↑

Properties

Label 2-640-128.109-c1-0-63
Degree 22
Conductor 640640
Sign −0.976+0.215i-0.976 + 0.215i
Analytic cond. 5.110425.11042
Root an. cond. 2.260622.26062
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  + (0.823 − 1.15i)2-s + (1.18 + 0.635i)3-s + (−0.645 − 1.89i)4-s + (−0.634 + 0.773i)5-s + (1.70 − 0.843i)6-s + (−4.07 − 2.72i)7-s + (−2.70 − 0.815i)8-s + (−0.658 − 0.985i)9-s + (0.366 + 1.36i)10-s + (−5.20 − 1.57i)11-s + (0.435 − 2.65i)12-s + (3.21 − 2.63i)13-s + (−6.48 + 2.44i)14-s + (−1.24 + 0.515i)15-s + (−3.16 + 2.44i)16-s + (3.95 + 1.64i)17-s + ⋯
L(s)  = 1  + (0.581 − 0.813i)2-s + (0.686 + 0.366i)3-s + (−0.322 − 0.946i)4-s + (−0.283 + 0.345i)5-s + (0.697 − 0.344i)6-s + (−1.53 − 1.02i)7-s + (−0.957 − 0.288i)8-s + (−0.219 − 0.328i)9-s + (0.116 + 0.431i)10-s + (−1.56 − 0.476i)11-s + (0.125 − 0.767i)12-s + (0.891 − 0.731i)13-s + (−1.73 + 0.653i)14-s + (−0.321 + 0.133i)15-s + (−0.791 + 0.610i)16-s + (0.960 + 0.397i)17-s + ⋯

Functional equation

Λ(s)=(640s/2 ΓC(s) L(s)=((−0.976+0.215i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.976 + 0.215i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(640s/2 ΓC(s+1/2) L(s)=((−0.976+0.215i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.976 + 0.215i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 640640    =    27⋅52^{7} \cdot 5
Sign: −0.976+0.215i-0.976 + 0.215i
Analytic conductor: 5.110425.11042
Root analytic conductor: 2.260622.26062
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ640(621,⋅)\chi_{640} (621, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 640, ( :1/2), −0.976+0.215i)(2,\ 640,\ (\ :1/2),\ -0.976 + 0.215i)

Particular Values

L(1)L(1) ≈\approx 0.133329−1.22255i0.133329 - 1.22255i
L(12)L(\frac12) ≈\approx 0.133329−1.22255i0.133329 - 1.22255i
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+(−0.823+1.15i)T 1 + (-0.823 + 1.15i)T
5 1+(0.634−0.773i)T 1 + (0.634 - 0.773i)T
good3 1+(−1.18−0.635i)T+(1.66+2.49i)T2 1 + (-1.18 - 0.635i)T + (1.66 + 2.49i)T^{2}
7 1+(4.07+2.72i)T+(2.67+6.46i)T2 1 + (4.07 + 2.72i)T + (2.67 + 6.46i)T^{2}
11 1+(5.20+1.57i)T+(9.14+6.11i)T2 1 + (5.20 + 1.57i)T + (9.14 + 6.11i)T^{2}
13 1+(−3.21+2.63i)T+(2.53−12.7i)T2 1 + (-3.21 + 2.63i)T + (2.53 - 12.7i)T^{2}
17 1+(−3.95−1.64i)T+(12.0+12.0i)T2 1 + (-3.95 - 1.64i)T + (12.0 + 12.0i)T^{2}
19 1+(0.725−7.36i)T+(−18.6−3.70i)T2 1 + (0.725 - 7.36i)T + (-18.6 - 3.70i)T^{2}
23 1+(0.125+0.0249i)T+(21.2+8.80i)T2 1 + (0.125 + 0.0249i)T + (21.2 + 8.80i)T^{2}
29 1+(0.673+2.21i)T+(−24.1+16.1i)T2 1 + (0.673 + 2.21i)T + (-24.1 + 16.1i)T^{2}
31 1+(−6.08+6.08i)T−31iT2 1 + (-6.08 + 6.08i)T - 31iT^{2}
37 1+(2.21−0.217i)T+(36.2−7.21i)T2 1 + (2.21 - 0.217i)T + (36.2 - 7.21i)T^{2}
41 1+(−2.29+11.5i)T+(−37.8−15.6i)T2 1 + (-2.29 + 11.5i)T + (-37.8 - 15.6i)T^{2}
43 1+(1.39−0.745i)T+(23.8−35.7i)T2 1 + (1.39 - 0.745i)T + (23.8 - 35.7i)T^{2}
47 1+(−2.64+6.39i)T+(−33.2−33.2i)T2 1 + (-2.64 + 6.39i)T + (-33.2 - 33.2i)T^{2}
53 1+(−1.26+4.17i)T+(−44.0−29.4i)T2 1 + (-1.26 + 4.17i)T + (-44.0 - 29.4i)T^{2}
59 1+(1.98+1.62i)T+(11.5+57.8i)T2 1 + (1.98 + 1.62i)T + (11.5 + 57.8i)T^{2}
61 1+(0.230−0.431i)T+(−33.8−50.7i)T2 1 + (0.230 - 0.431i)T + (-33.8 - 50.7i)T^{2}
67 1+(4.49−8.40i)T+(−37.2−55.7i)T2 1 + (4.49 - 8.40i)T + (-37.2 - 55.7i)T^{2}
71 1+(−3.72+5.57i)T+(−27.1−65.5i)T2 1 + (-3.72 + 5.57i)T + (-27.1 - 65.5i)T^{2}
73 1+(−3.99+2.67i)T+(27.9−67.4i)T2 1 + (-3.99 + 2.67i)T + (27.9 - 67.4i)T^{2}
79 1+(2.35+5.69i)T+(−55.8+55.8i)T2 1 + (2.35 + 5.69i)T + (-55.8 + 55.8i)T^{2}
83 1+(6.14+0.605i)T+(81.4+16.1i)T2 1 + (6.14 + 0.605i)T + (81.4 + 16.1i)T^{2}
89 1+(2.48−0.494i)T+(82.2−34.0i)T2 1 + (2.48 - 0.494i)T + (82.2 - 34.0i)T^{2}
97 1+(1.35−1.35i)T−97iT2 1 + (1.35 - 1.35i)T - 97iT^{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

−10.28010006349828799301153543596, −9.777040220500500792859341644845, −8.503806691494550901997262566078, −7.68478297833541464014902334665, −6.22614288097513270996430455462, −5.67447187367798370110536679174, −3.82468646294150955989788406262, −3.57380600132465621610164205031, −2.70384711750067323790100521499, −0.48101149615792726524400469515, 2.70263923327728635300213512438, 3.10061309992106238358794899013, 4.72136070917921146273347353691, 5.55895532031661480005499963756, 6.57063464432237178493167061170, 7.39832742693370488501814966539, 8.343927097388516228014576851229, 8.922340875272599660743467386861, 9.736771426986255949513466332547, 11.14858746639544451352611397790

Graph of the ZZ-function along the critical line