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Properties

Label 2-3e6-81.22-c1-0-3
Degree $2$
Conductor $729$
Sign $0.999 + 0.0280i$
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  + (−0.947 − 1.00i)2-s + (0.00549 − 0.0943i)4-s + (−4.01 + 0.468i)5-s + (−3.17 − 1.59i)7-s + (−2.21 + 1.85i)8-s + (4.26 + 3.58i)10-s + (0.0643 + 0.149i)11-s + (0.339 + 0.0805i)13-s + (1.40 + 4.70i)14-s + (3.77 + 0.441i)16-s + (−0.388 + 2.20i)17-s + (−0.215 − 1.22i)19-s + (0.0221 + 0.380i)20-s + (0.0888 − 0.205i)22-s + (2.70 − 1.35i)23-s + ⋯
L(s)  = 1  + (−0.669 − 0.709i)2-s + (0.00274 − 0.0471i)4-s + (−1.79 + 0.209i)5-s + (−1.20 − 0.603i)7-s + (−0.783 + 0.657i)8-s + (1.35 + 1.13i)10-s + (0.0194 + 0.0449i)11-s + (0.0942 + 0.0223i)13-s + (0.376 + 1.25i)14-s + (0.944 + 0.110i)16-s + (−0.0941 + 0.533i)17-s + (−0.0494 − 0.280i)19-s + (0.00496 + 0.0851i)20-s + (0.0189 − 0.0439i)22-s + (0.564 − 0.283i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0280i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0280i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(729\)    =    \(3^{6}\)
Sign: $0.999 + 0.0280i$
Analytic conductor: \(5.82109\)
Root analytic conductor: \(2.41269\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{729} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 729,\ (\ :1/2),\ 0.999 + 0.0280i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.341949 - 0.00479339i\)
\(L(\frac12)\) \(\approx\) \(0.341949 - 0.00479339i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.947 + 1.00i)T + (-0.116 + 1.99i)T^{2} \)
5 \( 1 + (4.01 - 0.468i)T + (4.86 - 1.15i)T^{2} \)
7 \( 1 + (3.17 + 1.59i)T + (4.18 + 5.61i)T^{2} \)
11 \( 1 + (-0.0643 - 0.149i)T + (-7.54 + 8.00i)T^{2} \)
13 \( 1 + (-0.339 - 0.0805i)T + (11.6 + 5.83i)T^{2} \)
17 \( 1 + (0.388 - 2.20i)T + (-15.9 - 5.81i)T^{2} \)
19 \( 1 + (0.215 + 1.22i)T + (-17.8 + 6.49i)T^{2} \)
23 \( 1 + (-2.70 + 1.35i)T + (13.7 - 18.4i)T^{2} \)
29 \( 1 + (-2.03 + 6.80i)T + (-24.2 - 15.9i)T^{2} \)
31 \( 1 + (-0.722 - 0.475i)T + (12.2 + 28.4i)T^{2} \)
37 \( 1 + (-4.23 + 1.54i)T + (28.3 - 23.7i)T^{2} \)
41 \( 1 + (4.85 - 5.14i)T + (-2.38 - 40.9i)T^{2} \)
43 \( 1 + (5.63 - 7.56i)T + (-12.3 - 41.1i)T^{2} \)
47 \( 1 + (3.65 - 2.40i)T + (18.6 - 43.1i)T^{2} \)
53 \( 1 + (-2.32 + 4.02i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.00 + 4.65i)T + (-40.4 - 42.9i)T^{2} \)
61 \( 1 + (-0.536 - 9.21i)T + (-60.5 + 7.08i)T^{2} \)
67 \( 1 + (-3.72 - 12.4i)T + (-55.9 + 36.8i)T^{2} \)
71 \( 1 + (1.23 + 1.03i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 + (8.94 - 7.50i)T + (12.6 - 71.8i)T^{2} \)
79 \( 1 + (-4.94 - 5.24i)T + (-4.59 + 78.8i)T^{2} \)
83 \( 1 + (-1.57 - 1.67i)T + (-4.82 + 82.8i)T^{2} \)
89 \( 1 + (-8.92 + 7.49i)T + (15.4 - 87.6i)T^{2} \)
97 \( 1 + (-4.17 - 0.487i)T + (94.3 + 22.3i)T^{2} \)
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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

−10.32980803316419300802080728929, −9.785778255347879210534086837731, −8.682107663042385709831428015693, −8.021671939630197709009525798566, −6.99255226879061746817815365521, −6.22007510248436195563726401658, −4.60810110360631422061739404362, −3.62524599897153560971908365684, −2.79618688657500485617101495538, −0.790051723536358806652794815420, 0.34196124480672552762479631217, 3.12258991594832654148822720549, 3.64619360826934731447230964651, 5.01557028530841908706943056655, 6.38348541684216721370677354170, 7.08306991137213129826801715105, 7.78589843108567363989422137901, 8.696256113534500990544861744518, 9.092701741892568387829092259898, 10.20352461876239690534244237080

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