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Properties

Label 2-720-45.2-c1-0-20
Degree 22
Conductor 720720
Sign 0.348+0.937i0.348 + 0.937i
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  + (−1.10 − 1.33i)3-s + (0.792 − 2.09i)5-s + (1.05 − 0.283i)7-s + (−0.548 + 2.94i)9-s + (5.44 + 3.14i)11-s + (3.34 + 0.896i)13-s + (−3.66 + 1.25i)15-s + (3.14 − 3.14i)17-s − 1.55i·19-s + (−1.55 − 1.09i)21-s + (−0.258 + 0.965i)23-s + (−3.74 − 3.31i)25-s + (4.53 − 2.53i)27-s + (−1.57 + 2.72i)29-s + (−2.22 − 3.85i)31-s + ⋯
L(s)  = 1  + (−0.639 − 0.769i)3-s + (0.354 − 0.935i)5-s + (0.400 − 0.107i)7-s + (−0.182 + 0.983i)9-s + (1.64 + 0.948i)11-s + (0.928 + 0.248i)13-s + (−0.945 + 0.325i)15-s + (0.763 − 0.763i)17-s − 0.355i·19-s + (−0.338 − 0.239i)21-s + (−0.0539 + 0.201i)23-s + (−0.748 − 0.663i)25-s + (0.872 − 0.487i)27-s + (−0.292 + 0.505i)29-s + (−0.399 − 0.692i)31-s + ⋯

Functional equation

Λ(s)=(720s/2 ΓC(s) L(s)=((0.348+0.937i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.348 + 0.937i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(720s/2 ΓC(s+1/2) L(s)=((0.348+0.937i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.348 + 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 720720    =    24⋅32⋅52^{4} \cdot 3^{2} \cdot 5
Sign: 0.348+0.937i0.348 + 0.937i
Analytic conductor: 5.749225.74922
Root analytic conductor: 2.397752.39775
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ720(497,⋅)\chi_{720} (497, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 720, ( :1/2), 0.348+0.937i)(2,\ 720,\ (\ :1/2),\ 0.348 + 0.937i)

Particular Values

L(1)L(1) ≈\approx 1.25455−0.871609i1.25455 - 0.871609i
L(12)L(\frac12) ≈\approx 1.25455−0.871609i1.25455 - 0.871609i
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+(1.10+1.33i)T 1 + (1.10 + 1.33i)T
5 1+(−0.792+2.09i)T 1 + (-0.792 + 2.09i)T
good7 1+(−1.05+0.283i)T+(6.06−3.5i)T2 1 + (-1.05 + 0.283i)T + (6.06 - 3.5i)T^{2}
11 1+(−5.44−3.14i)T+(5.5+9.52i)T2 1 + (-5.44 - 3.14i)T + (5.5 + 9.52i)T^{2}
13 1+(−3.34−0.896i)T+(11.2+6.5i)T2 1 + (-3.34 - 0.896i)T + (11.2 + 6.5i)T^{2}
17 1+(−3.14+3.14i)T−17iT2 1 + (-3.14 + 3.14i)T - 17iT^{2}
19 1+1.55iT−19T2 1 + 1.55iT - 19T^{2}
23 1+(0.258−0.965i)T+(−19.9−11.5i)T2 1 + (0.258 - 0.965i)T + (-19.9 - 11.5i)T^{2}
29 1+(1.57−2.72i)T+(−14.5−25.1i)T2 1 + (1.57 - 2.72i)T + (-14.5 - 25.1i)T^{2}
31 1+(2.22+3.85i)T+(−15.5+26.8i)T2 1 + (2.22 + 3.85i)T + (-15.5 + 26.8i)T^{2}
37 1+(3+3i)T+37iT2 1 + (3 + 3i)T + 37iT^{2}
41 1+(3.39−1.96i)T+(20.5−35.5i)T2 1 + (3.39 - 1.96i)T + (20.5 - 35.5i)T^{2}
43 1+(0.896+3.34i)T+(−37.2+21.5i)T2 1 + (0.896 + 3.34i)T + (-37.2 + 21.5i)T^{2}
47 1+(2.32+8.69i)T+(−40.7+23.5i)T2 1 + (2.32 + 8.69i)T + (-40.7 + 23.5i)T^{2}
53 1+(−6.61−6.61i)T+53iT2 1 + (-6.61 - 6.61i)T + 53iT^{2}
59 1+(−5.90−10.2i)T+(−29.5+51.0i)T2 1 + (-5.90 - 10.2i)T + (-29.5 + 51.0i)T^{2}
61 1+(−2.72+4.71i)T+(−30.5−52.8i)T2 1 + (-2.72 + 4.71i)T + (-30.5 - 52.8i)T^{2}
67 1+(0.978−3.65i)T+(−58.0−33.5i)T2 1 + (0.978 - 3.65i)T + (-58.0 - 33.5i)T^{2}
71 1+0.635iT−71T2 1 + 0.635iT - 71T^{2}
73 1+(−2.89+2.89i)T−73iT2 1 + (-2.89 + 2.89i)T - 73iT^{2}
79 1+(2.12+1.22i)T+(39.5+68.4i)T2 1 + (2.12 + 1.22i)T + (39.5 + 68.4i)T^{2}
83 1+(−0.531+0.142i)T+(71.8−41.5i)T2 1 + (-0.531 + 0.142i)T + (71.8 - 41.5i)T^{2}
89 1+2.36T+89T2 1 + 2.36T + 89T^{2}
97 1+(−10.7+2.89i)T+(84.0−48.5i)T2 1 + (-10.7 + 2.89i)T + (84.0 - 48.5i)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

−10.23084538520184278275259126385, −9.272493941762794124561985412105, −8.601245977567894423327363144588, −7.48050480391426009165806767622, −6.74109468099192778747557768988, −5.77212304504437341149225672687, −4.92793852246534286196181039502, −3.91931703893039862341005045040, −1.90167968012703380951272818188, −1.08388334578722462950625355891, 1.40001341998620602543489607870, 3.38560982027414113342567854533, 3.85920943645874940210307653602, 5.35570501066652036097349244571, 6.18054489515975045897298305336, 6.65442490233123946395454229886, 8.153022016405837514825485963616, 8.975361609142908143903474006617, 9.906573231578370639008988589306, 10.60595857366989085188840666337

Graph of the ZZ-function along the critical line