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Properties

Label 2-3e6-243.151-c1-0-1
Degree 22
Conductor 729729
Sign −0.538−0.842i-0.538 - 0.842i
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  + (1.83 − 0.510i)2-s + (1.39 − 0.841i)4-s + (−2.94 − 0.114i)5-s + (−2.67 + 0.417i)7-s + (−0.484 + 0.513i)8-s + (−5.46 + 1.29i)10-s + (−0.589 + 4.31i)11-s + (1.84 + 2.69i)13-s + (−4.68 + 2.13i)14-s + (−2.14 + 4.07i)16-s + (−4.70 − 2.36i)17-s + (0.0335 − 0.575i)19-s + (−4.20 + 2.32i)20-s + (1.12 + 8.21i)22-s + (−2.84 − 7.36i)23-s + ⋯
L(s)  = 1  + (1.29 − 0.361i)2-s + (0.697 − 0.420i)4-s + (−1.31 − 0.0511i)5-s + (−1.00 + 0.157i)7-s + (−0.171 + 0.181i)8-s + (−1.72 + 0.409i)10-s + (−0.177 + 1.30i)11-s + (0.511 + 0.746i)13-s + (−1.25 + 0.569i)14-s + (−0.536 + 1.01i)16-s + (−1.14 − 0.572i)17-s + (0.00769 − 0.132i)19-s + (−0.940 + 0.519i)20-s + (0.239 + 1.75i)22-s + (−0.592 − 1.53i)23-s + ⋯

Functional equation

Λ(s)=(729s/2 ΓC(s) L(s)=((−0.538−0.842i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.538 - 0.842i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(729s/2 ΓC(s+1/2) L(s)=((−0.538−0.842i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.538 - 0.842i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 729729    =    363^{6}
Sign: −0.538−0.842i-0.538 - 0.842i
Analytic conductor: 5.821095.82109
Root analytic conductor: 2.412692.41269
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ729(613,⋅)\chi_{729} (613, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 729, ( :1/2), −0.538−0.842i)(2,\ 729,\ (\ :1/2),\ -0.538 - 0.842i)

Particular Values

L(1)L(1) ≈\approx 0.399009+0.728388i0.399009 + 0.728388i
L(12)L(\frac12) ≈\approx 0.399009+0.728388i0.399009 + 0.728388i
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+(−1.83+0.510i)T+(1.71−1.03i)T2 1 + (-1.83 + 0.510i)T + (1.71 - 1.03i)T^{2}
5 1+(2.94+0.114i)T+(4.98+0.387i)T2 1 + (2.94 + 0.114i)T + (4.98 + 0.387i)T^{2}
7 1+(2.67−0.417i)T+(6.66−2.13i)T2 1 + (2.67 - 0.417i)T + (6.66 - 2.13i)T^{2}
11 1+(0.589−4.31i)T+(−10.5−2.94i)T2 1 + (0.589 - 4.31i)T + (-10.5 - 2.94i)T^{2}
13 1+(−1.84−2.69i)T+(−4.68+12.1i)T2 1 + (-1.84 - 2.69i)T + (-4.68 + 12.1i)T^{2}
17 1+(4.70+2.36i)T+(10.1+13.6i)T2 1 + (4.70 + 2.36i)T + (10.1 + 13.6i)T^{2}
19 1+(−0.0335+0.575i)T+(−18.8−2.20i)T2 1 + (-0.0335 + 0.575i)T + (-18.8 - 2.20i)T^{2}
23 1+(2.84+7.36i)T+(−17.0+15.4i)T2 1 + (2.84 + 7.36i)T + (-17.0 + 15.4i)T^{2}
29 1+(−0.495+0.353i)T+(9.38−27.4i)T2 1 + (-0.495 + 0.353i)T + (9.38 - 27.4i)T^{2}
31 1+(−5.02−5.75i)T+(−4.19+30.7i)T2 1 + (-5.02 - 5.75i)T + (-4.19 + 30.7i)T^{2}
37 1+(−1.86+2.50i)T+(−10.6−35.4i)T2 1 + (-1.86 + 2.50i)T + (-10.6 - 35.4i)T^{2}
41 1+(−2.17−8.45i)T+(−35.8+19.8i)T2 1 + (-2.17 - 8.45i)T + (-35.8 + 19.8i)T^{2}
43 1+(−0.623+0.565i)T+(4.16−42.7i)T2 1 + (-0.623 + 0.565i)T + (4.16 - 42.7i)T^{2}
47 1+(5.83−6.68i)T+(−6.36−46.5i)T2 1 + (5.83 - 6.68i)T + (-6.36 - 46.5i)T^{2}
53 1+(2.35+13.3i)T+(−49.8+18.1i)T2 1 + (2.35 + 13.3i)T + (-49.8 + 18.1i)T^{2}
59 1+(6.01+2.45i)T+(42.1+41.3i)T2 1 + (6.01 + 2.45i)T + (42.1 + 41.3i)T^{2}
61 1+(−4.92−2.97i)T+(28.4+53.9i)T2 1 + (-4.92 - 2.97i)T + (28.4 + 53.9i)T^{2}
67 1+(7.89+5.64i)T+(21.6+63.3i)T2 1 + (7.89 + 5.64i)T + (21.6 + 63.3i)T^{2}
71 1+(0.635−2.12i)T+(−59.3−39.0i)T2 1 + (0.635 - 2.12i)T + (-59.3 - 39.0i)T^{2}
73 1+(5.07+1.20i)T+(65.2+32.7i)T2 1 + (5.07 + 1.20i)T + (65.2 + 32.7i)T^{2}
79 1+(−8.20−8.04i)T+(1.53+78.9i)T2 1 + (-8.20 - 8.04i)T + (1.53 + 78.9i)T^{2}
83 1+(3.22−12.5i)T+(−72.6−40.0i)T2 1 + (3.22 - 12.5i)T + (-72.6 - 40.0i)T^{2}
89 1+(1.18+3.94i)T+(−74.3+48.9i)T2 1 + (1.18 + 3.94i)T + (-74.3 + 48.9i)T^{2}
97 1+(4.78−0.185i)T+(96.7−7.51i)T2 1 + (4.78 - 0.185i)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

−11.12416451383753354706521747290, −9.959441970844358182300496917015, −8.959119447884363859958893001523, −8.069847142707966981507871510442, −6.80370921282931591436321959975, −6.36069177766074974930789357329, −4.66751561536893385813620328753, −4.43738371910145522110474319995, −3.36014696114512785849730245534, −2.35919205136858386488206651789, 0.26576634062613349735267451467, 3.07320540737956303202203609442, 3.63420687764808119394126732497, 4.38404617777074681005215367667, 5.77065207579773059229778344192, 6.22798994224766678585922646448, 7.32751850478936638643202538130, 8.158749230173064021925089954067, 9.111136949166655233518736873257, 10.29976204939121383367561171248

Graph of the ZZ-function along the critical line