Properties

Label 2-3e6-81.67-c1-0-1
Degree $2$
Conductor $729$
Sign $-0.783 + 0.621i$
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  + (−0.890 + 0.585i)2-s + (−0.341 + 0.792i)4-s + (−0.585 + 0.620i)5-s + (0.284 + 0.0332i)7-s + (−0.530 − 3.00i)8-s + (0.158 − 0.896i)10-s + (0.519 − 1.73i)11-s + (−0.325 + 5.58i)13-s + (−0.272 + 0.137i)14-s + (1.04 + 1.11i)16-s + (−4.42 + 1.61i)17-s + (−1.75 − 0.638i)19-s + (−0.291 − 0.676i)20-s + (0.553 + 1.84i)22-s + (−1.34 + 0.157i)23-s + ⋯
L(s)  = 1  + (−0.629 + 0.414i)2-s + (−0.170 + 0.396i)4-s + (−0.262 + 0.277i)5-s + (0.107 + 0.0125i)7-s + (−0.187 − 1.06i)8-s + (0.0499 − 0.283i)10-s + (0.156 − 0.522i)11-s + (−0.0901 + 1.54i)13-s + (−0.0729 + 0.0366i)14-s + (0.262 + 0.278i)16-s + (−1.07 + 0.390i)17-s + (−0.402 − 0.146i)19-s + (−0.0652 − 0.151i)20-s + (0.117 + 0.394i)22-s + (−0.280 + 0.0328i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.783 + 0.621i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.783 + 0.621i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(729\)    =    \(3^{6}\)
Sign: $-0.783 + 0.621i$
Analytic conductor: \(5.82109\)
Root analytic conductor: \(2.41269\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{729} (28, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 729,\ (\ :1/2),\ -0.783 + 0.621i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0584509 - 0.167652i\)
\(L(\frac12)\) \(\approx\) \(0.0584509 - 0.167652i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.890 - 0.585i)T + (0.792 - 1.83i)T^{2} \)
5 \( 1 + (0.585 - 0.620i)T + (-0.290 - 4.99i)T^{2} \)
7 \( 1 + (-0.284 - 0.0332i)T + (6.81 + 1.61i)T^{2} \)
11 \( 1 + (-0.519 + 1.73i)T + (-9.19 - 6.04i)T^{2} \)
13 \( 1 + (0.325 - 5.58i)T + (-12.9 - 1.50i)T^{2} \)
17 \( 1 + (4.42 - 1.61i)T + (13.0 - 10.9i)T^{2} \)
19 \( 1 + (1.75 + 0.638i)T + (14.5 + 12.2i)T^{2} \)
23 \( 1 + (1.34 - 0.157i)T + (22.3 - 5.30i)T^{2} \)
29 \( 1 + (5.35 + 2.68i)T + (17.3 + 23.2i)T^{2} \)
31 \( 1 + (2.54 + 3.41i)T + (-8.89 + 29.6i)T^{2} \)
37 \( 1 + (5.46 + 4.58i)T + (6.42 + 36.4i)T^{2} \)
41 \( 1 + (9.58 + 6.30i)T + (16.2 + 37.6i)T^{2} \)
43 \( 1 + (-12.3 + 2.93i)T + (38.4 - 19.2i)T^{2} \)
47 \( 1 + (-1.22 + 1.64i)T + (-13.4 - 45.0i)T^{2} \)
53 \( 1 + (-1.54 + 2.66i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.361 + 1.20i)T + (-49.2 + 32.4i)T^{2} \)
61 \( 1 + (-1.09 - 2.54i)T + (-41.8 + 44.3i)T^{2} \)
67 \( 1 + (5.61 - 2.81i)T + (40.0 - 53.7i)T^{2} \)
71 \( 1 + (2.24 - 12.7i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (1.11 + 6.32i)T + (-68.5 + 24.9i)T^{2} \)
79 \( 1 + (5.95 - 3.91i)T + (31.2 - 72.5i)T^{2} \)
83 \( 1 + (11.3 - 7.44i)T + (32.8 - 76.2i)T^{2} \)
89 \( 1 + (-0.0750 - 0.425i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (6.92 + 7.34i)T + (-5.64 + 96.8i)T^{2} \)
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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

−10.98551399020905989926445802321, −9.792184809970784250807028081558, −8.940388379812004766504233123760, −8.561800240920141545714244139939, −7.31536113277446880443664201017, −6.91073551933895164965318490413, −5.81826557562837480331738778854, −4.28860755785417684556505391696, −3.66831886749124636691597481456, −2.03341952648700399818625471553, 0.11105255139586935055389527065, 1.63419162339504205580274680054, 2.92506186663719302374755592112, 4.45230378207027191563118229064, 5.25217379661845067988525872469, 6.29050590573060072897389095048, 7.51338907253072408396432414462, 8.375691357196147276515767877713, 9.033707847539300364898026316547, 9.971518195178423997421572330685

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