↑

Properties

Label 2-5850-5.4-c1-0-54
Degree 22
Conductor 58505850
Sign 0.447+0.894i0.447 + 0.894i
Analytic cond. 46.712446.7124
Root an. cond. 6.834656.83465
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − i·2-s − 4-s + 2i·7-s + i·8-s + 4·11-s − i·13-s + 2·14-s + 16-s + 6·19-s − 4i·22-s − 4i·23-s − 26-s − 2i·28-s − 8·29-s − 2·31-s − i·32-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s + 0.755i·7-s + 0.353i·8-s + 1.20·11-s − 0.277i·13-s + 0.534·14-s + 0.250·16-s + 1.37·19-s − 0.852i·22-s − 0.834i·23-s − 0.196·26-s − 0.377i·28-s − 1.48·29-s − 0.359·31-s − 0.176i·32-s + ⋯

Functional equation

Λ(s)=(5850s/2 ΓC(s) L(s)=((0.447+0.894i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(5850s/2 ΓC(s+1/2) L(s)=((0.447+0.894i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 58505850    =    2⋅32⋅52⋅132 \cdot 3^{2} \cdot 5^{2} \cdot 13
Sign: 0.447+0.894i0.447 + 0.894i
Analytic conductor: 46.712446.7124
Root analytic conductor: 6.834656.83465
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ5850(5149,⋅)\chi_{5850} (5149, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 5850, ( :1/2), 0.447+0.894i)(2,\ 5850,\ (\ :1/2),\ 0.447 + 0.894i)

Particular Values

L(1)L(1) ≈\approx 1.9817787871.981778787
L(12)L(\frac12) ≈\approx 1.9817787871.981778787
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+iT 1 + iT
3 1 1
5 1 1
13 1+iT 1 + iT
good7 1−2iT−7T2 1 - 2iT - 7T^{2}
11 1−4T+11T2 1 - 4T + 11T^{2}
17 1−17T2 1 - 17T^{2}
19 1−6T+19T2 1 - 6T + 19T^{2}
23 1+4iT−23T2 1 + 4iT - 23T^{2}
29 1+8T+29T2 1 + 8T + 29T^{2}
31 1+2T+31T2 1 + 2T + 31T^{2}
37 1+6iT−37T2 1 + 6iT - 37T^{2}
41 1+6T+41T2 1 + 6T + 41T^{2}
43 1+8iT−43T2 1 + 8iT - 43T^{2}
47 1−8iT−47T2 1 - 8iT - 47T^{2}
53 1+12iT−53T2 1 + 12iT - 53T^{2}
59 1−4T+59T2 1 - 4T + 59T^{2}
61 1−10T+61T2 1 - 10T + 61T^{2}
67 1−2iT−67T2 1 - 2iT - 67T^{2}
71 1−16T+71T2 1 - 16T + 71T^{2}
73 1−14iT−73T2 1 - 14iT - 73T^{2}
79 1−4T+79T2 1 - 4T + 79T^{2}
83 1−12iT−83T2 1 - 12iT - 83T^{2}
89 1+6T+89T2 1 + 6T + 89T^{2}
97 1−10iT−97T2 1 - 10iT - 97T^{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

−8.201715606930708252025037642233, −7.23854486688122120611865192190, −6.59951484815488178626689415255, −5.50562291979258512182371338605, −5.28782373922338305537407371163, −4.00212743948860978137508466672, −3.58487593539234385744396251027, −2.54774098240635433239818409795, −1.77433640158321144240438325485, −0.68105099931596257592621609714, 0.884521569734502184556520834614, 1.77393214030762998308744781617, 3.35807442193009513728454564044, 3.77886063266321019197301658509, 4.67415203493844251435354007373, 5.42678464495939484324141187873, 6.17813753322531809211104249985, 6.95309731478338360853701405256, 7.37122253526833356456622698365, 8.046351333288953364648310047215

Graph of the ZZ-function along the critical line