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Properties

Label 2-3e6-27.7-c1-0-10
Degree $2$
Conductor $729$
Sign $-0.597 - 0.802i$
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  + (−1.99 + 0.726i)2-s + (1.92 − 1.61i)4-s + (0.359 + 2.03i)5-s + (3.71 + 3.11i)7-s + (−0.547 + 0.949i)8-s + (−2.20 − 3.81i)10-s + (−0.720 + 4.08i)11-s + (1.14 + 0.415i)13-s + (−9.68 − 3.52i)14-s + (−0.469 + 2.66i)16-s + (1.18 + 2.04i)17-s + (0.919 − 1.59i)19-s + (3.99 + 3.34i)20-s + (−1.53 − 8.68i)22-s + (3.29 − 2.76i)23-s + ⋯
L(s)  = 1  + (−1.41 + 0.513i)2-s + (0.963 − 0.808i)4-s + (0.160 + 0.912i)5-s + (1.40 + 1.17i)7-s + (−0.193 + 0.335i)8-s + (−0.695 − 1.20i)10-s + (−0.217 + 1.23i)11-s + (0.316 + 0.115i)13-s + (−2.58 − 0.941i)14-s + (−0.117 + 0.665i)16-s + (0.286 + 0.496i)17-s + (0.210 − 0.365i)19-s + (0.892 + 0.748i)20-s + (−0.326 − 1.85i)22-s + (0.687 − 0.576i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.597 - 0.802i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.597 - 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(729\)    =    \(3^{6}\)
Sign: $-0.597 - 0.802i$
Analytic conductor: \(5.82109\)
Root analytic conductor: \(2.41269\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{729} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 729,\ (\ :1/2),\ -0.597 - 0.802i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.403674 + 0.803782i\)
\(L(\frac12)\) \(\approx\) \(0.403674 + 0.803782i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.99 - 0.726i)T + (1.53 - 1.28i)T^{2} \)
5 \( 1 + (-0.359 - 2.03i)T + (-4.69 + 1.71i)T^{2} \)
7 \( 1 + (-3.71 - 3.11i)T + (1.21 + 6.89i)T^{2} \)
11 \( 1 + (0.720 - 4.08i)T + (-10.3 - 3.76i)T^{2} \)
13 \( 1 + (-1.14 - 0.415i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (-1.18 - 2.04i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.919 + 1.59i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.29 + 2.76i)T + (3.99 - 22.6i)T^{2} \)
29 \( 1 + (-2.80 + 1.01i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (-1.12 + 0.947i)T + (5.38 - 30.5i)T^{2} \)
37 \( 1 + (4.48 + 7.76i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.12 - 0.773i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (0.952 - 5.40i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (5.50 + 4.61i)T + (8.16 + 46.2i)T^{2} \)
53 \( 1 - 6.32T + 53T^{2} \)
59 \( 1 + (0.0455 + 0.258i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (3.41 + 2.86i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (3.88 + 1.41i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (-1.54 - 2.67i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (6.38 - 11.0i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.27 - 1.55i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (7.94 - 2.89i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 + (-8.48 + 14.6i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.887 - 5.03i)T + (-91.1 - 33.1i)T^{2} \)
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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

−10.54918239999547335716984767909, −9.721237974720832975408356035359, −8.827385501212033852625896074407, −8.229873369143875005231612307600, −7.37491021410942897152087841957, −6.66506125902201740836914952582, −5.59019802494969078708767839880, −4.47740701278347380431746483790, −2.58419635992268454445859372086, −1.61657102127382554144289596392, 0.870426376688650794305197492475, 1.47071709179171851696291556603, 3.20792301084338657274745262102, 4.66918509926147703812682351686, 5.43407607821265678077940808982, 7.05901378864879168463242014099, 7.939110142393985828852388521194, 8.440738640960861496531787613249, 9.105407315039499176224497416799, 10.18666147015616875492751585511

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