Properties

Label 2-3e6-81.34-c1-0-26
Degree $2$
Conductor $729$
Sign $-0.515 + 0.856i$
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  + (−0.361 + 0.839i)2-s + (0.799 + 0.847i)4-s + (−0.221 − 3.80i)5-s + (−0.706 − 0.167i)7-s + (−2.71 + 0.989i)8-s + (3.27 + 1.19i)10-s + (−2.24 − 1.47i)11-s + (−4.57 − 0.534i)13-s + (0.396 − 0.532i)14-s + (0.0182 − 0.313i)16-s + (0.692 − 0.581i)17-s + (−1.12 − 0.940i)19-s + (3.04 − 3.22i)20-s + (2.05 − 1.35i)22-s + (−3.79 + 0.899i)23-s + ⋯
L(s)  = 1  + (−0.255 + 0.593i)2-s + (0.399 + 0.423i)4-s + (−0.0990 − 1.70i)5-s + (−0.267 − 0.0633i)7-s + (−0.960 + 0.349i)8-s + (1.03 + 0.376i)10-s + (−0.678 − 0.446i)11-s + (−1.26 − 0.148i)13-s + (0.105 − 0.142i)14-s + (0.00455 − 0.0782i)16-s + (0.168 − 0.140i)17-s + (−0.257 − 0.215i)19-s + (0.680 − 0.721i)20-s + (0.438 − 0.288i)22-s + (−0.791 + 0.187i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.515 + 0.856i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.515 + 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(729\)    =    \(3^{6}\)
Sign: $-0.515 + 0.856i$
Analytic conductor: \(5.82109\)
Root analytic conductor: \(2.41269\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{729} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 729,\ (\ :1/2),\ -0.515 + 0.856i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.223776 - 0.396032i\)
\(L(\frac12)\) \(\approx\) \(0.223776 - 0.396032i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.361 - 0.839i)T + (-1.37 - 1.45i)T^{2} \)
5 \( 1 + (0.221 + 3.80i)T + (-4.96 + 0.580i)T^{2} \)
7 \( 1 + (0.706 + 0.167i)T + (6.25 + 3.14i)T^{2} \)
11 \( 1 + (2.24 + 1.47i)T + (4.35 + 10.1i)T^{2} \)
13 \( 1 + (4.57 + 0.534i)T + (12.6 + 2.99i)T^{2} \)
17 \( 1 + (-0.692 + 0.581i)T + (2.95 - 16.7i)T^{2} \)
19 \( 1 + (1.12 + 0.940i)T + (3.29 + 18.7i)T^{2} \)
23 \( 1 + (3.79 - 0.899i)T + (20.5 - 10.3i)T^{2} \)
29 \( 1 + (3.06 + 4.12i)T + (-8.31 + 27.7i)T^{2} \)
31 \( 1 + (2.86 - 9.56i)T + (-25.9 - 17.0i)T^{2} \)
37 \( 1 + (0.348 + 1.97i)T + (-34.7 + 12.6i)T^{2} \)
41 \( 1 + (-2.59 - 6.02i)T + (-28.1 + 29.8i)T^{2} \)
43 \( 1 + (-6.51 + 3.27i)T + (25.6 - 34.4i)T^{2} \)
47 \( 1 + (1.28 + 4.28i)T + (-39.2 + 25.8i)T^{2} \)
53 \( 1 + (3.43 + 5.95i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.590 - 0.388i)T + (23.3 - 54.1i)T^{2} \)
61 \( 1 + (-1.83 + 1.94i)T + (-3.54 - 60.8i)T^{2} \)
67 \( 1 + (-8.91 + 11.9i)T + (-19.2 - 64.1i)T^{2} \)
71 \( 1 + (-9.19 - 3.34i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (-15.0 + 5.48i)T + (55.9 - 46.9i)T^{2} \)
79 \( 1 + (-1.89 + 4.38i)T + (-54.2 - 57.4i)T^{2} \)
83 \( 1 + (-2.86 + 6.65i)T + (-56.9 - 60.3i)T^{2} \)
89 \( 1 + (7.00 - 2.54i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (-0.258 + 4.44i)T + (-96.3 - 11.2i)T^{2} \)
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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

−9.759262571592090802661621585883, −9.123675033042036809691403525786, −8.160902701428594162764704587385, −7.80187596180294660829459575726, −6.68268614462371533508137856393, −5.53478526912845567902966178666, −4.90880446347950120216232688778, −3.57300504141739393547834034835, −2.17547443863161850551943578535, −0.22540807488295927499912379532, 2.19941847556343693050417130978, 2.70793001989437875583687045151, 3.90477810473037758160022887158, 5.50987730348814375945223640276, 6.39107397896939386600807118724, 7.17777623957165066489620447752, 7.86145118214310587464375354375, 9.613092546723450408266676147266, 9.813216635547208442081234543733, 10.81107402111262626684080246521

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