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Properties

Label 2-3e6-81.67-c1-0-16
Degree 22
Conductor 729729
Sign 0.907−0.419i0.907 - 0.419i
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  + (0.652 − 0.429i)2-s + (−0.550 + 1.27i)4-s + (1.01 − 1.07i)5-s + (3.77 + 0.441i)7-s + (0.459 + 2.60i)8-s + (0.200 − 1.13i)10-s + (−1.44 + 4.84i)11-s + (0.261 − 4.48i)13-s + (2.65 − 1.33i)14-s + (−0.485 − 0.515i)16-s + (−4.30 + 1.56i)17-s + (4.19 + 1.52i)19-s + (0.814 + 1.88i)20-s + (1.13 + 3.78i)22-s + (3.43 − 0.401i)23-s + ⋯
L(s)  = 1  + (0.461 − 0.303i)2-s + (−0.275 + 0.637i)4-s + (0.454 − 0.481i)5-s + (1.42 + 0.166i)7-s + (0.162 + 0.922i)8-s + (0.0635 − 0.360i)10-s + (−0.437 + 1.45i)11-s + (0.0724 − 1.24i)13-s + (0.709 − 0.356i)14-s + (−0.121 − 0.128i)16-s + (−1.04 + 0.380i)17-s + (0.962 + 0.350i)19-s + (0.182 + 0.422i)20-s + (0.241 + 0.806i)22-s + (0.715 − 0.0836i)23-s + ⋯

Functional equation

Λ(s)=(729s/2 ΓC(s) L(s)=((0.907−0.419i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.907 - 0.419i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(729s/2 ΓC(s+1/2) L(s)=((0.907−0.419i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.907 - 0.419i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 729729    =    363^{6}
Sign: 0.907−0.419i0.907 - 0.419i
Analytic conductor: 5.821095.82109
Root analytic conductor: 2.412692.41269
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ729(28,⋅)\chi_{729} (28, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 729, ( :1/2), 0.907−0.419i)(2,\ 729,\ (\ :1/2),\ 0.907 - 0.419i)

Particular Values

L(1)L(1) ≈\approx 2.13303+0.469004i2.13303 + 0.469004i
L(12)L(\frac12) ≈\approx 2.13303+0.469004i2.13303 + 0.469004i
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+(−0.652+0.429i)T+(0.792−1.83i)T2 1 + (-0.652 + 0.429i)T + (0.792 - 1.83i)T^{2}
5 1+(−1.01+1.07i)T+(−0.290−4.99i)T2 1 + (-1.01 + 1.07i)T + (-0.290 - 4.99i)T^{2}
7 1+(−3.77−0.441i)T+(6.81+1.61i)T2 1 + (-3.77 - 0.441i)T + (6.81 + 1.61i)T^{2}
11 1+(1.44−4.84i)T+(−9.19−6.04i)T2 1 + (1.44 - 4.84i)T + (-9.19 - 6.04i)T^{2}
13 1+(−0.261+4.48i)T+(−12.9−1.50i)T2 1 + (-0.261 + 4.48i)T + (-12.9 - 1.50i)T^{2}
17 1+(4.30−1.56i)T+(13.0−10.9i)T2 1 + (4.30 - 1.56i)T + (13.0 - 10.9i)T^{2}
19 1+(−4.19−1.52i)T+(14.5+12.2i)T2 1 + (-4.19 - 1.52i)T + (14.5 + 12.2i)T^{2}
23 1+(−3.43+0.401i)T+(22.3−5.30i)T2 1 + (-3.43 + 0.401i)T + (22.3 - 5.30i)T^{2}
29 1+(−0.583−0.293i)T+(17.3+23.2i)T2 1 + (-0.583 - 0.293i)T + (17.3 + 23.2i)T^{2}
31 1+(−0.393−0.527i)T+(−8.89+29.6i)T2 1 + (-0.393 - 0.527i)T + (-8.89 + 29.6i)T^{2}
37 1+(−0.766−0.642i)T+(6.42+36.4i)T2 1 + (-0.766 - 0.642i)T + (6.42 + 36.4i)T^{2}
41 1+(−0.570−0.375i)T+(16.2+37.6i)T2 1 + (-0.570 - 0.375i)T + (16.2 + 37.6i)T^{2}
43 1+(−8.16+1.93i)T+(38.4−19.2i)T2 1 + (-8.16 + 1.93i)T + (38.4 - 19.2i)T^{2}
47 1+(−4.73+6.36i)T+(−13.4−45.0i)T2 1 + (-4.73 + 6.36i)T + (-13.4 - 45.0i)T^{2}
53 1+(2.07−3.59i)T+(−26.5−45.8i)T2 1 + (2.07 - 3.59i)T + (-26.5 - 45.8i)T^{2}
59 1+(1.51+5.04i)T+(−49.2+32.4i)T2 1 + (1.51 + 5.04i)T + (-49.2 + 32.4i)T^{2}
61 1+(2.68+6.23i)T+(−41.8+44.3i)T2 1 + (2.68 + 6.23i)T + (-41.8 + 44.3i)T^{2}
67 1+(−4.73+2.37i)T+(40.0−53.7i)T2 1 + (-4.73 + 2.37i)T + (40.0 - 53.7i)T^{2}
71 1+(1.06−6.03i)T+(−66.7−24.2i)T2 1 + (1.06 - 6.03i)T + (-66.7 - 24.2i)T^{2}
73 1+(0.764+4.33i)T+(−68.5+24.9i)T2 1 + (0.764 + 4.33i)T + (-68.5 + 24.9i)T^{2}
79 1+(9.39−6.17i)T+(31.2−72.5i)T2 1 + (9.39 - 6.17i)T + (31.2 - 72.5i)T^{2}
83 1+(1.35−0.891i)T+(32.8−76.2i)T2 1 + (1.35 - 0.891i)T + (32.8 - 76.2i)T^{2}
89 1+(0.181+1.02i)T+(−83.6+30.4i)T2 1 + (0.181 + 1.02i)T + (-83.6 + 30.4i)T^{2}
97 1+(3.42+3.62i)T+(−5.64+96.8i)T2 1 + (3.42 + 3.62i)T + (-5.64 + 96.8i)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

−10.65795229996278987006624207402, −9.544615892547079100857035332967, −8.645102529220618484564256754217, −7.905388026894487888535578812516, −7.23240341419865954025932925950, −5.46671511583230642925673296929, −5.00809813149367326728010469077, −4.17388829366212532653414151648, −2.72349597666272633688659943408, −1.66259339929559840598405799377, 1.14282462103448127595297095064, 2.57942787182437347095882414864, 4.16361503700700928034007597134, 4.94213075669194995492254737609, 5.81130449196698595867822294569, 6.64364972009758406998538310872, 7.58822401350265633838165170564, 8.756438864149266988804320287260, 9.348737939752917283645954191613, 10.58341059247748032480933202499

Graph of the ZZ-function along the critical line