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Properties

Label 2-720-20.19-c2-0-20
Degree 22
Conductor 720720
Sign 0.799+0.599i0.799 + 0.599i
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  + (4 + 3i)5-s − 24i·13-s − 30i·17-s + (7 + 24i)25-s + 40·29-s + 24i·37-s + 80·41-s − 49·49-s − 90i·53-s + 22·61-s + (72 − 96i)65-s + 96i·73-s + (90 − 120i)85-s + 160·89-s − 144i·97-s + ⋯
L(s)  = 1  + (0.800 + 0.600i)5-s − 1.84i·13-s − 1.76i·17-s + (0.280 + 0.959i)25-s + 1.37·29-s + 0.648i·37-s + 1.95·41-s − 0.999·49-s − 1.69i·53-s + 0.360·61-s + (1.10 − 1.47i)65-s + 1.31i·73-s + (1.05 − 1.41i)85-s + 1.79·89-s − 1.48i·97-s + ⋯

Functional equation

Λ(s)=(720s/2 ΓC(s) L(s)=((0.799+0.599i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.799 + 0.599i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(720s/2 ΓC(s+1) L(s)=((0.799+0.599i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.799 + 0.599i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 720720    =    24⋅32⋅52^{4} \cdot 3^{2} \cdot 5
Sign: 0.799+0.599i0.799 + 0.599i
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), 0.799+0.599i)(2,\ 720,\ (\ :1),\ 0.799 + 0.599i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 2.0626055782.062605578
L(12)L(\frac12) ≈\approx 2.0626055782.062605578
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+(−4−3i)T 1 + (-4 - 3i)T
good7 1+49T2 1 + 49T^{2}
11 1−121T2 1 - 121T^{2}
13 1+24iT−169T2 1 + 24iT - 169T^{2}
17 1+30iT−289T2 1 + 30iT - 289T^{2}
19 1−361T2 1 - 361T^{2}
23 1+529T2 1 + 529T^{2}
29 1−40T+841T2 1 - 40T + 841T^{2}
31 1−961T2 1 - 961T^{2}
37 1−24iT−1.36e3T2 1 - 24iT - 1.36e3T^{2}
41 1−80T+1.68e3T2 1 - 80T + 1.68e3T^{2}
43 1+1.84e3T2 1 + 1.84e3T^{2}
47 1+2.20e3T2 1 + 2.20e3T^{2}
53 1+90iT−2.80e3T2 1 + 90iT - 2.80e3T^{2}
59 1−3.48e3T2 1 - 3.48e3T^{2}
61 1−22T+3.72e3T2 1 - 22T + 3.72e3T^{2}
67 1+4.48e3T2 1 + 4.48e3T^{2}
71 1−5.04e3T2 1 - 5.04e3T^{2}
73 1−96iT−5.32e3T2 1 - 96iT - 5.32e3T^{2}
79 1−6.24e3T2 1 - 6.24e3T^{2}
83 1+6.88e3T2 1 + 6.88e3T^{2}
89 1−160T+7.92e3T2 1 - 160T + 7.92e3T^{2}
97 1+144iT−9.40e3T2 1 + 144iT - 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

−10.06209609593778240058757712816, −9.509845417310531378175017399470, −8.352699868080623691552539794036, −7.46890873056342618583782140776, −6.57926182402594870502176612674, −5.61980897062721922562137901515, −4.85989479768828035598562976539, −3.20685489441831752310022918616, −2.53831365851164639560552336411, −0.797299987566476688261041837917, 1.33381644013624857409242843241, 2.31722429720029262513114357722, 3.99152190107949918441622133530, 4.74682511606462404615535684663, 6.02747037150496146758744486226, 6.50604283296682602516088397348, 7.79384236081318026139011622890, 8.807950454077778297592798289202, 9.291737438529966952632907849883, 10.25897143868498582204806844618

Graph of the ZZ-function along the critical line