↑

Properties

Label 2-728-13.4-c1-0-1
Degree 22
Conductor 728728
Sign −0.751−0.659i-0.751 - 0.659i
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  + (−0.509 + 0.883i)3-s − 2.02i·5-s + (−0.866 + 0.5i)7-s + (0.979 + 1.69i)9-s + (−5.06 − 2.92i)11-s + (−2.41 + 2.68i)13-s + (1.78 + 1.03i)15-s + (3.04 + 5.26i)17-s + (−0.610 + 0.352i)19-s − 1.01i·21-s + (−2.48 + 4.30i)23-s + 0.919·25-s − 5.05·27-s + (−4.92 + 8.53i)29-s − 4.81i·31-s + ⋯
L(s)  = 1  + (−0.294 + 0.509i)3-s − 0.903i·5-s + (−0.327 + 0.188i)7-s + (0.326 + 0.565i)9-s + (−1.52 − 0.881i)11-s + (−0.668 + 0.743i)13-s + (0.460 + 0.265i)15-s + (0.737 + 1.27i)17-s + (−0.139 + 0.0808i)19-s − 0.222i·21-s + (−0.517 + 0.897i)23-s + 0.183·25-s − 0.973·27-s + (−0.914 + 1.58i)29-s − 0.864i·31-s + ⋯

Functional equation

Λ(s)=(728s/2 ΓC(s) L(s)=((−0.751−0.659i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.751 - 0.659i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(728s/2 ΓC(s+1/2) L(s)=((−0.751−0.659i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.751 - 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 728728    =    23⋅7⋅132^{3} \cdot 7 \cdot 13
Sign: −0.751−0.659i-0.751 - 0.659i
Analytic conductor: 5.813105.81310
Root analytic conductor: 2.411032.41103
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ728(225,⋅)\chi_{728} (225, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 728, ( :1/2), −0.751−0.659i)(2,\ 728,\ (\ :1/2),\ -0.751 - 0.659i)

Particular Values

L(1)L(1) ≈\approx 0.200103+0.531798i0.200103 + 0.531798i
L(12)L(\frac12) ≈\approx 0.200103+0.531798i0.200103 + 0.531798i
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 1
7 1+(0.866−0.5i)T 1 + (0.866 - 0.5i)T
13 1+(2.41−2.68i)T 1 + (2.41 - 2.68i)T
good3 1+(0.509−0.883i)T+(−1.5−2.59i)T2 1 + (0.509 - 0.883i)T + (-1.5 - 2.59i)T^{2}
5 1+2.02iT−5T2 1 + 2.02iT - 5T^{2}
11 1+(5.06+2.92i)T+(5.5+9.52i)T2 1 + (5.06 + 2.92i)T + (5.5 + 9.52i)T^{2}
17 1+(−3.04−5.26i)T+(−8.5+14.7i)T2 1 + (-3.04 - 5.26i)T + (-8.5 + 14.7i)T^{2}
19 1+(0.610−0.352i)T+(9.5−16.4i)T2 1 + (0.610 - 0.352i)T + (9.5 - 16.4i)T^{2}
23 1+(2.48−4.30i)T+(−11.5−19.9i)T2 1 + (2.48 - 4.30i)T + (-11.5 - 19.9i)T^{2}
29 1+(4.92−8.53i)T+(−14.5−25.1i)T2 1 + (4.92 - 8.53i)T + (-14.5 - 25.1i)T^{2}
31 1+4.81iT−31T2 1 + 4.81iT - 31T^{2}
37 1+(−1.45−0.838i)T+(18.5+32.0i)T2 1 + (-1.45 - 0.838i)T + (18.5 + 32.0i)T^{2}
41 1+(1.89+1.09i)T+(20.5+35.5i)T2 1 + (1.89 + 1.09i)T + (20.5 + 35.5i)T^{2}
43 1+(−0.391−0.677i)T+(−21.5+37.2i)T2 1 + (-0.391 - 0.677i)T + (-21.5 + 37.2i)T^{2}
47 1−5.26iT−47T2 1 - 5.26iT - 47T^{2}
53 1+10.0T+53T2 1 + 10.0T + 53T^{2}
59 1+(5.03−2.90i)T+(29.5−51.0i)T2 1 + (5.03 - 2.90i)T + (29.5 - 51.0i)T^{2}
61 1+(3.48+6.03i)T+(−30.5+52.8i)T2 1 + (3.48 + 6.03i)T + (-30.5 + 52.8i)T^{2}
67 1+(−9.63−5.56i)T+(33.5+58.0i)T2 1 + (-9.63 - 5.56i)T + (33.5 + 58.0i)T^{2}
71 1+(10.7−6.22i)T+(35.5−61.4i)T2 1 + (10.7 - 6.22i)T + (35.5 - 61.4i)T^{2}
73 1+11.4iT−73T2 1 + 11.4iT - 73T^{2}
79 1−1.05T+79T2 1 - 1.05T + 79T^{2}
83 1+3.04iT−83T2 1 + 3.04iT - 83T^{2}
89 1+(1.34+0.775i)T+(44.5+77.0i)T2 1 + (1.34 + 0.775i)T + (44.5 + 77.0i)T^{2}
97 1+(−1.33+0.767i)T+(48.5−84.0i)T2 1 + (-1.33 + 0.767i)T + (48.5 - 84.0i)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

−10.65672006636119345169524707125, −9.924737762611939785505765879979, −9.094842683458798753708430707645, −8.131032156274490636291435827059, −7.48953060698649360396748519462, −5.95692814710090517617079326762, −5.31730820815345411938065788699, −4.51074581639131382542107280841, −3.30437609375132688338027008223, −1.77732907451972688249686981972, 0.28757343407275822317073358684, 2.33740334738522656758597591762, 3.19619734445304582112153965324, 4.67218786923450212878991788166, 5.67716397997850005968009950442, 6.72485270705195221624971040925, 7.38363845827066758305439172868, 7.947992014248597005626894001580, 9.560114367038726217958559007594, 10.07612984570341574322782992655

Graph of the ZZ-function along the critical line