↑

Properties

Label 2-728-728.675-c1-0-31
Degree 22
Conductor 728728
Sign −0.620+0.784i-0.620 + 0.784i
Analytic cond. 5.813105.81310
Root an. cond. 2.411032.41103
Motivic weight 11
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 − 1.22i)2-s + (−2.41 − 1.39i)3-s + (−0.999 + 1.73i)4-s + (−2.74 + 1.58i)5-s + 3.94i·6-s + (0.710 − 2.54i)7-s + 2.82·8-s + (2.39 + 4.15i)9-s + (3.88 + 2.24i)10-s + (4.83 − 2.79i)12-s + 3.60i·13-s + (−3.62 + 0.931i)14-s + 8.84·15-s + (−2.00 − 3.46i)16-s + (5.61 + 3.24i)17-s + (3.38 − 5.87i)18-s + ⋯
L(s)  = 1  + (−0.499 − 0.866i)2-s + (−1.39 − 0.805i)3-s + (−0.499 + 0.866i)4-s + (−1.22 + 0.708i)5-s + 1.61i·6-s + (0.268 − 0.963i)7-s + 0.999·8-s + (0.798 + 1.38i)9-s + (1.22 + 0.708i)10-s + (1.39 − 0.805i)12-s + 0.999i·13-s + (−0.968 + 0.248i)14-s + 2.28·15-s + (−0.500 − 0.866i)16-s + (1.36 + 0.786i)17-s + (0.798 − 1.38i)18-s + ⋯

Functional equation

Λ(s)=(728s/2 ΓC(s) L(s)=((−0.620+0.784i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.620 + 0.784i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(728s/2 ΓC(s+1/2) L(s)=((−0.620+0.784i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.620 + 0.784i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 728728    =    23⋅7⋅132^{3} \cdot 7 \cdot 13
Sign: −0.620+0.784i-0.620 + 0.784i
Analytic conductor: 5.813105.81310
Root analytic conductor: 2.411032.41103
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ728(675,⋅)\chi_{728} (675, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 728, ( :1/2), −0.620+0.784i)(2,\ 728,\ (\ :1/2),\ -0.620 + 0.784i)

Particular Values

L(1)L(1) ≈\approx 0.175380−0.362174i0.175380 - 0.362174i
L(12)L(\frac12) ≈\approx 0.175380−0.362174i0.175380 - 0.362174i
L(32)L(\frac{3}{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+(0.707+1.22i)T 1 + (0.707 + 1.22i)T
7 1+(−0.710+2.54i)T 1 + (-0.710 + 2.54i)T
13 1−3.60iT 1 - 3.60iT
good3 1+(2.41+1.39i)T+(1.5+2.59i)T2 1 + (2.41 + 1.39i)T + (1.5 + 2.59i)T^{2}
5 1+(2.74−1.58i)T+(2.5−4.33i)T2 1 + (2.74 - 1.58i)T + (2.5 - 4.33i)T^{2}
11 1+(5.5+9.52i)T2 1 + (5.5 + 9.52i)T^{2}
17 1+(−5.61−3.24i)T+(8.5+14.7i)T2 1 + (-5.61 - 3.24i)T + (8.5 + 14.7i)T^{2}
19 1+(−9.5+16.4i)T2 1 + (-9.5 + 16.4i)T^{2}
23 1+(11.5−19.9i)T2 1 + (11.5 - 19.9i)T^{2}
29 1−29T2 1 - 29T^{2}
31 1+(9.48+5.47i)T+(15.5+26.8i)T2 1 + (9.48 + 5.47i)T + (15.5 + 26.8i)T^{2}
37 1+(5.06+8.76i)T+(−18.5+32.0i)T2 1 + (5.06 + 8.76i)T + (-18.5 + 32.0i)T^{2}
41 1+41T2 1 + 41T^{2}
43 1−13.1T+43T2 1 - 13.1T + 43T^{2}
47 1+(2.34−1.35i)T+(23.5−40.7i)T2 1 + (2.34 - 1.35i)T + (23.5 - 40.7i)T^{2}
53 1+(26.5+45.8i)T2 1 + (26.5 + 45.8i)T^{2}
59 1+(−29.5−51.0i)T2 1 + (-29.5 - 51.0i)T^{2}
61 1+(−30.5+52.8i)T2 1 + (-30.5 + 52.8i)T^{2}
67 1+(33.5+58.0i)T2 1 + (33.5 + 58.0i)T^{2}
71 1−15.7T+71T2 1 - 15.7T + 71T^{2}
73 1+(−36.5−63.2i)T2 1 + (-36.5 - 63.2i)T^{2}
79 1+(39.5−68.4i)T2 1 + (39.5 - 68.4i)T^{2}
83 1+83T2 1 + 83T^{2}
89 1+(−44.5+77.0i)T2 1 + (-44.5 + 77.0i)T^{2}
97 1+97T2 1 + 97T^{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

−10.68041190942427509803948030651, −9.486613400494767032136238800714, −8.063669915299760872325666110571, −7.39878825323980076821745352622, −6.98897136947671177415222251097, −5.64201355244744292916018676958, −4.26351809838524161211604088700, −3.61344301492177075435747829750, −1.75774673889826632871080382800, −0.45829451794364199343323664764, 0.849486795646886939645714661023, 3.62697820487122897670744888853, 4.94627667959251472910048633083, 5.20143329358368631194174233318, 6.03255250347620464395119594944, 7.31697855354850006980722452295, 8.091607957025637178421662561060, 8.943676678921623024166771633497, 9.799579724458806372625686435682, 10.67375261011096382397958325951

Graph of the ZZ-function along the critical line