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Properties

Label 2-3e6-27.13-c1-0-23
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  + (0.274 + 1.55i)2-s + (−0.461 + 0.168i)4-s + (−1.28 − 1.07i)5-s + (−2.61 − 0.950i)7-s + (1.19 + 2.06i)8-s + (1.32 − 2.29i)10-s + (3.18 − 2.66i)11-s + (1.19 − 6.77i)13-s + (0.761 − 4.32i)14-s + (−3.63 + 3.04i)16-s + (0.488 − 0.845i)17-s + (−1.34 − 2.32i)19-s + (0.775 + 0.282i)20-s + (5.01 + 4.21i)22-s + (1.51 − 0.551i)23-s + ⋯
L(s)  = 1  + (0.193 + 1.09i)2-s + (−0.230 + 0.0840i)4-s + (−0.575 − 0.482i)5-s + (−0.987 − 0.359i)7-s + (0.420 + 0.729i)8-s + (0.418 − 0.725i)10-s + (0.958 − 0.804i)11-s + (0.331 − 1.87i)13-s + (0.203 − 1.15i)14-s + (−0.908 + 0.761i)16-s + (0.118 − 0.205i)17-s + (−0.308 − 0.533i)19-s + (0.173 + 0.0630i)20-s + (1.07 + 0.897i)22-s + (0.316 − 0.115i)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(82,⋅)\chi_{729} (82, \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 1.43479−0.0835672i1.43479 - 0.0835672i
L(12)L(\frac12) ≈\approx 1.43479−0.0835672i1.43479 - 0.0835672i
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+(−0.274−1.55i)T+(−1.87+0.684i)T2 1 + (-0.274 - 1.55i)T + (-1.87 + 0.684i)T^{2}
5 1+(1.28+1.07i)T+(0.868+4.92i)T2 1 + (1.28 + 1.07i)T + (0.868 + 4.92i)T^{2}
7 1+(2.61+0.950i)T+(5.36+4.49i)T2 1 + (2.61 + 0.950i)T + (5.36 + 4.49i)T^{2}
11 1+(−3.18+2.66i)T+(1.91−10.8i)T2 1 + (-3.18 + 2.66i)T + (1.91 - 10.8i)T^{2}
13 1+(−1.19+6.77i)T+(−12.2−4.44i)T2 1 + (-1.19 + 6.77i)T + (-12.2 - 4.44i)T^{2}
17 1+(−0.488+0.845i)T+(−8.5−14.7i)T2 1 + (-0.488 + 0.845i)T + (-8.5 - 14.7i)T^{2}
19 1+(1.34+2.32i)T+(−9.5+16.4i)T2 1 + (1.34 + 2.32i)T + (-9.5 + 16.4i)T^{2}
23 1+(−1.51+0.551i)T+(17.6−14.7i)T2 1 + (-1.51 + 0.551i)T + (17.6 - 14.7i)T^{2}
29 1+(−1.42−8.10i)T+(−27.2+9.91i)T2 1 + (-1.42 - 8.10i)T + (-27.2 + 9.91i)T^{2}
31 1+(0.981−0.357i)T+(23.7−19.9i)T2 1 + (0.981 - 0.357i)T + (23.7 - 19.9i)T^{2}
37 1+(−0.654+1.13i)T+(−18.5−32.0i)T2 1 + (-0.654 + 1.13i)T + (-18.5 - 32.0i)T^{2}
41 1+(−0.841+4.77i)T+(−38.5−14.0i)T2 1 + (-0.841 + 4.77i)T + (-38.5 - 14.0i)T^{2}
43 1+(−7.53+6.32i)T+(7.46−42.3i)T2 1 + (-7.53 + 6.32i)T + (7.46 - 42.3i)T^{2}
47 1+(−11.7−4.27i)T+(36.0+30.2i)T2 1 + (-11.7 - 4.27i)T + (36.0 + 30.2i)T^{2}
53 1+7.34T+53T2 1 + 7.34T + 53T^{2}
59 1+(6.93+5.81i)T+(10.2+58.1i)T2 1 + (6.93 + 5.81i)T + (10.2 + 58.1i)T^{2}
61 1+(−1.20−0.439i)T+(46.7+39.2i)T2 1 + (-1.20 - 0.439i)T + (46.7 + 39.2i)T^{2}
67 1+(0.806−4.57i)T+(−62.9−22.9i)T2 1 + (0.806 - 4.57i)T + (-62.9 - 22.9i)T^{2}
71 1+(−2.81+4.87i)T+(−35.5−61.4i)T2 1 + (-2.81 + 4.87i)T + (-35.5 - 61.4i)T^{2}
73 1+(−2.28−3.95i)T+(−36.5+63.2i)T2 1 + (-2.28 - 3.95i)T + (-36.5 + 63.2i)T^{2}
79 1+(−0.808−4.58i)T+(−74.2+27.0i)T2 1 + (-0.808 - 4.58i)T + (-74.2 + 27.0i)T^{2}
83 1+(1.00+5.67i)T+(−77.9+28.3i)T2 1 + (1.00 + 5.67i)T + (-77.9 + 28.3i)T^{2}
89 1+(−2.27−3.93i)T+(−44.5+77.0i)T2 1 + (-2.27 - 3.93i)T + (-44.5 + 77.0i)T^{2}
97 1+(−6.56+5.51i)T+(16.8−95.5i)T2 1 + (-6.56 + 5.51i)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

−10.55115514700079913905855009523, −9.170000445578031718834031374460, −8.500304141866003071289269027540, −7.65196142478523416393973280998, −6.83045178010266691419378714044, −6.02225761879833696798445725750, −5.24087138243292057514720644106, −4.00796864580223280477754803134, −2.99640499258973248179294375490, −0.73145251538387940281589159058, 1.60562938697263097818521600661, 2.72743771537326474042167179693, 3.88671489359827757663981382466, 4.29054841486621335523643122532, 6.24997036199589835030902002406, 6.76654655199948731335929363382, 7.70106403637083899927317665676, 9.322106914989496476700348676822, 9.448853068026077320347973608975, 10.58663808475116933059323227955

Graph of the ZZ-function along the critical line