↑

Properties

Label 2-490-1.1-c1-0-14
Degree 22
Conductor 490490
Sign −1-1
Analytic cond. 3.912663.91266
Root an. cond. 1.978041.97804
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  + 2-s − 2·3-s + 4-s − 5-s − 2·6-s + 8-s + 9-s − 10-s − 4·11-s − 2·12-s − 2·13-s + 2·15-s + 16-s − 8·17-s + 18-s + 6·19-s − 20-s − 4·22-s − 4·23-s − 2·24-s + 25-s − 2·26-s + 4·27-s − 6·29-s + 2·30-s − 4·31-s + 32-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.447·5-s − 0.816·6-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 1.20·11-s − 0.577·12-s − 0.554·13-s + 0.516·15-s + 1/4·16-s − 1.94·17-s + 0.235·18-s + 1.37·19-s − 0.223·20-s − 0.852·22-s − 0.834·23-s − 0.408·24-s + 1/5·25-s − 0.392·26-s + 0.769·27-s − 1.11·29-s + 0.365·30-s − 0.718·31-s + 0.176·32-s + ⋯

Functional equation

Λ(s)=(490s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(490s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 490490    =    2⋅5⋅722 \cdot 5 \cdot 7^{2}
Sign: −1-1
Analytic conductor: 3.912663.91266
Root analytic conductor: 1.978041.97804
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 490, ( :1/2), −1)(2,\ 490,\ (\ :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−T 1 - T
5 1+T 1 + T
7 1 1
good3 1+2T+pT2 1 + 2 T + p T^{2}
11 1+4T+pT2 1 + 4 T + p T^{2}
13 1+2T+pT2 1 + 2 T + p T^{2}
17 1+8T+pT2 1 + 8 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+6T+pT2 1 + 6 T + p T^{2}
31 1+4T+pT2 1 + 4 T + p T^{2}
37 1+10T+pT2 1 + 10 T + p T^{2}
41 1+4T+pT2 1 + 4 T + p T^{2}
43 1−4T+pT2 1 - 4 T + p T^{2}
47 1+4T+pT2 1 + 4 T + p T^{2}
53 1−10T+pT2 1 - 10 T + p T^{2}
59 1−14T+pT2 1 - 14 T + p T^{2}
61 1−10T+pT2 1 - 10 T + p T^{2}
67 1+4T+pT2 1 + 4 T + p T^{2}
71 1−12T+pT2 1 - 12 T + p T^{2}
73 1+4T+pT2 1 + 4 T + p T^{2}
79 1−4T+pT2 1 - 4 T + p T^{2}
83 1−2T+pT2 1 - 2 T + p T^{2}
89 1+8T+pT2 1 + 8 T + p T^{2}
97 1+pT2 1 + 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

−10.86614564966969779050895065772, −9.987626462691323536290955576580, −8.621807622472038281841661682648, −7.43586792817004186337731069491, −6.72584879403071822637349996832, −5.47531989167733320735048693294, −5.07475421442181782846073138217, −3.82742742653431763007701470093, −2.35089027266828839682953321286, 0, 2.35089027266828839682953321286, 3.82742742653431763007701470093, 5.07475421442181782846073138217, 5.47531989167733320735048693294, 6.72584879403071822637349996832, 7.43586792817004186337731069491, 8.621807622472038281841661682648, 9.987626462691323536290955576580, 10.86614564966969779050895065772

Graph of the ZZ-function along the critical line