↑

Properties

Label 2-22e2-11.10-c2-0-11
Degree 22
Conductor 484484
Sign −0.372+0.927i-0.372 + 0.927i
Analytic cond. 13.188013.1880
Root an. cond. 3.631533.63153
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  − 4.43·3-s + 6.56·5-s − 11.5i·7-s + 10.6·9-s + 5.05i·13-s − 29.1·15-s + 8.56i·17-s − 10.8i·19-s + 51.0i·21-s + 11.7·23-s + 18.0·25-s − 7.47·27-s − 20.2i·29-s − 36.4·31-s − 75.5i·35-s + ⋯
L(s)  = 1  − 1.47·3-s + 1.31·5-s − 1.64i·7-s + 1.18·9-s + 0.389i·13-s − 1.94·15-s + 0.504i·17-s − 0.570i·19-s + 2.43i·21-s + 0.510·23-s + 0.721·25-s − 0.276·27-s − 0.697i·29-s − 1.17·31-s − 2.15i·35-s + ⋯

Functional equation

Λ(s)=(484s/2 ΓC(s) L(s)=((−0.372+0.927i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 484 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.372 + 0.927i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(484s/2 ΓC(s+1) L(s)=((−0.372+0.927i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 484 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.372 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 484484    =    22⋅1122^{2} \cdot 11^{2}
Sign: −0.372+0.927i-0.372 + 0.927i
Analytic conductor: 13.188013.1880
Root analytic conductor: 3.631533.63153
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ484(241,⋅)\chi_{484} (241, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 484, ( :1), −0.372+0.927i)(2,\ 484,\ (\ :1),\ -0.372 + 0.927i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.98111915780.9811191578
L(12)L(\frac12) ≈\approx 0.98111915780.9811191578
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)
bad2 1 1
11 1 1
good3 1+4.43T+9T2 1 + 4.43T + 9T^{2}
5 1−6.56T+25T2 1 - 6.56T + 25T^{2}
7 1+11.5iT−49T2 1 + 11.5iT - 49T^{2}
13 1−5.05iT−169T2 1 - 5.05iT - 169T^{2}
17 1−8.56iT−289T2 1 - 8.56iT - 289T^{2}
19 1+10.8iT−361T2 1 + 10.8iT - 361T^{2}
23 1−11.7T+529T2 1 - 11.7T + 529T^{2}
29 1+20.2iT−841T2 1 + 20.2iT - 841T^{2}
31 1+36.4T+961T2 1 + 36.4T + 961T^{2}
37 1+36.8T+1.36e3T2 1 + 36.8T + 1.36e3T^{2}
41 1+42.3iT−1.68e3T2 1 + 42.3iT - 1.68e3T^{2}
43 1+72.4iT−1.84e3T2 1 + 72.4iT - 1.84e3T^{2}
47 1+10.9T+2.20e3T2 1 + 10.9T + 2.20e3T^{2}
53 1−72.3T+2.80e3T2 1 - 72.3T + 2.80e3T^{2}
59 1−4.47T+3.48e3T2 1 - 4.47T + 3.48e3T^{2}
61 1+108.iT−3.72e3T2 1 + 108. iT - 3.72e3T^{2}
67 1+37.6T+4.48e3T2 1 + 37.6T + 4.48e3T^{2}
71 1+53.6T+5.04e3T2 1 + 53.6T + 5.04e3T^{2}
73 1−24.9iT−5.32e3T2 1 - 24.9iT - 5.32e3T^{2}
79 1+71.2iT−6.24e3T2 1 + 71.2iT - 6.24e3T^{2}
83 1+11.5iT−6.88e3T2 1 + 11.5iT - 6.88e3T^{2}
89 1+75.6T+7.92e3T2 1 + 75.6T + 7.92e3T^{2}
97 1−2.98T+9.40e3T2 1 - 2.98T + 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

−10.53986895209614651394800334115, −10.01321718344032432542646329171, −8.926191241860100055354530520760, −7.28430934516203665417317863677, −6.71428603210220761508290121020, −5.78459495889418125656315867711, −4.97846393406741564087569133275, −3.85733576990677011411810571631, −1.78819701777217851936192339916, −0.49134607663686539118241760768, 1.52212080844269529631277462917, 2.82070078764230200512506984891, 4.90559303342721170960610400305, 5.65000161639164385107107706129, 5.95996612902976166460769862948, 7.02308696374759842574199556503, 8.583684602554701720995334796139, 9.429163713417495942077310812540, 10.20819994161950083741146085016, 11.11809886385823390104844774280

Graph of the ZZ-function along the critical line