↑

Properties

Label 2-459-1.1-c1-0-2
Degree 22
Conductor 459459
Sign 11
Analytic cond. 3.665133.66513
Root an. cond. 1.914451.91445
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s − 3·5-s + 2·7-s + 3·11-s + 2·13-s + 4·16-s − 17-s + 5·19-s + 6·20-s + 4·25-s − 4·28-s + 3·29-s + 8·31-s − 6·35-s + 8·37-s − 6·41-s − 4·43-s − 6·44-s + 6·47-s − 3·49-s − 4·52-s − 12·53-s − 9·55-s + 12·59-s − 10·61-s − 8·64-s − 6·65-s + ⋯
L(s)  = 1  − 4-s − 1.34·5-s + 0.755·7-s + 0.904·11-s + 0.554·13-s + 16-s − 0.242·17-s + 1.14·19-s + 1.34·20-s + 4/5·25-s − 0.755·28-s + 0.557·29-s + 1.43·31-s − 1.01·35-s + 1.31·37-s − 0.937·41-s − 0.609·43-s − 0.904·44-s + 0.875·47-s − 3/7·49-s − 0.554·52-s − 1.64·53-s − 1.21·55-s + 1.56·59-s − 1.28·61-s − 64-s − 0.744·65-s + ⋯

Functional equation

Λ(s)=(459s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 459 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(459s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 459 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 459459    =    33⋅173^{3} \cdot 17
Sign: 11
Analytic conductor: 3.665133.66513
Root analytic conductor: 1.914451.91445
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 459, ( :1/2), 1)(2,\ 459,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.0148288921.014828892
L(12)L(\frac12) ≈\approx 1.0148288921.014828892
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)
bad3 1 1
17 1+T 1 + T
good2 1+pT2 1 + p T^{2}
5 1+3T+pT2 1 + 3 T + p T^{2}
7 1−2T+pT2 1 - 2 T + p T^{2}
11 1−3T+pT2 1 - 3 T + p T^{2}
13 1−2T+pT2 1 - 2 T + p T^{2}
19 1−5T+pT2 1 - 5 T + p T^{2}
23 1+pT2 1 + p T^{2}
29 1−3T+pT2 1 - 3 T + p T^{2}
31 1−8T+pT2 1 - 8 T + p T^{2}
37 1−8T+pT2 1 - 8 T + p T^{2}
41 1+6T+pT2 1 + 6 T + p T^{2}
43 1+4T+pT2 1 + 4 T + p T^{2}
47 1−6T+pT2 1 - 6 T + p T^{2}
53 1+12T+pT2 1 + 12 T + p T^{2}
59 1−12T+pT2 1 - 12 T + p T^{2}
61 1+10T+pT2 1 + 10 T + p T^{2}
67 1−5T+pT2 1 - 5 T + p T^{2}
71 1−15T+pT2 1 - 15 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 1−6T+pT2 1 - 6 T + p T^{2}
89 1+pT2 1 + p T^{2}
97 1−14T+pT2 1 - 14 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

−11.38093182139037659842566820047, −10.10278302014150851075341421887, −9.109300011117320213307134485881, −8.256713755120548244557004000097, −7.74542291215750154959689622231, −6.44257768768837718607648442666, −5.01146682876845260799537247768, −4.25748270915351505319381939602, −3.37512184396728216346992291538, −1.01021305890200635033573864009, 1.01021305890200635033573864009, 3.37512184396728216346992291538, 4.25748270915351505319381939602, 5.01146682876845260799537247768, 6.44257768768837718607648442666, 7.74542291215750154959689622231, 8.256713755120548244557004000097, 9.109300011117320213307134485881, 10.10278302014150851075341421887, 11.38093182139037659842566820047

Graph of the ZZ-function along the critical line