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Properties

Label 2-3e6-243.103-c1-0-13
Degree 22
Conductor 729729
Sign 0.774+0.632i0.774 + 0.632i
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  + (0.129 + 0.0359i)2-s + (−1.69 − 1.02i)4-s + (2.51 − 0.0976i)5-s + (4.08 + 0.638i)7-s + (−0.366 − 0.388i)8-s + (0.328 + 0.0778i)10-s + (−0.654 − 4.78i)11-s + (−0.189 + 0.276i)13-s + (0.504 + 0.229i)14-s + (1.81 + 3.44i)16-s + (−1.56 + 0.787i)17-s + (0.123 + 2.12i)19-s + (−4.36 − 2.41i)20-s + (0.0877 − 0.642i)22-s + (0.620 − 1.60i)23-s + ⋯
L(s)  = 1  + (0.0913 + 0.0254i)2-s + (−0.848 − 0.512i)4-s + (1.12 − 0.0436i)5-s + (1.54 + 0.241i)7-s + (−0.129 − 0.137i)8-s + (0.103 + 0.0246i)10-s + (−0.197 − 1.44i)11-s + (−0.0526 + 0.0767i)13-s + (0.134 + 0.0613i)14-s + (0.453 + 0.860i)16-s + (−0.380 + 0.190i)17-s + (0.0283 + 0.487i)19-s + (−0.977 − 0.539i)20-s + (0.0186 − 0.136i)22-s + (0.129 − 0.335i)23-s + ⋯

Functional equation

Λ(s)=(729s/2 ΓC(s) L(s)=((0.774+0.632i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.774 + 0.632i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(729s/2 ΓC(s+1/2) L(s)=((0.774+0.632i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.774 + 0.632i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 729729    =    363^{6}
Sign: 0.774+0.632i0.774 + 0.632i
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), 0.774+0.632i)(2,\ 729,\ (\ :1/2),\ 0.774 + 0.632i)

Particular Values

L(1)L(1) ≈\approx 1.71859−0.612069i1.71859 - 0.612069i
L(12)L(\frac12) ≈\approx 1.71859−0.612069i1.71859 - 0.612069i
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+(−0.129−0.0359i)T+(1.71+1.03i)T2 1 + (-0.129 - 0.0359i)T + (1.71 + 1.03i)T^{2}
5 1+(−2.51+0.0976i)T+(4.98−0.387i)T2 1 + (-2.51 + 0.0976i)T + (4.98 - 0.387i)T^{2}
7 1+(−4.08−0.638i)T+(6.66+2.13i)T2 1 + (-4.08 - 0.638i)T + (6.66 + 2.13i)T^{2}
11 1+(0.654+4.78i)T+(−10.5+2.94i)T2 1 + (0.654 + 4.78i)T + (-10.5 + 2.94i)T^{2}
13 1+(0.189−0.276i)T+(−4.68−12.1i)T2 1 + (0.189 - 0.276i)T + (-4.68 - 12.1i)T^{2}
17 1+(1.56−0.787i)T+(10.1−13.6i)T2 1 + (1.56 - 0.787i)T + (10.1 - 13.6i)T^{2}
19 1+(−0.123−2.12i)T+(−18.8+2.20i)T2 1 + (-0.123 - 2.12i)T + (-18.8 + 2.20i)T^{2}
23 1+(−0.620+1.60i)T+(−17.0−15.4i)T2 1 + (-0.620 + 1.60i)T + (-17.0 - 15.4i)T^{2}
29 1+(−0.238−0.170i)T+(9.38+27.4i)T2 1 + (-0.238 - 0.170i)T + (9.38 + 27.4i)T^{2}
31 1+(−5.27+6.04i)T+(−4.19−30.7i)T2 1 + (-5.27 + 6.04i)T + (-4.19 - 30.7i)T^{2}
37 1+(−6.18−8.31i)T+(−10.6+35.4i)T2 1 + (-6.18 - 8.31i)T + (-10.6 + 35.4i)T^{2}
41 1+(−1.90+7.40i)T+(−35.8−19.8i)T2 1 + (-1.90 + 7.40i)T + (-35.8 - 19.8i)T^{2}
43 1+(0.684+0.620i)T+(4.16+42.7i)T2 1 + (0.684 + 0.620i)T + (4.16 + 42.7i)T^{2}
47 1+(4.38+5.02i)T+(−6.36+46.5i)T2 1 + (4.38 + 5.02i)T + (-6.36 + 46.5i)T^{2}
53 1+(−2.13+12.1i)T+(−49.8−18.1i)T2 1 + (-2.13 + 12.1i)T + (-49.8 - 18.1i)T^{2}
59 1+(−1.52+0.623i)T+(42.1−41.3i)T2 1 + (-1.52 + 0.623i)T + (42.1 - 41.3i)T^{2}
61 1+(0.475−0.287i)T+(28.4−53.9i)T2 1 + (0.475 - 0.287i)T + (28.4 - 53.9i)T^{2}
67 1+(−6.22+4.45i)T+(21.6−63.3i)T2 1 + (-6.22 + 4.45i)T + (21.6 - 63.3i)T^{2}
71 1+(−3.03−10.1i)T+(−59.3+39.0i)T2 1 + (-3.03 - 10.1i)T + (-59.3 + 39.0i)T^{2}
73 1+(10.8−2.56i)T+(65.2−32.7i)T2 1 + (10.8 - 2.56i)T + (65.2 - 32.7i)T^{2}
79 1+(11.3−11.1i)T+(1.53−78.9i)T2 1 + (11.3 - 11.1i)T + (1.53 - 78.9i)T^{2}
83 1+(−3.20−12.4i)T+(−72.6+40.0i)T2 1 + (-3.20 - 12.4i)T + (-72.6 + 40.0i)T^{2}
89 1+(2.10−7.03i)T+(−74.3−48.9i)T2 1 + (2.10 - 7.03i)T + (-74.3 - 48.9i)T^{2}
97 1+(−0.615−0.0238i)T+(96.7+7.51i)T2 1 + (-0.615 - 0.0238i)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

−10.22404243816809609994246016874, −9.486326382223987648672467377861, −8.404196976016670643724120888904, −8.226527984912673572930521328114, −6.44936482129508131903783035868, −5.62919584862181432307008890293, −5.09405448190912865738837788852, −4.00400555150647635157867434122, −2.32352429063195538382966788947, −1.11068006733504424329624811257, 1.53510033750068317411054984549, 2.68778828665969101427304520475, 4.50665261944917320018404642480, 4.73839279605340879924598354195, 5.83708367643159024824743556239, 7.22306781207494882375331179824, 7.87260314373811646589038507867, 8.871200675633341326579994674156, 9.584635116728913271160218149471, 10.34384516610544995604965773426

Graph of the ZZ-function along the critical line