Properties

Label 2-3e6-81.34-c1-0-28
Degree $2$
Conductor $729$
Sign $-0.906 - 0.421i$
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  + (0.837 − 1.94i)2-s + (−1.69 − 1.79i)4-s + (0.0261 + 0.448i)5-s + (−2.37 − 0.562i)7-s + (−0.937 + 0.341i)8-s + (0.892 + 0.324i)10-s + (−2.86 − 1.88i)11-s + (−3.02 − 0.353i)13-s + (−3.08 + 4.13i)14-s + (0.164 − 2.82i)16-s + (1.96 − 1.65i)17-s + (−5.90 − 4.95i)19-s + (0.761 − 0.807i)20-s + (−6.05 + 3.98i)22-s + (−5.55 + 1.31i)23-s + ⋯
L(s)  = 1  + (0.592 − 1.37i)2-s + (−0.848 − 0.898i)4-s + (0.0116 + 0.200i)5-s + (−0.897 − 0.212i)7-s + (−0.331 + 0.120i)8-s + (0.282 + 0.102i)10-s + (−0.862 − 0.567i)11-s + (−0.838 − 0.0979i)13-s + (−0.823 + 1.10i)14-s + (0.0412 − 0.707i)16-s + (0.477 − 0.400i)17-s + (−1.35 − 1.13i)19-s + (0.170 − 0.180i)20-s + (−1.29 + 0.848i)22-s + (−1.15 + 0.274i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.906 - 0.421i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.906 - 0.421i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(729\)    =    \(3^{6}\)
Sign: $-0.906 - 0.421i$
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: \((2,\ 729,\ (\ :1/2),\ -0.906 - 0.421i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.255054 + 1.15435i\)
\(L(\frac12)\) \(\approx\) \(0.255054 + 1.15435i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.837 + 1.94i)T + (-1.37 - 1.45i)T^{2} \)
5 \( 1 + (-0.0261 - 0.448i)T + (-4.96 + 0.580i)T^{2} \)
7 \( 1 + (2.37 + 0.562i)T + (6.25 + 3.14i)T^{2} \)
11 \( 1 + (2.86 + 1.88i)T + (4.35 + 10.1i)T^{2} \)
13 \( 1 + (3.02 + 0.353i)T + (12.6 + 2.99i)T^{2} \)
17 \( 1 + (-1.96 + 1.65i)T + (2.95 - 16.7i)T^{2} \)
19 \( 1 + (5.90 + 4.95i)T + (3.29 + 18.7i)T^{2} \)
23 \( 1 + (5.55 - 1.31i)T + (20.5 - 10.3i)T^{2} \)
29 \( 1 + (-1.87 - 2.51i)T + (-8.31 + 27.7i)T^{2} \)
31 \( 1 + (-2.39 + 8.01i)T + (-25.9 - 17.0i)T^{2} \)
37 \( 1 + (-0.0238 - 0.135i)T + (-34.7 + 12.6i)T^{2} \)
41 \( 1 + (-4.79 - 11.1i)T + (-28.1 + 29.8i)T^{2} \)
43 \( 1 + (-4.05 + 2.03i)T + (25.6 - 34.4i)T^{2} \)
47 \( 1 + (-0.745 - 2.48i)T + (-39.2 + 25.8i)T^{2} \)
53 \( 1 + (-0.184 - 0.319i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-8.82 + 5.80i)T + (23.3 - 54.1i)T^{2} \)
61 \( 1 + (-1.55 + 1.64i)T + (-3.54 - 60.8i)T^{2} \)
67 \( 1 + (-0.831 + 1.11i)T + (-19.2 - 64.1i)T^{2} \)
71 \( 1 + (-3.35 - 1.22i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (-4.75 + 1.73i)T + (55.9 - 46.9i)T^{2} \)
79 \( 1 + (4.06 - 9.43i)T + (-54.2 - 57.4i)T^{2} \)
83 \( 1 + (0.387 - 0.898i)T + (-56.9 - 60.3i)T^{2} \)
89 \( 1 + (8.14 - 2.96i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (-0.123 + 2.12i)T + (-96.3 - 11.2i)T^{2} \)
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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

−10.01577636926495186279135839857, −9.615425182850095240338548698388, −8.293506804573466592670444638329, −7.23530614932420934960087356756, −6.19219918138748642414410825914, −5.03257168802291776439724292686, −4.13503703564118570632558025792, −2.97139070506149242046329149154, −2.42552715720312674551853479171, −0.47091906163159818240756067552, 2.30642102168440111653144128522, 3.82990625812455413425573883837, 4.77001420282740175509725507017, 5.68778673991607113002213789295, 6.40882609423304126755592030356, 7.25261282078262607607679077003, 8.071088138947104974326950639638, 8.834289971469260290048940394067, 10.11666686563825904031632688775, 10.49777556505801660090325554079

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