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Properties

Label 2-720-9.7-c1-0-20
Degree 22
Conductor 720720
Sign 0.00922+0.999i0.00922 + 0.999i
Analytic cond. 5.749225.74922
Root an. cond. 2.397752.39775
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.724 − 1.57i)3-s + (0.5 + 0.866i)5-s + (1.72 − 2.98i)7-s + (−1.94 − 2.28i)9-s + (1.72 − 0.158i)15-s − 2·17-s + 6.89·19-s + (−3.44 − 4.87i)21-s + (−3.72 − 6.45i)23-s + (−0.499 + 0.866i)25-s + (−5.00 + 1.41i)27-s + (0.949 − 1.64i)29-s + (0.550 + 0.953i)31-s + 3.44·35-s − 6·37-s + ⋯
L(s)  = 1  + (0.418 − 0.908i)3-s + (0.223 + 0.387i)5-s + (0.651 − 1.12i)7-s + (−0.649 − 0.760i)9-s + (0.445 − 0.0410i)15-s − 0.485·17-s + 1.58·19-s + (−0.752 − 1.06i)21-s + (−0.776 − 1.34i)23-s + (−0.0999 + 0.173i)25-s + (−0.962 + 0.272i)27-s + (0.176 − 0.305i)29-s + (0.0988 + 0.171i)31-s + 0.583·35-s − 0.986·37-s + ⋯

Functional equation

Λ(s)=(720s/2 ΓC(s) L(s)=((0.00922+0.999i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00922 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(720s/2 ΓC(s+1/2) L(s)=((0.00922+0.999i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.00922 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 720720    =    24⋅32⋅52^{4} \cdot 3^{2} \cdot 5
Sign: 0.00922+0.999i0.00922 + 0.999i
Analytic conductor: 5.749225.74922
Root analytic conductor: 2.397752.39775
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ720(241,⋅)\chi_{720} (241, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 720, ( :1/2), 0.00922+0.999i)(2,\ 720,\ (\ :1/2),\ 0.00922 + 0.999i)

Particular Values

L(1)L(1) ≈\approx 1.29837−1.28644i1.29837 - 1.28644i
L(12)L(\frac12) ≈\approx 1.29837−1.28644i1.29837 - 1.28644i
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
3 1+(−0.724+1.57i)T 1 + (-0.724 + 1.57i)T
5 1+(−0.5−0.866i)T 1 + (-0.5 - 0.866i)T
good7 1+(−1.72+2.98i)T+(−3.5−6.06i)T2 1 + (-1.72 + 2.98i)T + (-3.5 - 6.06i)T^{2}
11 1+(−5.5−9.52i)T2 1 + (-5.5 - 9.52i)T^{2}
13 1+(−6.5+11.2i)T2 1 + (-6.5 + 11.2i)T^{2}
17 1+2T+17T2 1 + 2T + 17T^{2}
19 1−6.89T+19T2 1 - 6.89T + 19T^{2}
23 1+(3.72+6.45i)T+(−11.5+19.9i)T2 1 + (3.72 + 6.45i)T + (-11.5 + 19.9i)T^{2}
29 1+(−0.949+1.64i)T+(−14.5−25.1i)T2 1 + (-0.949 + 1.64i)T + (-14.5 - 25.1i)T^{2}
31 1+(−0.550−0.953i)T+(−15.5+26.8i)T2 1 + (-0.550 - 0.953i)T + (-15.5 + 26.8i)T^{2}
37 1+6T+37T2 1 + 6T + 37T^{2}
41 1+(−4.94−8.57i)T+(−20.5+35.5i)T2 1 + (-4.94 - 8.57i)T + (-20.5 + 35.5i)T^{2}
43 1+(−5.89+10.2i)T+(−21.5−37.2i)T2 1 + (-5.89 + 10.2i)T + (-21.5 - 37.2i)T^{2}
47 1+(4.72−8.18i)T+(−23.5−40.7i)T2 1 + (4.72 - 8.18i)T + (-23.5 - 40.7i)T^{2}
53 1−7.79T+53T2 1 - 7.79T + 53T^{2}
59 1+(0.550+0.953i)T+(−29.5+51.0i)T2 1 + (0.550 + 0.953i)T + (-29.5 + 51.0i)T^{2}
61 1+(−1.5+2.59i)T+(−30.5−52.8i)T2 1 + (-1.5 + 2.59i)T + (-30.5 - 52.8i)T^{2}
67 1+(6.62+11.4i)T+(−33.5+58.0i)T2 1 + (6.62 + 11.4i)T + (-33.5 + 58.0i)T^{2}
71 1+9.79T+71T2 1 + 9.79T + 71T^{2}
73 1−13.7T+73T2 1 - 13.7T + 73T^{2}
79 1+(3.44−5.97i)T+(−39.5−68.4i)T2 1 + (3.44 - 5.97i)T + (-39.5 - 68.4i)T^{2}
83 1+(2.72−4.71i)T+(−41.5−71.8i)T2 1 + (2.72 - 4.71i)T + (-41.5 - 71.8i)T^{2}
89 1−2.79T+89T2 1 - 2.79T + 89T^{2}
97 1+(1−1.73i)T+(−48.5−84.0i)T2 1 + (1 - 1.73i)T + (-48.5 - 84.0i)T^{2}
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   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

−10.26079918218444508237033054458, −9.286671787870741268903462382014, −8.247626688955354888665877254979, −7.54942886089865946097834693420, −6.87981915569236881268368140640, −5.95429030302776918113362440536, −4.63879131146943997356033840944, −3.47539106898845180001855828377, −2.26823031383768460065273737564, −0.962778779675070972594813309367, 1.86822181613396461112825376592, 3.03881253096132605292838542846, 4.23081924586108941434091526201, 5.34128921405890531874754445330, 5.68136778805612990920863920969, 7.37066044645943103308851492332, 8.286359980126810791528012763339, 9.018392504470202848595215541374, 9.581452996931094808019792354208, 10.48885653339331397485250377547

Graph of the ZZ-function along the critical line