Properties

Label 2-3e6-243.151-c1-0-13
Degree $2$
Conductor $729$
Sign $0.0820 + 0.996i$
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  + (−1.16 + 0.323i)2-s + (−0.467 + 0.282i)4-s + (1.06 + 0.0411i)5-s + (−5.09 + 0.796i)7-s + (2.10 − 2.23i)8-s + (−1.24 + 0.295i)10-s + (−0.460 + 3.36i)11-s + (1.07 + 1.56i)13-s + (5.65 − 2.57i)14-s + (−1.21 + 2.30i)16-s + (1.45 + 0.730i)17-s + (0.362 − 6.23i)19-s + (−0.507 + 0.279i)20-s + (−0.554 − 4.06i)22-s + (−1.67 − 4.33i)23-s + ⋯
L(s)  = 1  + (−0.821 + 0.228i)2-s + (−0.233 + 0.141i)4-s + (0.474 + 0.0183i)5-s + (−1.92 + 0.301i)7-s + (0.744 − 0.789i)8-s + (−0.393 + 0.0932i)10-s + (−0.138 + 1.01i)11-s + (0.298 + 0.434i)13-s + (1.51 − 0.687i)14-s + (−0.304 + 0.577i)16-s + (0.353 + 0.177i)17-s + (0.0832 − 1.42i)19-s + (−0.113 + 0.0625i)20-s + (−0.118 − 0.865i)22-s + (−0.349 − 0.903i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0820 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0820 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(729\)    =    \(3^{6}\)
Sign: $0.0820 + 0.996i$
Analytic conductor: \(5.82109\)
Root analytic conductor: \(2.41269\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{729} (613, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 729,\ (\ :1/2),\ 0.0820 + 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.243904 - 0.224652i\)
\(L(\frac12)\) \(\approx\) \(0.243904 - 0.224652i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.16 - 0.323i)T + (1.71 - 1.03i)T^{2} \)
5 \( 1 + (-1.06 - 0.0411i)T + (4.98 + 0.387i)T^{2} \)
7 \( 1 + (5.09 - 0.796i)T + (6.66 - 2.13i)T^{2} \)
11 \( 1 + (0.460 - 3.36i)T + (-10.5 - 2.94i)T^{2} \)
13 \( 1 + (-1.07 - 1.56i)T + (-4.68 + 12.1i)T^{2} \)
17 \( 1 + (-1.45 - 0.730i)T + (10.1 + 13.6i)T^{2} \)
19 \( 1 + (-0.362 + 6.23i)T + (-18.8 - 2.20i)T^{2} \)
23 \( 1 + (1.67 + 4.33i)T + (-17.0 + 15.4i)T^{2} \)
29 \( 1 + (-5.05 + 3.61i)T + (9.38 - 27.4i)T^{2} \)
31 \( 1 + (3.36 + 3.85i)T + (-4.19 + 30.7i)T^{2} \)
37 \( 1 + (3.97 - 5.33i)T + (-10.6 - 35.4i)T^{2} \)
41 \( 1 + (1.40 + 5.44i)T + (-35.8 + 19.8i)T^{2} \)
43 \( 1 + (-3.72 + 3.38i)T + (4.16 - 42.7i)T^{2} \)
47 \( 1 + (-3.71 + 4.25i)T + (-6.36 - 46.5i)T^{2} \)
53 \( 1 + (-0.178 - 1.01i)T + (-49.8 + 18.1i)T^{2} \)
59 \( 1 + (7.85 + 3.20i)T + (42.1 + 41.3i)T^{2} \)
61 \( 1 + (4.86 + 2.93i)T + (28.4 + 53.9i)T^{2} \)
67 \( 1 + (2.72 + 1.94i)T + (21.6 + 63.3i)T^{2} \)
71 \( 1 + (-4.36 + 14.5i)T + (-59.3 - 39.0i)T^{2} \)
73 \( 1 + (-3.76 - 0.892i)T + (65.2 + 32.7i)T^{2} \)
79 \( 1 + (9.19 + 9.01i)T + (1.53 + 78.9i)T^{2} \)
83 \( 1 + (-1.47 + 5.72i)T + (-72.6 - 40.0i)T^{2} \)
89 \( 1 + (-0.365 - 1.22i)T + (-74.3 + 48.9i)T^{2} \)
97 \( 1 + (11.0 - 0.427i)T + (96.7 - 7.51i)T^{2} \)
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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

−9.964017395766527171891005219170, −9.316665869936017382668274338645, −8.807711618830692745738167375321, −7.55033478674608381916220183741, −6.75377252747560566270518308139, −6.09631298527584732708165810565, −4.65889441775433967583027874118, −3.56729433612705074216355868005, −2.28477357379281491957153827626, −0.24982651994966467906470292472, 1.23495226827746904417891576440, 2.97545345019189576065131419604, 3.85269586268414010317539771041, 5.60801381013410491635408600039, 5.99865426731045609383599750252, 7.25599054507636846448840325335, 8.234342361197484694247247797015, 9.113345466733858899518256293408, 9.811983599810337850689215785022, 10.25613417754133180412057178969

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