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Properties

Label 2-720-80.59-c0-0-0
Degree 22
Conductor 720720
Sign −0.382−0.923i-0.382 - 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: −0.382−0.923i-0.382 - 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 1.1725922301.172592230
L(12)L(\frac12) ≈\approx 1.1725922301.172592230
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

−10.97317434809376828717867320608, −10.18655162890125860536696265526, −8.873034328356344227895818577097, −7.970036923532743868076785153349, −7.45237123227378816983632751686, −6.36693237660249894171426637406, −5.74710953651620593703347171559, −4.32768987079081907197032935784, −3.71103122058905116286250460304, −2.51040726453255872723452292787, 1.13445124268816426143868428178, 2.83457145538099995172097371699, 3.76971133479913460119449386306, 4.95770460266557193651478883978, 5.34522420386373569232417709751, 6.85744546489755928801315468096, 7.62033086590413236406189592231, 9.042360657172713694227520769634, 9.383612710503227533419508223958, 10.58622698184824029020416472264

Graph of the ZZ-function along the critical line