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Properties

Label 2-720-20.3-c3-0-23
Degree 22
Conductor 720720
Sign 0.930+0.365i0.930 + 0.365i
Analytic cond. 42.481342.4813
Root an. cond. 6.517776.51777
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2 − 11i)5-s + (55 + 55i)13-s + (5 − 5i)17-s + (−117 + 44i)25-s + 284i·29-s + (305 − 305i)37-s + 472·41-s − 343i·49-s + (−545 − 545i)53-s + 468·61-s + (495 − 715i)65-s + (845 + 845i)73-s + (−65 − 45i)85-s − 176i·89-s + (1.20e3 − 1.20e3i)97-s + ⋯
L(s)  = 1  + (−0.178 − 0.983i)5-s + (1.17 + 1.17i)13-s + (0.0713 − 0.0713i)17-s + (−0.936 + 0.351i)25-s + 1.81i·29-s + (1.35 − 1.35i)37-s + 1.79·41-s − i·49-s + (−1.41 − 1.41i)53-s + 0.982·61-s + (0.944 − 1.36i)65-s + (1.35 + 1.35i)73-s + (−0.0829 − 0.0574i)85-s − 0.209i·89-s + (1.26 − 1.26i)97-s + ⋯

Functional equation

Λ(s)=(720s/2 ΓC(s) L(s)=((0.930+0.365i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.930 + 0.365i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(720s/2 ΓC(s+3/2) L(s)=((0.930+0.365i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.930 + 0.365i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 720720    =    24⋅32⋅52^{4} \cdot 3^{2} \cdot 5
Sign: 0.930+0.365i0.930 + 0.365i
Analytic conductor: 42.481342.4813
Root analytic conductor: 6.517776.51777
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ720(703,⋅)\chi_{720} (703, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 720, ( :3/2), 0.930+0.365i)(2,\ 720,\ (\ :3/2),\ 0.930 + 0.365i)

Particular Values

L(2)L(2) ≈\approx 2.0193363242.019336324
L(12)L(\frac12) ≈\approx 2.0193363242.019336324
L(52)L(\frac{5}{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+(2+11i)T 1 + (2 + 11i)T
good7 1+343iT2 1 + 343iT^{2}
11 1−1.33e3T2 1 - 1.33e3T^{2}
13 1+(−55−55i)T+2.19e3iT2 1 + (-55 - 55i)T + 2.19e3iT^{2}
17 1+(−5+5i)T−4.91e3iT2 1 + (-5 + 5i)T - 4.91e3iT^{2}
19 1+6.85e3T2 1 + 6.85e3T^{2}
23 1−1.21e4iT2 1 - 1.21e4iT^{2}
29 1−284iT−2.43e4T2 1 - 284iT - 2.43e4T^{2}
31 1−2.97e4T2 1 - 2.97e4T^{2}
37 1+(−305+305i)T−5.06e4iT2 1 + (-305 + 305i)T - 5.06e4iT^{2}
41 1−472T+6.89e4T2 1 - 472T + 6.89e4T^{2}
43 1−7.95e4iT2 1 - 7.95e4iT^{2}
47 1+1.03e5iT2 1 + 1.03e5iT^{2}
53 1+(545+545i)T+1.48e5iT2 1 + (545 + 545i)T + 1.48e5iT^{2}
59 1+2.05e5T2 1 + 2.05e5T^{2}
61 1−468T+2.26e5T2 1 - 468T + 2.26e5T^{2}
67 1+3.00e5iT2 1 + 3.00e5iT^{2}
71 1−3.57e5T2 1 - 3.57e5T^{2}
73 1+(−845−845i)T+3.89e5iT2 1 + (-845 - 845i)T + 3.89e5iT^{2}
79 1+4.93e5T2 1 + 4.93e5T^{2}
83 1−5.71e5iT2 1 - 5.71e5iT^{2}
89 1+176iT−7.04e5T2 1 + 176iT - 7.04e5T^{2}
97 1+(−1.20e3+1.20e3i)T−9.12e5iT2 1 + (-1.20e3 + 1.20e3i)T - 9.12e5iT^{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

−9.752539448161338722711781985564, −9.021038256292121561963168621617, −8.434774163510553721743460744860, −7.40367563835019125982729736841, −6.38090626175692947072100751376, −5.42365980623551616013159577834, −4.41691765596952084384104064892, −3.58793458086556101685615658429, −1.93469832411392515147513232177, −0.827088644139587172965063481575, 0.844783072068396638577751644417, 2.50309463173715131180934963601, 3.40502973650737013732085776326, 4.43262118835111899270036255911, 5.94069092019349706860256077472, 6.30309606982893415500692894033, 7.70037538277589803902575063093, 8.035207237350253545276539554942, 9.327099308316052791764916078768, 10.17090362423516289352292614470

Graph of the ZZ-function along the critical line