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Properties

Label 2-3e6-243.103-c1-0-1
Degree 22
Conductor 729729
Sign 0.0820−0.996i0.0820 - 0.996i
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  + (−1.16 − 0.323i)2-s + (−0.467 − 0.282i)4-s + (1.06 − 0.0411i)5-s + (−5.09 − 0.796i)7-s + (2.10 + 2.23i)8-s + (−1.24 − 0.295i)10-s + (−0.460 − 3.36i)11-s + (1.07 − 1.56i)13-s + (5.65 + 2.57i)14-s + (−1.21 − 2.30i)16-s + (1.45 − 0.730i)17-s + (0.362 + 6.23i)19-s + (−0.507 − 0.279i)20-s + (−0.554 + 4.06i)22-s + (−1.67 + 4.33i)23-s + ⋯
L(s)  = 1  + (−0.821 − 0.228i)2-s + (−0.233 − 0.141i)4-s + (0.474 − 0.0183i)5-s + (−1.92 − 0.301i)7-s + (0.744 + 0.789i)8-s + (−0.393 − 0.0932i)10-s + (−0.138 − 1.01i)11-s + (0.298 − 0.434i)13-s + (1.51 + 0.687i)14-s + (−0.304 − 0.577i)16-s + (0.353 − 0.177i)17-s + (0.0832 + 1.42i)19-s + (−0.113 − 0.0625i)20-s + (−0.118 + 0.865i)22-s + (−0.349 + 0.903i)23-s + ⋯

Functional equation

Λ(s)=(729s/2 ΓC(s) L(s)=((0.0820−0.996i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0820 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(729s/2 ΓC(s+1/2) L(s)=((0.0820−0.996i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0820 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 729729    =    363^{6}
Sign: 0.0820−0.996i0.0820 - 0.996i
Analytic conductor: 5.821095.82109
Root analytic conductor: 2.412692.41269
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ729(685,⋅)\chi_{729} (685, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 729, ( :1/2), 0.0820−0.996i)(2,\ 729,\ (\ :1/2),\ 0.0820 - 0.996i)

Particular Values

L(1)L(1) ≈\approx 0.243904+0.224652i0.243904 + 0.224652i
L(12)L(\frac12) ≈\approx 0.243904+0.224652i0.243904 + 0.224652i
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+(1.16+0.323i)T+(1.71+1.03i)T2 1 + (1.16 + 0.323i)T + (1.71 + 1.03i)T^{2}
5 1+(−1.06+0.0411i)T+(4.98−0.387i)T2 1 + (-1.06 + 0.0411i)T + (4.98 - 0.387i)T^{2}
7 1+(5.09+0.796i)T+(6.66+2.13i)T2 1 + (5.09 + 0.796i)T + (6.66 + 2.13i)T^{2}
11 1+(0.460+3.36i)T+(−10.5+2.94i)T2 1 + (0.460 + 3.36i)T + (-10.5 + 2.94i)T^{2}
13 1+(−1.07+1.56i)T+(−4.68−12.1i)T2 1 + (-1.07 + 1.56i)T + (-4.68 - 12.1i)T^{2}
17 1+(−1.45+0.730i)T+(10.1−13.6i)T2 1 + (-1.45 + 0.730i)T + (10.1 - 13.6i)T^{2}
19 1+(−0.362−6.23i)T+(−18.8+2.20i)T2 1 + (-0.362 - 6.23i)T + (-18.8 + 2.20i)T^{2}
23 1+(1.67−4.33i)T+(−17.0−15.4i)T2 1 + (1.67 - 4.33i)T + (-17.0 - 15.4i)T^{2}
29 1+(−5.05−3.61i)T+(9.38+27.4i)T2 1 + (-5.05 - 3.61i)T + (9.38 + 27.4i)T^{2}
31 1+(3.36−3.85i)T+(−4.19−30.7i)T2 1 + (3.36 - 3.85i)T + (-4.19 - 30.7i)T^{2}
37 1+(3.97+5.33i)T+(−10.6+35.4i)T2 1 + (3.97 + 5.33i)T + (-10.6 + 35.4i)T^{2}
41 1+(1.40−5.44i)T+(−35.8−19.8i)T2 1 + (1.40 - 5.44i)T + (-35.8 - 19.8i)T^{2}
43 1+(−3.72−3.38i)T+(4.16+42.7i)T2 1 + (-3.72 - 3.38i)T + (4.16 + 42.7i)T^{2}
47 1+(−3.71−4.25i)T+(−6.36+46.5i)T2 1 + (-3.71 - 4.25i)T + (-6.36 + 46.5i)T^{2}
53 1+(−0.178+1.01i)T+(−49.8−18.1i)T2 1 + (-0.178 + 1.01i)T + (-49.8 - 18.1i)T^{2}
59 1+(7.85−3.20i)T+(42.1−41.3i)T2 1 + (7.85 - 3.20i)T + (42.1 - 41.3i)T^{2}
61 1+(4.86−2.93i)T+(28.4−53.9i)T2 1 + (4.86 - 2.93i)T + (28.4 - 53.9i)T^{2}
67 1+(2.72−1.94i)T+(21.6−63.3i)T2 1 + (2.72 - 1.94i)T + (21.6 - 63.3i)T^{2}
71 1+(−4.36−14.5i)T+(−59.3+39.0i)T2 1 + (-4.36 - 14.5i)T + (-59.3 + 39.0i)T^{2}
73 1+(−3.76+0.892i)T+(65.2−32.7i)T2 1 + (-3.76 + 0.892i)T + (65.2 - 32.7i)T^{2}
79 1+(9.19−9.01i)T+(1.53−78.9i)T2 1 + (9.19 - 9.01i)T + (1.53 - 78.9i)T^{2}
83 1+(−1.47−5.72i)T+(−72.6+40.0i)T2 1 + (-1.47 - 5.72i)T + (-72.6 + 40.0i)T^{2}
89 1+(−0.365+1.22i)T+(−74.3−48.9i)T2 1 + (-0.365 + 1.22i)T + (-74.3 - 48.9i)T^{2}
97 1+(11.0+0.427i)T+(96.7+7.51i)T2 1 + (11.0 + 0.427i)T + (96.7 + 7.51i)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

−10.25613417754133180412057178969, −9.811983599810337850689215785022, −9.113345466733858899518256293408, −8.234342361197484694247247797015, −7.25599054507636846448840325335, −5.99865426731045609383599750252, −5.60801381013410491635408600039, −3.85269586268414010317539771041, −2.97545345019189576065131419604, −1.23495226827746904417891576440, 0.24982651994966467906470292472, 2.28477357379281491957153827626, 3.56729433612705074216355868005, 4.65889441775433967583027874118, 6.09631298527584732708165810565, 6.75377252747560566270518308139, 7.55033478674608381916220183741, 8.807711618830692745738167375321, 9.316665869936017382668274338645, 9.964017395766527171891005219170

Graph of the ZZ-function along the critical line