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Properties

Label 2-3e6-81.34-c1-0-4
Degree $2$
Conductor $729$
Sign $-0.710 - 0.703i$
Analytic cond. $5.82109$
Root an. cond. $2.41269$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.415 + 0.962i)2-s + (0.618 + 0.655i)4-s + (0.0443 + 0.761i)5-s + (2.82 + 0.668i)7-s + (−2.85 + 1.04i)8-s + (−0.751 − 0.273i)10-s + (1.36 + 0.899i)11-s + (−4.07 − 0.475i)13-s + (−1.81 + 2.43i)14-s + (0.0806 − 1.38i)16-s + (−4.52 + 3.79i)17-s + (4.77 + 4.00i)19-s + (−0.471 + 0.499i)20-s + (−1.43 + 0.942i)22-s + (3.75 − 0.891i)23-s + ⋯
L(s)  = 1  + (−0.293 + 0.680i)2-s + (0.309 + 0.327i)4-s + (0.0198 + 0.340i)5-s + (1.06 + 0.252i)7-s + (−1.01 + 0.367i)8-s + (−0.237 − 0.0865i)10-s + (0.412 + 0.271i)11-s + (−1.12 − 0.131i)13-s + (−0.484 + 0.651i)14-s + (0.0201 − 0.346i)16-s + (−1.09 + 0.921i)17-s + (1.09 + 0.919i)19-s + (−0.105 + 0.111i)20-s + (−0.305 + 0.200i)22-s + (0.783 − 0.185i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.710 - 0.703i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.710 - 0.703i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(729\)    =    \(3^{6}\)
Sign: $-0.710 - 0.703i$
Analytic conductor: \(5.82109\)
Root analytic conductor: \(2.41269\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{729} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 729,\ (\ :1/2),\ -0.710 - 0.703i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.539814 + 1.31170i\)
\(L(\frac12)\) \(\approx\) \(0.539814 + 1.31170i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
good2 \( 1 + (0.415 - 0.962i)T + (-1.37 - 1.45i)T^{2} \)
5 \( 1 + (-0.0443 - 0.761i)T + (-4.96 + 0.580i)T^{2} \)
7 \( 1 + (-2.82 - 0.668i)T + (6.25 + 3.14i)T^{2} \)
11 \( 1 + (-1.36 - 0.899i)T + (4.35 + 10.1i)T^{2} \)
13 \( 1 + (4.07 + 0.475i)T + (12.6 + 2.99i)T^{2} \)
17 \( 1 + (4.52 - 3.79i)T + (2.95 - 16.7i)T^{2} \)
19 \( 1 + (-4.77 - 4.00i)T + (3.29 + 18.7i)T^{2} \)
23 \( 1 + (-3.75 + 0.891i)T + (20.5 - 10.3i)T^{2} \)
29 \( 1 + (1.97 + 2.64i)T + (-8.31 + 27.7i)T^{2} \)
31 \( 1 + (0.166 - 0.556i)T + (-25.9 - 17.0i)T^{2} \)
37 \( 1 + (0.349 + 1.98i)T + (-34.7 + 12.6i)T^{2} \)
41 \( 1 + (0.581 + 1.34i)T + (-28.1 + 29.8i)T^{2} \)
43 \( 1 + (6.99 - 3.51i)T + (25.6 - 34.4i)T^{2} \)
47 \( 1 + (-3.37 - 11.2i)T + (-39.2 + 25.8i)T^{2} \)
53 \( 1 + (0.600 + 1.03i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.89 - 3.87i)T + (23.3 - 54.1i)T^{2} \)
61 \( 1 + (-8.34 + 8.84i)T + (-3.54 - 60.8i)T^{2} \)
67 \( 1 + (1.39 - 1.88i)T + (-19.2 - 64.1i)T^{2} \)
71 \( 1 + (-10.2 - 3.72i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (12.9 - 4.71i)T + (55.9 - 46.9i)T^{2} \)
79 \( 1 + (1.70 - 3.95i)T + (-54.2 - 57.4i)T^{2} \)
83 \( 1 + (-6.82 + 15.8i)T + (-56.9 - 60.3i)T^{2} \)
89 \( 1 + (3.38 - 1.23i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (0.637 - 10.9i)T + (-96.3 - 11.2i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.81889512627624714892322149493, −9.689109635755132081290047158327, −8.780102568033645598154139924727, −8.009708407361055019444032688638, −7.30865687787628809318608808767, −6.52726572148591889780104589527, −5.48932737051989986920897838927, −4.47188652493866194590310136592, −3.05543567843004754245375927319, −1.89022733498820266027110135669, 0.810656811490687312280178763235, 2.04488266264265682527017234320, 3.14860591017911852194172784158, 4.75963322464705571960846345479, 5.25640656779156786358357492584, 6.79887547660310197813889284966, 7.31757614504041374314159602666, 8.725041670120908113368027289172, 9.237199736916279597034388336200, 10.12724531049968240598644208777

Graph of the $Z$-function along the critical line