↑

Properties

Label 2-224-224.125-c0-0-0
Degree 22
Conductor 224224
Sign 0.831−0.555i0.831 - 0.555i
Analytic cond. 0.1117900.111790
Root an. cond. 0.3343500.334350
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 + 0.707i)2-s − 1.00i·4-s + (0.707 − 0.707i)7-s + (0.707 + 0.707i)8-s + (0.707 + 0.707i)9-s + (−0.707 + 0.292i)11-s + 1.00i·14-s − 1.00·16-s − 1.00·18-s + (0.292 − 0.707i)22-s + (−1 − i)23-s + (−0.707 + 0.707i)25-s + (−0.707 − 0.707i)28-s + (0.707 + 0.292i)29-s + (0.707 − 0.707i)32-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s − 1.00i·4-s + (0.707 − 0.707i)7-s + (0.707 + 0.707i)8-s + (0.707 + 0.707i)9-s + (−0.707 + 0.292i)11-s + 1.00i·14-s − 1.00·16-s − 1.00·18-s + (0.292 − 0.707i)22-s + (−1 − i)23-s + (−0.707 + 0.707i)25-s + (−0.707 − 0.707i)28-s + (0.707 + 0.292i)29-s + (0.707 − 0.707i)32-s + ⋯

Functional equation

Λ(s)=(224s/2 ΓC(s) L(s)=((0.831−0.555i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.831 - 0.555i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(224s/2 ΓC(s) L(s)=((0.831−0.555i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.831 - 0.555i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 224224    =    25⋅72^{5} \cdot 7
Sign: 0.831−0.555i0.831 - 0.555i
Analytic conductor: 0.1117900.111790
Root analytic conductor: 0.3343500.334350
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ224(125,⋅)\chi_{224} (125, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 224, ( :0), 0.831−0.555i)(2,\ 224,\ (\ :0),\ 0.831 - 0.555i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.55263854690.5526385469
L(12)L(\frac12) ≈\approx 0.55263854690.5526385469
L(1)L(1) 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+(0.707−0.707i)T 1 + (0.707 - 0.707i)T
7 1+(−0.707+0.707i)T 1 + (-0.707 + 0.707i)T
good3 1+(−0.707−0.707i)T2 1 + (-0.707 - 0.707i)T^{2}
5 1+(0.707−0.707i)T2 1 + (0.707 - 0.707i)T^{2}
11 1+(0.707−0.292i)T+(0.707−0.707i)T2 1 + (0.707 - 0.292i)T + (0.707 - 0.707i)T^{2}
13 1+(0.707+0.707i)T2 1 + (0.707 + 0.707i)T^{2}
17 1+T2 1 + T^{2}
19 1+(0.707+0.707i)T2 1 + (0.707 + 0.707i)T^{2}
23 1+(1+i)T+iT2 1 + (1 + i)T + iT^{2}
29 1+(−0.707−0.292i)T+(0.707+0.707i)T2 1 + (-0.707 - 0.292i)T + (0.707 + 0.707i)T^{2}
31 1−T2 1 - T^{2}
37 1+(0.707+1.70i)T+(−0.707+0.707i)T2 1 + (0.707 + 1.70i)T + (-0.707 + 0.707i)T^{2}
41 1−iT2 1 - iT^{2}
43 1+(1.70−0.707i)T+(0.707−0.707i)T2 1 + (1.70 - 0.707i)T + (0.707 - 0.707i)T^{2}
47 1+T2 1 + T^{2}
53 1+(1.70−0.707i)T+(0.707−0.707i)T2 1 + (1.70 - 0.707i)T + (0.707 - 0.707i)T^{2}
59 1+(0.707−0.707i)T2 1 + (0.707 - 0.707i)T^{2}
61 1+(−0.707−0.707i)T2 1 + (-0.707 - 0.707i)T^{2}
67 1+(−1.70−0.707i)T+(0.707+0.707i)T2 1 + (-1.70 - 0.707i)T + (0.707 + 0.707i)T^{2}
71 1−iT2 1 - iT^{2}
73 1−iT2 1 - iT^{2}
79 1−1.41iT−T2 1 - 1.41iT - T^{2}
83 1+(0.707+0.707i)T2 1 + (0.707 + 0.707i)T^{2}
89 1+iT2 1 + iT^{2}
97 1−T2 1 - T^{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

−12.70719685740070954544584736102, −11.22085608121654554169311210613, −10.46067214844018528815526359447, −9.772348064972584875416691748883, −8.331101550051849999487007881591, −7.69871991938462507386883068052, −6.80257890134922510809578542478, −5.34425991198805266625831792784, −4.38902995657153048539593973232, −1.87327003141636038599724248583, 1.86659789505119545782081617086, 3.42045103322412422864168502433, 4.84629761488409957491405872450, 6.42454372208400273938871110723, 7.81307626176211525269068538871, 8.469677166565118245229208814431, 9.653087339370503317253723176989, 10.32268712461356583066495434374, 11.59556595055060028731739961768, 12.04135720120587371439196385071

Graph of the ZZ-function along the critical line