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Properties

Label 2-3e6-1.1-c1-0-23
Degree 22
Conductor 729729
Sign −1-1
Analytic cond. 5.821095.82109
Root an. cond. 2.412692.41269
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.415·2-s − 1.82·4-s + 2.21·5-s − 1.31·7-s + 1.59·8-s − 0.920·10-s − 5.21·11-s − 0.0180·13-s + 0.548·14-s + 2.99·16-s − 3.13·17-s + 0.417·19-s − 4.04·20-s + 2.16·22-s − 1.03·23-s − 0.0929·25-s + 0.00750·26-s + 2.41·28-s − 7.80·29-s + 3.72·31-s − 4.42·32-s + 1.30·34-s − 2.92·35-s + 4.42·37-s − 0.173·38-s + 3.52·40-s − 3.67·41-s + ⋯
L(s)  = 1  − 0.293·2-s − 0.913·4-s + 0.990·5-s − 0.498·7-s + 0.562·8-s − 0.291·10-s − 1.57·11-s − 0.00500·13-s + 0.146·14-s + 0.748·16-s − 0.759·17-s + 0.0957·19-s − 0.905·20-s + 0.461·22-s − 0.215·23-s − 0.0185·25-s + 0.00147·26-s + 0.455·28-s − 1.44·29-s + 0.669·31-s − 0.782·32-s + 0.223·34-s − 0.494·35-s + 0.727·37-s − 0.0281·38-s + 0.556·40-s − 0.573·41-s + ⋯

Functional equation

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

Invariants

Degree: 22
Conductor: 729729    =    363^{6}
Sign: −1-1
Analytic conductor: 5.821095.82109
Root analytic conductor: 2.412692.41269
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 729, ( :1/2), −1)(2,\ 729,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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+0.415T+2T2 1 + 0.415T + 2T^{2}
5 1−2.21T+5T2 1 - 2.21T + 5T^{2}
7 1+1.31T+7T2 1 + 1.31T + 7T^{2}
11 1+5.21T+11T2 1 + 5.21T + 11T^{2}
13 1+0.0180T+13T2 1 + 0.0180T + 13T^{2}
17 1+3.13T+17T2 1 + 3.13T + 17T^{2}
19 1−0.417T+19T2 1 - 0.417T + 19T^{2}
23 1+1.03T+23T2 1 + 1.03T + 23T^{2}
29 1+7.80T+29T2 1 + 7.80T + 29T^{2}
31 1−3.72T+31T2 1 - 3.72T + 31T^{2}
37 1−4.42T+37T2 1 - 4.42T + 37T^{2}
41 1+3.67T+41T2 1 + 3.67T + 41T^{2}
43 1+8.30T+43T2 1 + 8.30T + 43T^{2}
47 1+7.09T+47T2 1 + 7.09T + 47T^{2}
53 1+1.30T+53T2 1 + 1.30T + 53T^{2}
59 1+3.70T+59T2 1 + 3.70T + 59T^{2}
61 1−6.91T+61T2 1 - 6.91T + 61T^{2}
67 1+11.0T+67T2 1 + 11.0T + 67T^{2}
71 1+6.08T+71T2 1 + 6.08T + 71T^{2}
73 1+0.546T+73T2 1 + 0.546T + 73T^{2}
79 1−0.489T+79T2 1 - 0.489T + 79T^{2}
83 1−4.61T+83T2 1 - 4.61T + 83T^{2}
89 1+3.37T+89T2 1 + 3.37T + 89T^{2}
97 1−9.94T+97T2 1 - 9.94T + 97T^{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

−9.940461865239414781007964156773, −9.261173834202927770010241126741, −8.346106829826546679404691216687, −7.54203482616040937669375229235, −6.28681007110405631093527253163, −5.42684227220377653042652778140, −4.63087450589062558570813055946, −3.23478917790178859565313368717, −1.94560220139732442041610272342, 0, 1.94560220139732442041610272342, 3.23478917790178859565313368717, 4.63087450589062558570813055946, 5.42684227220377653042652778140, 6.28681007110405631093527253163, 7.54203482616040937669375229235, 8.346106829826546679404691216687, 9.261173834202927770010241126741, 9.940461865239414781007964156773

Graph of the ZZ-function along the critical line