↑

Properties

Label 2-45-15.14-c2-0-3
Degree 22
Conductor 4545
Sign 0.998+0.0578i0.998 + 0.0578i
Analytic cond. 1.226161.22616
Root an. cond. 1.107321.10732
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  + 2.64·2-s + 3.00·4-s + (−2.64 − 4.24i)5-s + 11.2i·7-s − 2.64·8-s + (−7.00 − 11.2i)10-s − 4.24i·11-s − 11.2i·13-s + 29.6i·14-s − 18.9·16-s + 10.5·17-s + 20·19-s + (−7.93 − 12.7i)20-s − 11.2i·22-s − 5.29·23-s + ⋯
L(s)  = 1  + 1.32·2-s + 0.750·4-s + (−0.529 − 0.848i)5-s + 1.60i·7-s − 0.330·8-s + (−0.700 − 1.12i)10-s − 0.385i·11-s − 0.863i·13-s + 2.12i·14-s − 1.18·16-s + 0.622·17-s + 1.05·19-s + (−0.396 − 0.636i)20-s − 0.510i·22-s − 0.230·23-s + ⋯

Functional equation

Λ(s)=(45s/2 ΓC(s) L(s)=((0.998+0.0578i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0578i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(45s/2 ΓC(s+1) L(s)=((0.998+0.0578i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.998 + 0.0578i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 4545    =    32⋅53^{2} \cdot 5
Sign: 0.998+0.0578i0.998 + 0.0578i
Analytic conductor: 1.226161.22616
Root analytic conductor: 1.107321.10732
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ45(44,⋅)\chi_{45} (44, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 45, ( :1), 0.998+0.0578i)(2,\ 45,\ (\ :1),\ 0.998 + 0.0578i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 1.77099−0.0512677i1.77099 - 0.0512677i
L(12)L(\frac12) ≈\approx 1.77099−0.0512677i1.77099 - 0.0512677i
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
5 1+(2.64+4.24i)T 1 + (2.64 + 4.24i)T
good2 1−2.64T+4T2 1 - 2.64T + 4T^{2}
7 1−11.2iT−49T2 1 - 11.2iT - 49T^{2}
11 1+4.24iT−121T2 1 + 4.24iT - 121T^{2}
13 1+11.2iT−169T2 1 + 11.2iT - 169T^{2}
17 1−10.5T+289T2 1 - 10.5T + 289T^{2}
19 1−20T+361T2 1 - 20T + 361T^{2}
23 1+5.29T+529T2 1 + 5.29T + 529T^{2}
29 1−8.48iT−841T2 1 - 8.48iT - 841T^{2}
31 1−26T+961T2 1 - 26T + 961T^{2}
37 1+33.6iT−1.36e3T2 1 + 33.6iT - 1.36e3T^{2}
41 1−55.1iT−1.68e3T2 1 - 55.1iT - 1.68e3T^{2}
43 1+22.4iT−1.84e3T2 1 + 22.4iT - 1.84e3T^{2}
47 1+21.1T+2.20e3T2 1 + 21.1T + 2.20e3T^{2}
53 1+84.6T+2.80e3T2 1 + 84.6T + 2.80e3T^{2}
59 1+46.6iT−3.48e3T2 1 + 46.6iT - 3.48e3T^{2}
61 1+22T+3.72e3T2 1 + 22T + 3.72e3T^{2}
67 1−89.7iT−4.48e3T2 1 - 89.7iT - 4.48e3T^{2}
71 1−50.9iT−5.04e3T2 1 - 50.9iT - 5.04e3T^{2}
73 1+67.3iT−5.32e3T2 1 + 67.3iT - 5.32e3T^{2}
79 1−14T+6.24e3T2 1 - 14T + 6.24e3T^{2}
83 1−74.0T+6.88e3T2 1 - 74.0T + 6.88e3T^{2}
89 1+89.0iT−7.92e3T2 1 + 89.0iT - 7.92e3T^{2}
97 1+22.4iT−9.40e3T2 1 + 22.4iT - 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

−15.49101780229463246507384250784, −14.45756399092038643472564129422, −13.09973144649670298051484729971, −12.28627678179624555870757487878, −11.57831972666020740592539212873, −9.341901056857532974839818189989, −8.140350270184260463931036343619, −5.86672440114740604001970269545, −5.01968940999722986554726678164, −3.17327027439020235664910501686, 3.42855380757235375829437411522, 4.54391709569482988669188469873, 6.52930374092958023186628093231, 7.58003047582494965399361367001, 9.895381696635400358434481100478, 11.20332124666384339933741293856, 12.21153878685675940118220140065, 13.75953545220165563995112091797, 14.09907473386160379182323141455, 15.27170911640132820099956290985

Graph of the ZZ-function along the critical line