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Properties

Label 2-3e6-27.25-c1-0-16
Degree 22
Conductor 729729
Sign 0.993−0.116i0.993 - 0.116i
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  + (−0.426 + 2.42i)2-s + (−3.79 − 1.38i)4-s + (−2.35 + 1.97i)5-s + (2.49 − 0.909i)7-s + (2.50 − 4.34i)8-s + (−3.78 − 6.55i)10-s + (−2.63 − 2.20i)11-s + (−0.580 − 3.29i)13-s + (1.13 + 6.43i)14-s + (3.25 + 2.72i)16-s + (−1.28 − 2.22i)17-s + (1.04 − 1.81i)19-s + (11.6 − 4.25i)20-s + (6.46 − 5.42i)22-s + (−0.502 − 0.182i)23-s + ⋯
L(s)  = 1  + (−0.301 + 1.71i)2-s + (−1.89 − 0.691i)4-s + (−1.05 + 0.885i)5-s + (0.944 − 0.343i)7-s + (0.886 − 1.53i)8-s + (−1.19 − 2.07i)10-s + (−0.793 − 0.665i)11-s + (−0.161 − 0.913i)13-s + (0.303 + 1.71i)14-s + (0.813 + 0.682i)16-s + (−0.311 − 0.540i)17-s + (0.240 − 0.416i)19-s + (2.61 − 0.951i)20-s + (1.37 − 1.15i)22-s + (−0.104 − 0.0381i)23-s + ⋯

Functional equation

Λ(s)=(729s/2 ΓC(s) L(s)=((0.993−0.116i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.116i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(729s/2 ΓC(s+1/2) L(s)=((0.993−0.116i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 - 0.116i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 729729    =    363^{6}
Sign: 0.993−0.116i0.993 - 0.116i
Analytic conductor: 5.821095.82109
Root analytic conductor: 2.412692.41269
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ729(649,⋅)\chi_{729} (649, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 729, ( :1/2), 0.993−0.116i)(2,\ 729,\ (\ :1/2),\ 0.993 - 0.116i)

Particular Values

L(1)L(1) ≈\approx 0.488638+0.0284599i0.488638 + 0.0284599i
L(12)L(\frac12) ≈\approx 0.488638+0.0284599i0.488638 + 0.0284599i
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+(0.426−2.42i)T+(−1.87−0.684i)T2 1 + (0.426 - 2.42i)T + (-1.87 - 0.684i)T^{2}
5 1+(2.35−1.97i)T+(0.868−4.92i)T2 1 + (2.35 - 1.97i)T + (0.868 - 4.92i)T^{2}
7 1+(−2.49+0.909i)T+(5.36−4.49i)T2 1 + (-2.49 + 0.909i)T + (5.36 - 4.49i)T^{2}
11 1+(2.63+2.20i)T+(1.91+10.8i)T2 1 + (2.63 + 2.20i)T + (1.91 + 10.8i)T^{2}
13 1+(0.580+3.29i)T+(−12.2+4.44i)T2 1 + (0.580 + 3.29i)T + (-12.2 + 4.44i)T^{2}
17 1+(1.28+2.22i)T+(−8.5+14.7i)T2 1 + (1.28 + 2.22i)T + (-8.5 + 14.7i)T^{2}
19 1+(−1.04+1.81i)T+(−9.5−16.4i)T2 1 + (-1.04 + 1.81i)T + (-9.5 - 16.4i)T^{2}
23 1+(0.502+0.182i)T+(17.6+14.7i)T2 1 + (0.502 + 0.182i)T + (17.6 + 14.7i)T^{2}
29 1+(−0.439+2.49i)T+(−27.2−9.91i)T2 1 + (-0.439 + 2.49i)T + (-27.2 - 9.91i)T^{2}
31 1+(7.24+2.63i)T+(23.7+19.9i)T2 1 + (7.24 + 2.63i)T + (23.7 + 19.9i)T^{2}
37 1+(−5.14−8.91i)T+(−18.5+32.0i)T2 1 + (-5.14 - 8.91i)T + (-18.5 + 32.0i)T^{2}
41 1+(0.848+4.81i)T+(−38.5+14.0i)T2 1 + (0.848 + 4.81i)T + (-38.5 + 14.0i)T^{2}
43 1+(−2.10−1.76i)T+(7.46+42.3i)T2 1 + (-2.10 - 1.76i)T + (7.46 + 42.3i)T^{2}
47 1+(−5.31+1.93i)T+(36.0−30.2i)T2 1 + (-5.31 + 1.93i)T + (36.0 - 30.2i)T^{2}
53 1−6.42T+53T2 1 - 6.42T + 53T^{2}
59 1+(−1.26+1.06i)T+(10.2−58.1i)T2 1 + (-1.26 + 1.06i)T + (10.2 - 58.1i)T^{2}
61 1+(13.5−4.91i)T+(46.7−39.2i)T2 1 + (13.5 - 4.91i)T + (46.7 - 39.2i)T^{2}
67 1+(1.02+5.78i)T+(−62.9+22.9i)T2 1 + (1.02 + 5.78i)T + (-62.9 + 22.9i)T^{2}
71 1+(7.40+12.8i)T+(−35.5+61.4i)T2 1 + (7.40 + 12.8i)T + (-35.5 + 61.4i)T^{2}
73 1+(0.940−1.62i)T+(−36.5−63.2i)T2 1 + (0.940 - 1.62i)T + (-36.5 - 63.2i)T^{2}
79 1+(−2.98+16.9i)T+(−74.2−27.0i)T2 1 + (-2.98 + 16.9i)T + (-74.2 - 27.0i)T^{2}
83 1+(−0.689+3.90i)T+(−77.9−28.3i)T2 1 + (-0.689 + 3.90i)T + (-77.9 - 28.3i)T^{2}
89 1+(2.54−4.41i)T+(−44.5−77.0i)T2 1 + (2.54 - 4.41i)T + (-44.5 - 77.0i)T^{2}
97 1+(8.14+6.83i)T+(16.8+95.5i)T2 1 + (8.14 + 6.83i)T + (16.8 + 95.5i)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

−10.42162844681983859357660440416, −9.168683068546513891248598512946, −8.204276216775787799708805390260, −7.64828319490062083662020211608, −7.28317172309611311079116292360, −6.12546698689150273333907353153, −5.21103084414943631865309084208, −4.32397111431135198842834675649, −2.98110128860672980408828884439, −0.30044551073686893832044183833, 1.39821524699995410657806521834, 2.38866165982343432024543814065, 3.90126081637754393294400078378, 4.46204165922005192941877541063, 5.36472288442885385514698113824, 7.34060587679714591712096478936, 8.181595817088048681002437708343, 8.835768682873583936755351759222, 9.563181708580393455938539686548, 10.67070560273723737466581606802

Graph of the ZZ-function along the critical line