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Properties

Label 2-3e6-81.31-c1-0-6
Degree 22
Conductor 729729
Sign 0.928−0.370i0.928 - 0.370i
Analytic cond. 5.821095.82109
Root an. cond. 2.412692.41269
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.588 − 1.36i)2-s + (−0.141 + 0.149i)4-s + (−0.0932 + 1.60i)5-s + (−1.85 + 0.438i)7-s + (−2.50 − 0.911i)8-s + (2.23 − 0.814i)10-s + (−3.38 + 2.22i)11-s + (1.67 − 0.196i)13-s + (1.68 + 2.26i)14-s + (0.254 + 4.36i)16-s + (4.14 + 3.47i)17-s + (3.70 − 3.10i)19-s + (−0.226 − 0.240i)20-s + (5.03 + 3.30i)22-s + (0.651 + 0.154i)23-s + ⋯
L(s)  = 1  + (−0.415 − 0.964i)2-s + (−0.0706 + 0.0749i)4-s + (−0.0416 + 0.715i)5-s + (−0.699 + 0.165i)7-s + (−0.885 − 0.322i)8-s + (0.707 − 0.257i)10-s + (−1.02 + 0.671i)11-s + (0.465 − 0.0544i)13-s + (0.450 + 0.605i)14-s + (0.0635 + 1.09i)16-s + (1.00 + 0.844i)17-s + (0.849 − 0.712i)19-s + (−0.0506 − 0.0537i)20-s + (1.07 + 0.705i)22-s + (0.135 + 0.0322i)23-s + ⋯

Functional equation

Λ(s)=(729s/2 ΓC(s) L(s)=((0.928−0.370i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.928 - 0.370i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(729s/2 ΓC(s+1/2) L(s)=((0.928−0.370i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.928 - 0.370i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 729729    =    363^{6}
Sign: 0.928−0.370i0.928 - 0.370i
Analytic conductor: 5.821095.82109
Root analytic conductor: 2.412692.41269
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ729(676,⋅)\chi_{729} (676, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 729, ( :1/2), 0.928−0.370i)(2,\ 729,\ (\ :1/2),\ 0.928 - 0.370i)

Particular Values

L(1)L(1) ≈\approx 0.864663+0.166255i0.864663 + 0.166255i
L(12)L(\frac12) ≈\approx 0.864663+0.166255i0.864663 + 0.166255i
L(32)L(\frac{3}{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)
bad3 1 1
good2 1+(0.588+1.36i)T+(−1.37+1.45i)T2 1 + (0.588 + 1.36i)T + (-1.37 + 1.45i)T^{2}
5 1+(0.0932−1.60i)T+(−4.96−0.580i)T2 1 + (0.0932 - 1.60i)T + (-4.96 - 0.580i)T^{2}
7 1+(1.85−0.438i)T+(6.25−3.14i)T2 1 + (1.85 - 0.438i)T + (6.25 - 3.14i)T^{2}
11 1+(3.38−2.22i)T+(4.35−10.1i)T2 1 + (3.38 - 2.22i)T + (4.35 - 10.1i)T^{2}
13 1+(−1.67+0.196i)T+(12.6−2.99i)T2 1 + (-1.67 + 0.196i)T + (12.6 - 2.99i)T^{2}
17 1+(−4.14−3.47i)T+(2.95+16.7i)T2 1 + (-4.14 - 3.47i)T + (2.95 + 16.7i)T^{2}
19 1+(−3.70+3.10i)T+(3.29−18.7i)T2 1 + (-3.70 + 3.10i)T + (3.29 - 18.7i)T^{2}
23 1+(−0.651−0.154i)T+(20.5+10.3i)T2 1 + (-0.651 - 0.154i)T + (20.5 + 10.3i)T^{2}
29 1+(2.85−3.83i)T+(−8.31−27.7i)T2 1 + (2.85 - 3.83i)T + (-8.31 - 27.7i)T^{2}
31 1+(−1.39−4.66i)T+(−25.9+17.0i)T2 1 + (-1.39 - 4.66i)T + (-25.9 + 17.0i)T^{2}
37 1+(2.07−11.7i)T+(−34.7−12.6i)T2 1 + (2.07 - 11.7i)T + (-34.7 - 12.6i)T^{2}
41 1+(4.00−9.28i)T+(−28.1−29.8i)T2 1 + (4.00 - 9.28i)T + (-28.1 - 29.8i)T^{2}
43 1+(−7.64−3.83i)T+(25.6+34.4i)T2 1 + (-7.64 - 3.83i)T + (25.6 + 34.4i)T^{2}
47 1+(−0.0578+0.193i)T+(−39.2−25.8i)T2 1 + (-0.0578 + 0.193i)T + (-39.2 - 25.8i)T^{2}
53 1+(−2.48+4.29i)T+(−26.5−45.8i)T2 1 + (-2.48 + 4.29i)T + (-26.5 - 45.8i)T^{2}
59 1+(−1.58−1.04i)T+(23.3+54.1i)T2 1 + (-1.58 - 1.04i)T + (23.3 + 54.1i)T^{2}
61 1+(−0.124−0.131i)T+(−3.54+60.8i)T2 1 + (-0.124 - 0.131i)T + (-3.54 + 60.8i)T^{2}
67 1+(7.95+10.6i)T+(−19.2+64.1i)T2 1 + (7.95 + 10.6i)T + (-19.2 + 64.1i)T^{2}
71 1+(−9.41+3.42i)T+(54.3−45.6i)T2 1 + (-9.41 + 3.42i)T + (54.3 - 45.6i)T^{2}
73 1+(10.9+3.97i)T+(55.9+46.9i)T2 1 + (10.9 + 3.97i)T + (55.9 + 46.9i)T^{2}
79 1+(1.45+3.37i)T+(−54.2+57.4i)T2 1 + (1.45 + 3.37i)T + (-54.2 + 57.4i)T^{2}
83 1+(2.17+5.03i)T+(−56.9+60.3i)T2 1 + (2.17 + 5.03i)T + (-56.9 + 60.3i)T^{2}
89 1+(6.65+2.42i)T+(68.1+57.2i)T2 1 + (6.65 + 2.42i)T + (68.1 + 57.2i)T^{2}
97 1+(−0.0778−1.33i)T+(−96.3+11.2i)T2 1 + (-0.0778 - 1.33i)T + (-96.3 + 11.2i)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

−10.36378982262180167537121534760, −9.935282999038440078815763604937, −9.042292409319618452748003112548, −7.954759934386691032698618632273, −6.90799151240319035122018373559, −6.13354548427263973850228833685, −4.99519101784562465543139386773, −3.25047554493592004060208978339, −2.93303807384907315832721379395, −1.42285862591785635012445385268, 0.54039497531238048568727637539, 2.73184848105866042909409875899, 3.80319990910919190793901709146, 5.55467083912137662410966473007, 5.67019045244194233922340216631, 7.09107493896957383112846086466, 7.67762094949207735417197374541, 8.515393224019527114505136673419, 9.249963019731520542684531132052, 10.07078129070551007336399055963

Graph of the ZZ-function along the critical line