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Properties

Label 2-720-20.19-c2-0-27
Degree 22
Conductor 720720
Sign 0.5+0.866i0.5 + 0.866i
Analytic cond. 19.618519.6185
Root an. cond. 4.429284.42928
Motivic weight 22
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5·5-s + 6.92·7-s − 13.8i·11-s − 24i·13-s + 24i·17-s − 27.7i·19-s − 34.6·23-s + 25·25-s − 10·29-s − 13.8i·31-s + 34.6·35-s + 24i·37-s + 34·41-s + 20.7·43-s − 6.92·47-s + ⋯
L(s)  = 1  + 5-s + 0.989·7-s − 1.25i·11-s − 1.84i·13-s + 1.41i·17-s − 1.45i·19-s − 1.50·23-s + 25-s − 0.344·29-s − 0.446i·31-s + 0.989·35-s + 0.648i·37-s + 0.829·41-s + 0.483·43-s − 0.147·47-s + ⋯

Functional equation

Λ(s)=(720s/2 ΓC(s) L(s)=((0.5+0.866i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.5 + 0.866i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(720s/2 ΓC(s+1) L(s)=((0.5+0.866i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.5 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 720720    =    24⋅32⋅52^{4} \cdot 3^{2} \cdot 5
Sign: 0.5+0.866i0.5 + 0.866i
Analytic conductor: 19.618519.6185
Root analytic conductor: 4.429284.42928
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ720(559,⋅)\chi_{720} (559, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 720, ( :1), 0.5+0.866i)(2,\ 720,\ (\ :1),\ 0.5 + 0.866i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 2.3279045842.327904584
L(12)L(\frac12) ≈\approx 2.3279045842.327904584
L(2)L(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
5 1−5T 1 - 5T
good7 1−6.92T+49T2 1 - 6.92T + 49T^{2}
11 1+13.8iT−121T2 1 + 13.8iT - 121T^{2}
13 1+24iT−169T2 1 + 24iT - 169T^{2}
17 1−24iT−289T2 1 - 24iT - 289T^{2}
19 1+27.7iT−361T2 1 + 27.7iT - 361T^{2}
23 1+34.6T+529T2 1 + 34.6T + 529T^{2}
29 1+10T+841T2 1 + 10T + 841T^{2}
31 1+13.8iT−961T2 1 + 13.8iT - 961T^{2}
37 1−24iT−1.36e3T2 1 - 24iT - 1.36e3T^{2}
41 1−34T+1.68e3T2 1 - 34T + 1.68e3T^{2}
43 1−20.7T+1.84e3T2 1 - 20.7T + 1.84e3T^{2}
47 1+6.92T+2.20e3T2 1 + 6.92T + 2.20e3T^{2}
53 1−48iT−2.80e3T2 1 - 48iT - 2.80e3T^{2}
59 1+13.8iT−3.48e3T2 1 + 13.8iT - 3.48e3T^{2}
61 1−70T+3.72e3T2 1 - 70T + 3.72e3T^{2}
67 1+90.0T+4.48e3T2 1 + 90.0T + 4.48e3T^{2}
71 1+55.4iT−5.04e3T2 1 + 55.4iT - 5.04e3T^{2}
73 1+48iT−5.32e3T2 1 + 48iT - 5.32e3T^{2}
79 1+41.5iT−6.24e3T2 1 + 41.5iT - 6.24e3T^{2}
83 1−90.0T+6.88e3T2 1 - 90.0T + 6.88e3T^{2}
89 1−14T+7.92e3T2 1 - 14T + 7.92e3T^{2}
97 1−96iT−9.40e3T2 1 - 96iT - 9.40e3T^{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

−10.31527281922936358209801944700, −9.122213539388492099825270098590, −8.305485871304281729564396758096, −7.71287702929072176432967726138, −6.15654455347882590867478682176, −5.75341672568722120469444311567, −4.73782485484767712413829166694, −3.34016834057295409023454527215, −2.18328405311313213166555257278, −0.835694250791680051347260399380, 1.66023917571681579179405076384, 2.20279777068237032636777196745, 4.09714730759728576549701523702, 4.86859941825498010978045105326, 5.86112292722128189501304316846, 6.89199608667027130683765194133, 7.64588040757718390205406262999, 8.788186468099153119674933020469, 9.623753923130955082304654617270, 10.09484095183783481428174532734

Graph of the ZZ-function along the critical line