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Properties

Label 2-3e6-729.4-c1-0-20
Degree $2$
Conductor $729$
Sign $-0.120 - 0.992i$
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  + (−1.93 − 0.0250i)2-s + (1.62 + 0.609i)3-s + (1.74 + 0.0450i)4-s + (0.582 + 0.178i)5-s + (−3.12 − 1.21i)6-s + (−1.25 + 3.13i)7-s + (0.498 + 0.0193i)8-s + (2.25 + 1.97i)9-s + (−1.12 − 0.359i)10-s + (1.10 + 0.626i)11-s + (2.79 + 1.13i)12-s + (−0.0673 − 0.150i)13-s + (2.51 − 6.04i)14-s + (0.835 + 0.643i)15-s + (−4.44 − 0.229i)16-s + (5.00 + 2.27i)17-s + ⋯
L(s)  = 1  + (−1.36 − 0.0176i)2-s + (0.936 + 0.351i)3-s + (0.870 + 0.0225i)4-s + (0.260 + 0.0797i)5-s + (−1.27 − 0.497i)6-s + (−0.475 + 1.18i)7-s + (0.176 + 0.00684i)8-s + (0.752 + 0.658i)9-s + (−0.354 − 0.113i)10-s + (0.332 + 0.188i)11-s + (0.807 + 0.327i)12-s + (−0.0186 − 0.0418i)13-s + (0.671 − 1.61i)14-s + (0.215 + 0.166i)15-s + (−1.11 − 0.0574i)16-s + (1.21 + 0.551i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.120 - 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.120 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(729\)    =    \(3^{6}\)
Sign: $-0.120 - 0.992i$
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: \((2,\ 729,\ (\ :1/2),\ -0.120 - 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.646708 + 0.730309i\)
\(L(\frac12)\) \(\approx\) \(0.646708 + 0.730309i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.62 - 0.609i)T \)
good2 \( 1 + (1.93 + 0.0250i)T + (1.99 + 0.0517i)T^{2} \)
5 \( 1 + (-0.582 - 0.178i)T + (4.14 + 2.80i)T^{2} \)
7 \( 1 + (1.25 - 3.13i)T + (-5.06 - 4.83i)T^{2} \)
11 \( 1 + (-1.10 - 0.626i)T + (5.62 + 9.45i)T^{2} \)
13 \( 1 + (0.0673 + 0.150i)T + (-8.67 + 9.68i)T^{2} \)
17 \( 1 + (-5.00 - 2.27i)T + (11.1 + 12.8i)T^{2} \)
19 \( 1 + (2.16 + 3.15i)T + (-6.84 + 17.7i)T^{2} \)
23 \( 1 + (2.08 - 6.35i)T + (-18.5 - 13.6i)T^{2} \)
29 \( 1 + (0.844 - 0.540i)T + (12.1 - 26.3i)T^{2} \)
31 \( 1 + (0.347 + 1.28i)T + (-26.7 + 15.6i)T^{2} \)
37 \( 1 + (-4.58 - 0.899i)T + (34.2 + 13.9i)T^{2} \)
41 \( 1 + (0.155 - 0.0589i)T + (30.7 - 27.1i)T^{2} \)
43 \( 1 + (6.18 - 2.71i)T + (29.1 - 31.6i)T^{2} \)
47 \( 1 + (-0.450 + 1.66i)T + (-40.5 - 23.7i)T^{2} \)
53 \( 1 + (0.285 - 0.302i)T + (-3.08 - 52.9i)T^{2} \)
59 \( 1 + (-2.32 + 0.394i)T + (55.6 - 19.4i)T^{2} \)
61 \( 1 + (-0.376 + 0.693i)T + (-33.1 - 51.1i)T^{2} \)
67 \( 1 + (-0.488 + 0.253i)T + (38.6 - 54.7i)T^{2} \)
71 \( 1 + (0.358 + 0.277i)T + (17.7 + 68.7i)T^{2} \)
73 \( 1 + (-0.379 + 0.121i)T + (59.3 - 42.4i)T^{2} \)
79 \( 1 + (2.90 - 15.3i)T + (-73.5 - 28.9i)T^{2} \)
83 \( 1 + (-1.16 - 7.12i)T + (-78.7 + 26.3i)T^{2} \)
89 \( 1 + (-7.22 - 2.95i)T + (63.5 + 62.3i)T^{2} \)
97 \( 1 + (2.10 - 9.16i)T + (-87.2 - 42.4i)T^{2} \)
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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

−10.02704921438632445274142355930, −9.690339941576873164033584878094, −9.032017933134618418603457607555, −8.256178272843673875123904336027, −7.62566242946815192649633307585, −6.47801447546455773739666903633, −5.33244741770348300655031961117, −3.92025446860099718299473690413, −2.66033910443397776261596122105, −1.67295639178645801502312083190, 0.73910310399773017409044795052, 1.90251862795184263886155217089, 3.38196166560348481419830563648, 4.37613130559423085258670922010, 6.18302116531524258037877669532, 7.12131307139348827226635284192, 7.72214861990522668854162126223, 8.463991334225720118003621894318, 9.315218505204765256651083988387, 10.05832945325872143588808091246

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