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Properties

Label 2-3e6-81.67-c1-0-7
Degree $2$
Conductor $729$
Sign $0.328 - 0.944i$
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  + (1.29 − 0.848i)2-s + (0.152 − 0.354i)4-s + (−0.349 + 0.370i)5-s + (−3.96 − 0.463i)7-s + (0.432 + 2.45i)8-s + (−0.136 + 0.774i)10-s + (−1.09 + 3.66i)11-s + (−0.181 + 3.10i)13-s + (−5.50 + 2.76i)14-s + (3.17 + 3.36i)16-s + (1.41 − 0.513i)17-s + (6.30 + 2.29i)19-s + (0.0778 + 0.180i)20-s + (1.69 + 5.66i)22-s + (−1.17 + 0.137i)23-s + ⋯
L(s)  = 1  + (0.912 − 0.600i)2-s + (0.0764 − 0.177i)4-s + (−0.156 + 0.165i)5-s + (−1.49 − 0.175i)7-s + (0.153 + 0.868i)8-s + (−0.0431 + 0.244i)10-s + (−0.331 + 1.10i)11-s + (−0.0502 + 0.862i)13-s + (−1.47 + 0.738i)14-s + (0.793 + 0.840i)16-s + (0.342 − 0.124i)17-s + (1.44 + 0.526i)19-s + (0.0173 + 0.0403i)20-s + (0.361 + 1.20i)22-s + (−0.245 + 0.0287i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.328 - 0.944i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.328 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(729\)    =    \(3^{6}\)
Sign: $0.328 - 0.944i$
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: \((2,\ 729,\ (\ :1/2),\ 0.328 - 0.944i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.25385 + 0.891891i\)
\(L(\frac12)\) \(\approx\) \(1.25385 + 0.891891i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.29 + 0.848i)T + (0.792 - 1.83i)T^{2} \)
5 \( 1 + (0.349 - 0.370i)T + (-0.290 - 4.99i)T^{2} \)
7 \( 1 + (3.96 + 0.463i)T + (6.81 + 1.61i)T^{2} \)
11 \( 1 + (1.09 - 3.66i)T + (-9.19 - 6.04i)T^{2} \)
13 \( 1 + (0.181 - 3.10i)T + (-12.9 - 1.50i)T^{2} \)
17 \( 1 + (-1.41 + 0.513i)T + (13.0 - 10.9i)T^{2} \)
19 \( 1 + (-6.30 - 2.29i)T + (14.5 + 12.2i)T^{2} \)
23 \( 1 + (1.17 - 0.137i)T + (22.3 - 5.30i)T^{2} \)
29 \( 1 + (6.17 + 3.10i)T + (17.3 + 23.2i)T^{2} \)
31 \( 1 + (0.0276 + 0.0371i)T + (-8.89 + 29.6i)T^{2} \)
37 \( 1 + (2.33 + 1.96i)T + (6.42 + 36.4i)T^{2} \)
41 \( 1 + (4.96 + 3.26i)T + (16.2 + 37.6i)T^{2} \)
43 \( 1 + (-0.231 + 0.0549i)T + (38.4 - 19.2i)T^{2} \)
47 \( 1 + (-2.82 + 3.80i)T + (-13.4 - 45.0i)T^{2} \)
53 \( 1 + (6.81 - 11.7i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.400 + 1.33i)T + (-49.2 + 32.4i)T^{2} \)
61 \( 1 + (0.124 + 0.288i)T + (-41.8 + 44.3i)T^{2} \)
67 \( 1 + (-4.90 + 2.46i)T + (40.0 - 53.7i)T^{2} \)
71 \( 1 + (-2.03 + 11.5i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (-2.70 - 15.3i)T + (-68.5 + 24.9i)T^{2} \)
79 \( 1 + (-11.6 + 7.69i)T + (31.2 - 72.5i)T^{2} \)
83 \( 1 + (-0.587 + 0.386i)T + (32.8 - 76.2i)T^{2} \)
89 \( 1 + (2.00 + 11.3i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (-8.92 - 9.46i)T + (-5.64 + 96.8i)T^{2} \)
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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

−10.65644944207629599352161720903, −9.737719162925162164225253186371, −9.208400096091682662963663396899, −7.69921949862425676379992700587, −7.09889789050897175833865326470, −5.91896794965763266574886765707, −4.97923517197411852794173029742, −3.82478444229656948492862939104, −3.25672748982893899122363383509, −2.00960465923601191135010000849, 0.58057922962240399209997424250, 3.09968287009453208064166035050, 3.56487943771476065912798403413, 5.04427329384364615022335524032, 5.73106584060197232666011093092, 6.44392587089160686989988127641, 7.35798586946739474871278331126, 8.386307780794884907735364972392, 9.534974743201309623012434447001, 10.04027921671883791455454788097

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