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Properties

Label 2-3e6-27.22-c1-0-4
Degree 22
Conductor 729729
Sign −0.993−0.116i-0.993 - 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  + (−1.62 + 1.36i)2-s + (0.436 − 2.47i)4-s + (1.94 − 0.708i)5-s + (0.841 + 4.77i)7-s + (0.547 + 0.949i)8-s + (−2.20 + 3.81i)10-s + (−3.89 − 1.41i)11-s + (−0.931 − 0.781i)13-s + (−7.89 − 6.62i)14-s + (2.53 + 0.924i)16-s + (−1.18 + 2.04i)17-s + (0.919 + 1.59i)19-s + (−0.904 − 5.13i)20-s + (8.28 − 3.01i)22-s + (−0.747 + 4.23i)23-s + ⋯
L(s)  = 1  + (−1.15 + 0.965i)2-s + (0.218 − 1.23i)4-s + (0.870 − 0.316i)5-s + (0.318 + 1.80i)7-s + (0.193 + 0.335i)8-s + (−0.695 + 1.20i)10-s + (−1.17 − 0.427i)11-s + (−0.258 − 0.216i)13-s + (−2.10 − 1.77i)14-s + (0.634 + 0.231i)16-s + (−0.286 + 0.496i)17-s + (0.210 + 0.365i)19-s + (−0.202 − 1.14i)20-s + (1.76 − 0.642i)22-s + (−0.155 + 0.883i)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: −0.993−0.116i-0.993 - 0.116i
Analytic conductor: 5.821095.82109
Root analytic conductor: 2.412692.41269
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ729(325,⋅)\chi_{729} (325, \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 0.0377385+0.647945i0.0377385 + 0.647945i
L(12)L(\frac12) ≈\approx 0.0377385+0.647945i0.0377385 + 0.647945i
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+(1.62−1.36i)T+(0.347−1.96i)T2 1 + (1.62 - 1.36i)T + (0.347 - 1.96i)T^{2}
5 1+(−1.94+0.708i)T+(3.83−3.21i)T2 1 + (-1.94 + 0.708i)T + (3.83 - 3.21i)T^{2}
7 1+(−0.841−4.77i)T+(−6.57+2.39i)T2 1 + (-0.841 - 4.77i)T + (-6.57 + 2.39i)T^{2}
11 1+(3.89+1.41i)T+(8.42+7.07i)T2 1 + (3.89 + 1.41i)T + (8.42 + 7.07i)T^{2}
13 1+(0.931+0.781i)T+(2.25+12.8i)T2 1 + (0.931 + 0.781i)T + (2.25 + 12.8i)T^{2}
17 1+(1.18−2.04i)T+(−8.5−14.7i)T2 1 + (1.18 - 2.04i)T + (-8.5 - 14.7i)T^{2}
19 1+(−0.919−1.59i)T+(−9.5+16.4i)T2 1 + (-0.919 - 1.59i)T + (-9.5 + 16.4i)T^{2}
23 1+(0.747−4.23i)T+(−21.6−7.86i)T2 1 + (0.747 - 4.23i)T + (-21.6 - 7.86i)T^{2}
29 1+(−2.28+1.91i)T+(5.03−28.5i)T2 1 + (-2.28 + 1.91i)T + (5.03 - 28.5i)T^{2}
31 1+(−0.255+1.45i)T+(−29.1−10.6i)T2 1 + (-0.255 + 1.45i)T + (-29.1 - 10.6i)T^{2}
37 1+(4.48−7.76i)T+(−18.5−32.0i)T2 1 + (4.48 - 7.76i)T + (-18.5 - 32.0i)T^{2}
41 1+(−1.73−1.45i)T+(7.11+40.3i)T2 1 + (-1.73 - 1.45i)T + (7.11 + 40.3i)T^{2}
43 1+(−5.15−1.87i)T+(32.9+27.6i)T2 1 + (-5.15 - 1.87i)T + (32.9 + 27.6i)T^{2}
47 1+(−1.24−7.07i)T+(−44.1+16.0i)T2 1 + (-1.24 - 7.07i)T + (-44.1 + 16.0i)T^{2}
53 1+6.32T+53T2 1 + 6.32T + 53T^{2}
59 1+(0.246−0.0896i)T+(45.1−37.9i)T2 1 + (0.246 - 0.0896i)T + (45.1 - 37.9i)T^{2}
61 1+(0.773+4.38i)T+(−57.3+20.8i)T2 1 + (0.773 + 4.38i)T + (-57.3 + 20.8i)T^{2}
67 1+(−3.16−2.65i)T+(11.6+65.9i)T2 1 + (-3.16 - 2.65i)T + (11.6 + 65.9i)T^{2}
71 1+(1.54−2.67i)T+(−35.5−61.4i)T2 1 + (1.54 - 2.67i)T + (-35.5 - 61.4i)T^{2}
73 1+(6.38+11.0i)T+(−36.5+63.2i)T2 1 + (6.38 + 11.0i)T + (-36.5 + 63.2i)T^{2}
79 1+(−3.48+2.92i)T+(13.7−77.7i)T2 1 + (-3.48 + 2.92i)T + (13.7 - 77.7i)T^{2}
83 1+(6.47−5.43i)T+(14.4−81.7i)T2 1 + (6.47 - 5.43i)T + (14.4 - 81.7i)T^{2}
89 1+(8.48+14.6i)T+(−44.5+77.0i)T2 1 + (8.48 + 14.6i)T + (-44.5 + 77.0i)T^{2}
97 1+(−4.80−1.74i)T+(74.3+62.3i)T2 1 + (-4.80 - 1.74i)T + (74.3 + 62.3i)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

−10.37666291627938600448051199947, −9.607367769169231515143703565430, −9.022561544791193190059808011700, −8.231265631056569464409791699684, −7.71623553710059514524835907024, −6.24060565356262178749155060105, −5.77136539848179671956095052659, −5.07050110358790078372625537769, −2.90446686712649650751191926120, −1.70149557297191593032543342730, 0.47482208794464278740935962647, 1.88612942088287916686372951353, 2.83170459323646686676324940354, 4.25041895048044688746810906203, 5.38354390184831735762426395815, 6.89556545176258099906813280677, 7.49153326810801626657855049981, 8.421933016176346380884952162050, 9.476830756582052017464804955929, 10.17592261836784970836961135286

Graph of the ZZ-function along the critical line