 10 T + p T^{2}
47 1+pT2 1 + p T^{2}
53 1−6T+pT2 1 - 6 T + p T^{2}
59 1−3T+pT2 1 - 3 T + p T^{2}
61 1+10T+pT2 1 + 10 T + p T^{2}
67 1−11T+pT2 1 - 11 T + p T^{2}
71 1−15T+pT2 1 - 15 T + p T^{2}
73 1−8T+pT2 1 - 8 T + p T^{2}
79 1+16T+pT2 1 + 16 T + p T^{2}
83 1+pT2 1 + p T^{2}
89 1−9T+pT2 1 - 9 T + p T^{2}
97 1+T+pT2 1 + T + p 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

−12.05917203220889978964010728308, −10.96102192801302376001650382810, −9.603034074754024658706636273886, −8.916975147724513941001822680471, −8.090908771111864109123836483338, −6.92292199886679346603140457685, −5.46804944393504386563865646601, −3.77762904601741073110269049747, −3.37641666123212985170587332393, 0, 3.37641666123212985170587332393, 3.77762904601741073110269049747, 5.46804944393504386563865646601, 6.92292199886679346603140457685, 8.090908771111864109123836483338, 8.916975147724513941001822680471, 9.603034074754024658706636273886, 10.96102192801302376001650382810, 12.05917203220889978964010728308

Graph of the ZZ-function along the critical line