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Properties

Label 2-720-60.23-c0-0-1
Degree 22
Conductor 720720
Sign 0.927+0.374i0.927 + 0.374i
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)5-s + (1 + i)13-s + (−1.41 − 1.41i)17-s − 1.00i·25-s + 1.41·29-s + (−1 + i)37-s + 1.41i·41-s − i·49-s + (−1.41 + 1.41i)53-s + 1.41·65-s + (−1 − i)73-s − 2.00·85-s − 1.41·89-s + (−1 + i)97-s + 1.41i·101-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)5-s + (1 + i)13-s + (−1.41 − 1.41i)17-s − 1.00i·25-s + 1.41·29-s + (−1 + i)37-s + 1.41i·41-s − i·49-s + (−1.41 + 1.41i)53-s + 1.41·65-s + (−1 − i)73-s − 2.00·85-s − 1.41·89-s + (−1 + i)97-s + 1.41i·101-s + ⋯

Functional equation

Λ(s)=(720s/2 ΓC(s) L(s)=((0.927+0.374i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 + 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(720s/2 ΓC(s) L(s)=((0.927+0.374i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 + 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 720720    =    24⋅32⋅52^{4} \cdot 3^{2} \cdot 5
Sign: 0.927+0.374i0.927 + 0.374i
Analytic conductor: 0.3593260.359326
Root analytic conductor: 0.5994380.599438
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ720(143,⋅)\chi_{720} (143, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 720, ( :0), 0.927+0.374i)(2,\ 720,\ (\ :0),\ 0.927 + 0.374i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.0663286611.066328661
L(12)L(\frac12) ≈\approx 1.0663286611.066328661
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 1
3 1 1
5 1+(−0.707+0.707i)T 1 + (-0.707 + 0.707i)T
good7 1+iT2 1 + iT^{2}
11 1+T2 1 + T^{2}
13 1+(−1−i)T+iT2 1 + (-1 - i)T + iT^{2}
17 1+(1.41+1.41i)T+iT2 1 + (1.41 + 1.41i)T + iT^{2}
19 1+T2 1 + T^{2}
23 1+iT2 1 + iT^{2}
29 1−1.41T+T2 1 - 1.41T + T^{2}
31 1−T2 1 - T^{2}
37 1+(1−i)T−iT2 1 + (1 - i)T - iT^{2}
41 1−1.41iT−T2 1 - 1.41iT - T^{2}
43 1−iT2 1 - iT^{2}
47 1−iT2 1 - iT^{2}
53 1+(1.41−1.41i)T−iT2 1 + (1.41 - 1.41i)T - iT^{2}
59 1−T2 1 - T^{2}
61 1+T2 1 + T^{2}
67 1+iT2 1 + iT^{2}
71 1+T2 1 + T^{2}
73 1+(1+i)T+iT2 1 + (1 + i)T + iT^{2}
79 1+T2 1 + T^{2}
83 1+iT2 1 + iT^{2}
89 1+1.41T+T2 1 + 1.41T + T^{2}
97 1+(1−i)T−iT2 1 + (1 - i)T - iT^{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

−10.53305327494656311680483445437, −9.522693860509059120906141963969, −8.953670645370411018823123301450, −8.222892346881707750128076094486, −6.81254689025469872329129787550, −6.28110856756256488084407364861, −5.00197457637456366638609138068, −4.36013492501150072990291401627, −2.79133556326799713206313610323, −1.48032047084430832667545811613, 1.79875325280363937898677624188, 3.03630434389988737926753874782, 4.11886355285127020876414119677, 5.50816170288864579214599052022, 6.25047255963373333986948750064, 6.99653012123343644775849492610, 8.248060925027745167681633201336, 8.861739567397791545992470839305, 10.01406325985155680674132560920, 10.71583165439973006758104148918

Graph of the ZZ-function along the critical line