↑

Properties

Label 2-1650-5.4-c3-0-34
Degree 22
Conductor 16501650
Sign −0.447−0.894i-0.447 - 0.894i
Analytic cond. 97.353197.3531
Root an. cond. 9.866779.86677
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2i·2-s + 3i·3-s − 4·4-s − 6·6-s − 14i·7-s − 8i·8-s − 9·9-s + 11·11-s − 12i·12-s + 80i·13-s + 28·14-s + 16·16-s − 30i·17-s − 18i·18-s − 56·19-s + ⋯
L(s)  = 1  + 0.707i·2-s + 0.577i·3-s − 0.5·4-s − 0.408·6-s − 0.755i·7-s − 0.353i·8-s − 0.333·9-s + 0.301·11-s − 0.288i·12-s + 1.70i·13-s + 0.534·14-s + 0.250·16-s − 0.428i·17-s − 0.235i·18-s − 0.676·19-s + ⋯

Functional equation

Λ(s)=(1650s/2 ΓC(s) L(s)=((−0.447−0.894i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(1650s/2 ΓC(s+3/2) L(s)=((−0.447−0.894i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1650 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 16501650    =    2⋅3⋅52⋅112 \cdot 3 \cdot 5^{2} \cdot 11
Sign: −0.447−0.894i-0.447 - 0.894i
Analytic conductor: 97.353197.3531
Root analytic conductor: 9.866779.86677
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ1650(199,⋅)\chi_{1650} (199, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1650, ( :3/2), −0.447−0.894i)(2,\ 1650,\ (\ :3/2),\ -0.447 - 0.894i)

Particular Values

L(2)L(2) ≈\approx 1.7288821401.728882140
L(12)L(\frac12) ≈\approx 1.7288821401.728882140
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 1−2iT 1 - 2iT
3 1−3iT 1 - 3iT
5 1 1
11 1−11T 1 - 11T
good7 1+14iT−343T2 1 + 14iT - 343T^{2}
13 1−80iT−2.19e3T2 1 - 80iT - 2.19e3T^{2}
17 1+30iT−4.91e3T2 1 + 30iT - 4.91e3T^{2}
19 1+56T+6.85e3T2 1 + 56T + 6.85e3T^{2}
23 1+126iT−1.21e4T2 1 + 126iT - 1.21e4T^{2}
29 1−222T+2.43e4T2 1 - 222T + 2.43e4T^{2}
31 1+16T+2.97e4T2 1 + 16T + 2.97e4T^{2}
37 1−106iT−5.06e4T2 1 - 106iT - 5.06e4T^{2}
41 1−114T+6.89e4T2 1 - 114T + 6.89e4T^{2}
43 1+52iT−7.95e4T2 1 + 52iT - 7.95e4T^{2}
47 1+246iT−1.03e5T2 1 + 246iT - 1.03e5T^{2}
53 1+264iT−1.48e5T2 1 + 264iT - 1.48e5T^{2}
59 1+264T+2.05e5T2 1 + 264T + 2.05e5T^{2}
61 1−92T+2.26e5T2 1 - 92T + 2.26e5T^{2}
67 1−796iT−3.00e5T2 1 - 796iT - 3.00e5T^{2}
71 1−426T+3.57e5T2 1 - 426T + 3.57e5T^{2}
73 1+1.17e3iT−3.89e5T2 1 + 1.17e3iT - 3.89e5T^{2}
79 1+842T+4.93e5T2 1 + 842T + 4.93e5T^{2}
83 1−852iT−5.71e5T2 1 - 852iT - 5.71e5T^{2}
89 1−1.06e3T+7.04e5T2 1 - 1.06e3T + 7.04e5T^{2}
97 1−1.28e3iT−9.12e5T2 1 - 1.28e3iT - 9.12e5T^{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

−9.048847157922289475938266027911, −8.643771129578116357788453522612, −7.60254680931525930687531715511, −6.67289744221264786608868768271, −6.35375210643377462112185657272, −4.95771324866655246549777336275, −4.40668024140145834066928344927, −3.70348970865567097435138853100, −2.29824213149477192691701130071, −0.818296200993271906033557257458, 0.50765480275322584495882246250, 1.56901505720109754669667902620, 2.63249424252351279769863459030, 3.33193754892157837471457175350, 4.53673405742854260207416583782, 5.61185215727687745192037078589, 6.08261916094949223670679463313, 7.30887240786027889982670477447, 8.143070786874588471232525199361, 8.689223038619894359470950896460

Graph of the ZZ-function along the critical line