↑

Properties

Label 2-33-11.10-c2-0-3
Degree 22
Conductor 3333
Sign −0.612+0.790i-0.612 + 0.790i
Analytic cond. 0.8991840.899184
Root an. cond. 0.9482530.948253
Motivic weight 22
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3.53i·2-s − 1.73·3-s − 8.46·4-s + 6.19·5-s + 6.11i·6-s − 2.58i·7-s + 15.7i·8-s + 2.99·9-s − 21.8i·10-s + (6.73 − 8.69i)11-s + 14.6·12-s + 23.7i·13-s − 9.12·14-s − 10.7·15-s + 21.7·16-s + 12.2i·17-s + ⋯
L(s)  = 1  − 1.76i·2-s − 0.577·3-s − 2.11·4-s + 1.23·5-s + 1.01i·6-s − 0.369i·7-s + 1.97i·8-s + 0.333·9-s − 2.18i·10-s + (0.612 − 0.790i)11-s + 1.22·12-s + 1.82i·13-s − 0.651·14-s − 0.715·15-s + 1.36·16-s + 0.719i·17-s + ⋯

Functional equation

Λ(s)=(33s/2 ΓC(s) L(s)=((−0.612+0.790i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.612 + 0.790i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(33s/2 ΓC(s+1) L(s)=((−0.612+0.790i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.612 + 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 3333    =    3⋅113 \cdot 11
Sign: −0.612+0.790i-0.612 + 0.790i
Analytic conductor: 0.8991840.899184
Root analytic conductor: 0.9482530.948253
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ33(10,⋅)\chi_{33} (10, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 33, ( :1), −0.612+0.790i)(2,\ 33,\ (\ :1),\ -0.612 + 0.790i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.404509−0.824515i0.404509 - 0.824515i
L(12)L(\frac12) ≈\approx 0.404509−0.824515i0.404509 - 0.824515i
L(2)L(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+1.73T 1 + 1.73T
11 1+(−6.73+8.69i)T 1 + (-6.73 + 8.69i)T
good2 1+3.53iT−4T2 1 + 3.53iT - 4T^{2}
5 1−6.19T+25T2 1 - 6.19T + 25T^{2}
7 1+2.58iT−49T2 1 + 2.58iT - 49T^{2}
13 1−23.7iT−169T2 1 - 23.7iT - 169T^{2}
17 1−12.2iT−289T2 1 - 12.2iT - 289T^{2}
19 1−3.27iT−361T2 1 - 3.27iT - 361T^{2}
23 1+14.3T+529T2 1 + 14.3T + 529T^{2}
29 1+38.5iT−841T2 1 + 38.5iT - 841T^{2}
31 1+11.1T+961T2 1 + 11.1T + 961T^{2}
37 1+12.5T+1.36e3T2 1 + 12.5T + 1.36e3T^{2}
41 1+1.38iT−1.68e3T2 1 + 1.38iT - 1.68e3T^{2}
43 1−23.9iT−1.84e3T2 1 - 23.9iT - 1.84e3T^{2}
47 1−19.8T+2.20e3T2 1 - 19.8T + 2.20e3T^{2}
53 1+12.0T+2.80e3T2 1 + 12.0T + 2.80e3T^{2}
59 1+62.7T+3.48e3T2 1 + 62.7T + 3.48e3T^{2}
61 1+21.3iT−3.72e3T2 1 + 21.3iT - 3.72e3T^{2}
67 1+34T+4.48e3T2 1 + 34T + 4.48e3T^{2}
71 1+69.2T+5.04e3T2 1 + 69.2T + 5.04e3T^{2}
73 1+39.9iT−5.32e3T2 1 + 39.9iT - 5.32e3T^{2}
79 1−97.6iT−6.24e3T2 1 - 97.6iT - 6.24e3T^{2}
83 1+71.9iT−6.88e3T2 1 + 71.9iT - 6.88e3T^{2}
89 1−107.T+7.92e3T2 1 - 107.T + 7.92e3T^{2}
97 1+166.T+9.40e3T2 1 + 166.T + 9.40e3T^{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

−16.66089905828759549416081151807, −14.09977699405630018722588118712, −13.49955232345092999990011623682, −12.10798662709275078093249925492, −11.14299715677443179220938044031, −10.01851215469821635860244540049, −9.069322791113611605464269711606, −6.15012025980602209124338683622, −4.15979940465240215847716423614, −1.75980946077415687813493024952, 5.14629586213630255833811611530, 6.01785272799938372427563424404, 7.34390685369888332716055361569, 9.027180936759949181677452379217, 10.20382368798121601978165258507, 12.52807099055674813784751832003, 13.66903638690475769431315642723, 14.80850459626339352154425597640, 15.78602266977772422524583427412, 16.98468076297493011220349814447

Graph of the ZZ-function along the critical line