Properties

Label 2-3e6-729.4-c1-0-11
Degree $2$
Conductor $729$
Sign $-0.967 + 0.252i$
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  + (−2.46 − 0.0318i)2-s + (−0.976 + 1.43i)3-s + (4.08 + 0.105i)4-s + (−4.03 − 1.23i)5-s + (2.45 − 3.49i)6-s + (−1.39 + 3.47i)7-s + (−5.14 − 0.199i)8-s + (−1.09 − 2.79i)9-s + (9.92 + 3.18i)10-s + (4.56 + 2.59i)11-s + (−4.14 + 5.74i)12-s + (1.64 + 3.67i)13-s + (3.55 − 8.53i)14-s + (5.71 − 4.56i)15-s + (4.52 + 0.234i)16-s + (3.12 + 1.42i)17-s + ⋯
L(s)  = 1  + (−1.74 − 0.0225i)2-s + (−0.563 + 0.826i)3-s + (2.04 + 0.0528i)4-s + (−1.80 − 0.553i)5-s + (1.00 − 1.42i)6-s + (−0.527 + 1.31i)7-s + (−1.81 − 0.0706i)8-s + (−0.364 − 0.931i)9-s + (3.13 + 1.00i)10-s + (1.37 + 0.783i)11-s + (−1.19 + 1.65i)12-s + (0.455 + 1.02i)13-s + (0.949 − 2.28i)14-s + (1.47 − 1.17i)15-s + (1.13 + 0.0585i)16-s + (0.757 + 0.344i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.967 + 0.252i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.967 + 0.252i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(729\)    =    \(3^{6}\)
Sign: $-0.967 + 0.252i$
Analytic conductor: \(5.82109\)
Root analytic conductor: \(2.41269\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{729} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 729,\ (\ :1/2),\ -0.967 + 0.252i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0310391 - 0.241463i\)
\(L(\frac12)\) \(\approx\) \(0.0310391 - 0.241463i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.976 - 1.43i)T \)
good2 \( 1 + (2.46 + 0.0318i)T + (1.99 + 0.0517i)T^{2} \)
5 \( 1 + (4.03 + 1.23i)T + (4.14 + 2.80i)T^{2} \)
7 \( 1 + (1.39 - 3.47i)T + (-5.06 - 4.83i)T^{2} \)
11 \( 1 + (-4.56 - 2.59i)T + (5.62 + 9.45i)T^{2} \)
13 \( 1 + (-1.64 - 3.67i)T + (-8.67 + 9.68i)T^{2} \)
17 \( 1 + (-3.12 - 1.42i)T + (11.1 + 12.8i)T^{2} \)
19 \( 1 + (-0.984 - 1.43i)T + (-6.84 + 17.7i)T^{2} \)
23 \( 1 + (1.40 - 4.29i)T + (-18.5 - 13.6i)T^{2} \)
29 \( 1 + (6.71 - 4.29i)T + (12.1 - 26.3i)T^{2} \)
31 \( 1 + (-0.0138 - 0.0510i)T + (-26.7 + 15.6i)T^{2} \)
37 \( 1 + (0.975 + 0.191i)T + (34.2 + 13.9i)T^{2} \)
41 \( 1 + (-2.62 + 0.995i)T + (30.7 - 27.1i)T^{2} \)
43 \( 1 + (-5.88 + 2.58i)T + (29.1 - 31.6i)T^{2} \)
47 \( 1 + (1.70 - 6.26i)T + (-40.5 - 23.7i)T^{2} \)
53 \( 1 + (0.0957 - 0.101i)T + (-3.08 - 52.9i)T^{2} \)
59 \( 1 + (-1.05 + 0.179i)T + (55.6 - 19.4i)T^{2} \)
61 \( 1 + (0.977 - 1.79i)T + (-33.1 - 51.1i)T^{2} \)
67 \( 1 + (-1.85 + 0.964i)T + (38.6 - 54.7i)T^{2} \)
71 \( 1 + (6.86 + 5.32i)T + (17.7 + 68.7i)T^{2} \)
73 \( 1 + (10.9 - 3.52i)T + (59.3 - 42.4i)T^{2} \)
79 \( 1 + (-1.95 + 10.2i)T + (-73.5 - 28.9i)T^{2} \)
83 \( 1 + (-2.22 - 13.6i)T + (-78.7 + 26.3i)T^{2} \)
89 \( 1 + (-3.74 - 1.53i)T + (63.5 + 62.3i)T^{2} \)
97 \( 1 + (2.71 - 11.8i)T + (-87.2 - 42.4i)T^{2} \)
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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

−10.86859691240847064133854359113, −9.558817615671612226215648394778, −9.202848804707401521346357085750, −8.667337906779025461898836840491, −7.62361797473942470492898264583, −6.78542705775538743569927159695, −5.72556223532591552006478532270, −4.26632457168129886447699310387, −3.42600354582644407494789515352, −1.41294869930096246083471621307, 0.35442709134322381603008642741, 0.952432849802220693141534417748, 3.07426980467628183685203469405, 4.05122825966765697479590342073, 6.12954787321521468945995296528, 6.90517232955361519426659756896, 7.49834047105366558517800491047, 7.982642105050199875108560642689, 8.826483266667072360115503346201, 10.13133662532289497553339826176

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