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Properties

Label 2-3e6-27.7-c1-0-6
Degree 22
Conductor 729729
Sign 0.802−0.597i0.802 - 0.597i
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  + (−1.20 + 0.439i)2-s + (−0.266 + 0.223i)4-s + (0.0775 + 0.439i)5-s + (−2.70 − 2.27i)7-s + (1.50 − 2.61i)8-s + (−0.286 − 0.497i)10-s + (−0.482 + 2.73i)11-s + (−3.09 − 1.12i)13-s + (4.26 + 1.55i)14-s + (−0.553 + 3.13i)16-s + (3.51 + 6.09i)17-s + (2.59 − 4.49i)19-s + (−0.118 − 0.0996i)20-s + (−0.620 − 3.51i)22-s + (5.57 − 4.67i)23-s + ⋯
L(s)  = 1  + (−0.854 + 0.310i)2-s + (−0.133 + 0.111i)4-s + (0.0346 + 0.196i)5-s + (−1.02 − 0.858i)7-s + (0.533 − 0.923i)8-s + (−0.0907 − 0.157i)10-s + (−0.145 + 0.825i)11-s + (−0.857 − 0.312i)13-s + (1.14 + 0.415i)14-s + (−0.138 + 0.784i)16-s + (0.853 + 1.47i)17-s + (0.594 − 1.03i)19-s + (−0.0265 − 0.0222i)20-s + (−0.132 − 0.750i)22-s + (1.16 − 0.975i)23-s + ⋯

Functional equation

Λ(s)=(729s/2 ΓC(s) L(s)=((0.802−0.597i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.802 - 0.597i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(729s/2 ΓC(s+1/2) L(s)=((0.802−0.597i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.802 - 0.597i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 729729    =    363^{6}
Sign: 0.802−0.597i0.802 - 0.597i
Analytic conductor: 5.821095.82109
Root analytic conductor: 2.412692.41269
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ729(163,⋅)\chi_{729} (163, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 729, ( :1/2), 0.802−0.597i)(2,\ 729,\ (\ :1/2),\ 0.802 - 0.597i)

Particular Values

L(1)L(1) ≈\approx 0.679275+0.225087i0.679275 + 0.225087i
L(12)L(\frac12) ≈\approx 0.679275+0.225087i0.679275 + 0.225087i
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+(1.20−0.439i)T+(1.53−1.28i)T2 1 + (1.20 - 0.439i)T + (1.53 - 1.28i)T^{2}
5 1+(−0.0775−0.439i)T+(−4.69+1.71i)T2 1 + (-0.0775 - 0.439i)T + (-4.69 + 1.71i)T^{2}
7 1+(2.70+2.27i)T+(1.21+6.89i)T2 1 + (2.70 + 2.27i)T + (1.21 + 6.89i)T^{2}
11 1+(0.482−2.73i)T+(−10.3−3.76i)T2 1 + (0.482 - 2.73i)T + (-10.3 - 3.76i)T^{2}
13 1+(3.09+1.12i)T+(9.95+8.35i)T2 1 + (3.09 + 1.12i)T + (9.95 + 8.35i)T^{2}
17 1+(−3.51−6.09i)T+(−8.5+14.7i)T2 1 + (-3.51 - 6.09i)T + (-8.5 + 14.7i)T^{2}
19 1+(−2.59+4.49i)T+(−9.5−16.4i)T2 1 + (-2.59 + 4.49i)T + (-9.5 - 16.4i)T^{2}
23 1+(−5.57+4.67i)T+(3.99−22.6i)T2 1 + (-5.57 + 4.67i)T + (3.99 - 22.6i)T^{2}
29 1+(3.40−1.23i)T+(22.2−18.6i)T2 1 + (3.40 - 1.23i)T + (22.2 - 18.6i)T^{2}
31 1+(1.48−1.24i)T+(5.38−30.5i)T2 1 + (1.48 - 1.24i)T + (5.38 - 30.5i)T^{2}
37 1+(−1.61−2.79i)T+(−18.5+32.0i)T2 1 + (-1.61 - 2.79i)T + (-18.5 + 32.0i)T^{2}
41 1+(−4.56−1.66i)T+(31.4+26.3i)T2 1 + (-4.56 - 1.66i)T + (31.4 + 26.3i)T^{2}
43 1+(1−5.67i)T+(−40.4−14.7i)T2 1 + (1 - 5.67i)T + (-40.4 - 14.7i)T^{2}
47 1+(−2.31−1.93i)T+(8.16+46.2i)T2 1 + (-2.31 - 1.93i)T + (8.16 + 46.2i)T^{2}
53 1−8.77T+53T2 1 - 8.77T + 53T^{2}
59 1+(−0.514−2.91i)T+(−55.4+20.1i)T2 1 + (-0.514 - 2.91i)T + (-55.4 + 20.1i)T^{2}
61 1+(−6.04−5.06i)T+(10.5+60.0i)T2 1 + (-6.04 - 5.06i)T + (10.5 + 60.0i)T^{2}
67 1+(−8.86−3.22i)T+(51.3+43.0i)T2 1 + (-8.86 - 3.22i)T + (51.3 + 43.0i)T^{2}
71 1+(2.65+4.59i)T+(−35.5+61.4i)T2 1 + (2.65 + 4.59i)T + (-35.5 + 61.4i)T^{2}
73 1+(−0.777+1.34i)T+(−36.5−63.2i)T2 1 + (-0.777 + 1.34i)T + (-36.5 - 63.2i)T^{2}
79 1+(−11.1+4.07i)T+(60.5−50.7i)T2 1 + (-11.1 + 4.07i)T + (60.5 - 50.7i)T^{2}
83 1+(−15.2+5.56i)T+(63.5−53.3i)T2 1 + (-15.2 + 5.56i)T + (63.5 - 53.3i)T^{2}
89 1+(9.21−15.9i)T+(−44.5−77.0i)T2 1 + (9.21 - 15.9i)T + (-44.5 - 77.0i)T^{2}
97 1+(−1.75+9.96i)T+(−91.1−33.1i)T2 1 + (-1.75 + 9.96i)T + (-91.1 - 33.1i)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

−10.23613214312298838295366406632, −9.661747669377858401354812157025, −8.857157595068361199209231254682, −7.78832107290762916404529719991, −7.12708747277902394542599839353, −6.53040784018095790970305354993, −4.99847828683696960396067708761, −3.96293570155094689995839303667, −2.86533934677251544714233457288, −0.856851872084023224857245447313, 0.75673667439372483522270822151, 2.42153069496494517170434371515, 3.46117418656772904146565702428, 5.26561989792676168036002211020, 5.54207975453760841746800612663, 7.03681072558805266378757326664, 7.86577905320871287402887953761, 9.018239789769627059613325031256, 9.378252914789455179128444531897, 9.975797277097905897177038779843

Graph of the ZZ-function along the critical line