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Properties

Label 2-720-80.59-c0-0-1
Degree 22
Conductor 720720
Sign 0.382+0.923i0.382 + 0.923i
Analytic cond. 0.3593260.359326
Root an. cond. 0.5994380.599438
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 − 0.707i)2-s + 1.00i·4-s + (0.707 − 0.707i)5-s + (0.707 − 0.707i)8-s − 1.00·10-s − 1.00·16-s − 1.41i·17-s + (1 + i)19-s + (0.707 + 0.707i)20-s + 1.41i·23-s − 1.00i·25-s − 2i·31-s + (0.707 + 0.707i)32-s + (−1.00 + 1.00i)34-s − 1.41i·38-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)2-s + 1.00i·4-s + (0.707 − 0.707i)5-s + (0.707 − 0.707i)8-s − 1.00·10-s − 1.00·16-s − 1.41i·17-s + (1 + i)19-s + (0.707 + 0.707i)20-s + 1.41i·23-s − 1.00i·25-s − 2i·31-s + (0.707 + 0.707i)32-s + (−1.00 + 1.00i)34-s − 1.41i·38-s + ⋯

Functional equation

Λ(s)=(720s/2 ΓC(s) L(s)=((0.382+0.923i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(720s/2 ΓC(s) L(s)=((0.382+0.923i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 720720    =    24⋅32⋅52^{4} \cdot 3^{2} \cdot 5
Sign: 0.382+0.923i0.382 + 0.923i
Analytic conductor: 0.3593260.359326
Root analytic conductor: 0.5994380.599438
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ720(379,⋅)\chi_{720} (379, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 720, ( :0), 0.382+0.923i)(2,\ 720,\ (\ :0),\ 0.382 + 0.923i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.76494218310.7649421831
L(12)L(\frac12) ≈\approx 0.76494218310.7649421831
L(1)L(1) 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+0.707i)T 1 + (0.707 + 0.707i)T
3 1 1
5 1+(−0.707+0.707i)T 1 + (-0.707 + 0.707i)T
good7 1−T2 1 - T^{2}
11 1−iT2 1 - iT^{2}
13 1+iT2 1 + iT^{2}
17 1+1.41iT−T2 1 + 1.41iT - T^{2}
19 1+(−1−i)T+iT2 1 + (-1 - i)T + iT^{2}
23 1−1.41iT−T2 1 - 1.41iT - T^{2}
29 1−iT2 1 - iT^{2}
31 1+2iT−T2 1 + 2iT - T^{2}
37 1−iT2 1 - iT^{2}
41 1−T2 1 - T^{2}
43 1+iT2 1 + iT^{2}
47 1+1.41T+T2 1 + 1.41T + T^{2}
53 1−iT2 1 - iT^{2}
59 1−iT2 1 - iT^{2}
61 1+(1−i)T−iT2 1 + (1 - i)T - iT^{2}
67 1−iT2 1 - iT^{2}
71 1+T2 1 + T^{2}
73 1+T2 1 + T^{2}
79 1−T2 1 - T^{2}
83 1+(1.41−1.41i)T−iT2 1 + (1.41 - 1.41i)T - iT^{2}
89 1−T2 1 - T^{2}
97 1−T2 1 - 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

−10.16082418165504980075734466990, −9.607022466584688324558734475436, −9.085113752990866780665547868187, −7.963615193229931851982675290965, −7.32067924072005349420657053653, −5.94177936700939655715355141630, −4.96641370577638982848358959965, −3.74322955803488103635970183538, −2.50138346045873189197283043621, −1.24946640468036301038987573849, 1.65089928933564993302833260922, 3.01702673398197661037316773425, 4.69447148435520395844182353373, 5.69949412272248130018087159535, 6.55672195386414276608856355551, 7.13931642202132727536844916255, 8.280599784901348970192876446868, 8.993184408357590596372335303342, 9.927025037500139583340724238653, 10.56172363104129373175838486318

Graph of the ZZ-function along the critical line