Properties

Label 2-3e6-81.67-c1-0-16
Degree $2$
Conductor $729$
Sign $0.907 - 0.419i$
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  + (0.652 − 0.429i)2-s + (−0.550 + 1.27i)4-s + (1.01 − 1.07i)5-s + (3.77 + 0.441i)7-s + (0.459 + 2.60i)8-s + (0.200 − 1.13i)10-s + (−1.44 + 4.84i)11-s + (0.261 − 4.48i)13-s + (2.65 − 1.33i)14-s + (−0.485 − 0.515i)16-s + (−4.30 + 1.56i)17-s + (4.19 + 1.52i)19-s + (0.814 + 1.88i)20-s + (1.13 + 3.78i)22-s + (3.43 − 0.401i)23-s + ⋯
L(s)  = 1  + (0.461 − 0.303i)2-s + (−0.275 + 0.637i)4-s + (0.454 − 0.481i)5-s + (1.42 + 0.166i)7-s + (0.162 + 0.922i)8-s + (0.0635 − 0.360i)10-s + (−0.437 + 1.45i)11-s + (0.0724 − 1.24i)13-s + (0.709 − 0.356i)14-s + (−0.121 − 0.128i)16-s + (−1.04 + 0.380i)17-s + (0.962 + 0.350i)19-s + (0.182 + 0.422i)20-s + (0.241 + 0.806i)22-s + (0.715 − 0.0836i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.907 - 0.419i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.907 - 0.419i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(729\)    =    \(3^{6}\)
Sign: $0.907 - 0.419i$
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: \((2,\ 729,\ (\ :1/2),\ 0.907 - 0.419i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.13303 + 0.469004i\)
\(L(\frac12)\) \(\approx\) \(2.13303 + 0.469004i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.652 + 0.429i)T + (0.792 - 1.83i)T^{2} \)
5 \( 1 + (-1.01 + 1.07i)T + (-0.290 - 4.99i)T^{2} \)
7 \( 1 + (-3.77 - 0.441i)T + (6.81 + 1.61i)T^{2} \)
11 \( 1 + (1.44 - 4.84i)T + (-9.19 - 6.04i)T^{2} \)
13 \( 1 + (-0.261 + 4.48i)T + (-12.9 - 1.50i)T^{2} \)
17 \( 1 + (4.30 - 1.56i)T + (13.0 - 10.9i)T^{2} \)
19 \( 1 + (-4.19 - 1.52i)T + (14.5 + 12.2i)T^{2} \)
23 \( 1 + (-3.43 + 0.401i)T + (22.3 - 5.30i)T^{2} \)
29 \( 1 + (-0.583 - 0.293i)T + (17.3 + 23.2i)T^{2} \)
31 \( 1 + (-0.393 - 0.527i)T + (-8.89 + 29.6i)T^{2} \)
37 \( 1 + (-0.766 - 0.642i)T + (6.42 + 36.4i)T^{2} \)
41 \( 1 + (-0.570 - 0.375i)T + (16.2 + 37.6i)T^{2} \)
43 \( 1 + (-8.16 + 1.93i)T + (38.4 - 19.2i)T^{2} \)
47 \( 1 + (-4.73 + 6.36i)T + (-13.4 - 45.0i)T^{2} \)
53 \( 1 + (2.07 - 3.59i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.51 + 5.04i)T + (-49.2 + 32.4i)T^{2} \)
61 \( 1 + (2.68 + 6.23i)T + (-41.8 + 44.3i)T^{2} \)
67 \( 1 + (-4.73 + 2.37i)T + (40.0 - 53.7i)T^{2} \)
71 \( 1 + (1.06 - 6.03i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (0.764 + 4.33i)T + (-68.5 + 24.9i)T^{2} \)
79 \( 1 + (9.39 - 6.17i)T + (31.2 - 72.5i)T^{2} \)
83 \( 1 + (1.35 - 0.891i)T + (32.8 - 76.2i)T^{2} \)
89 \( 1 + (0.181 + 1.02i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (3.42 + 3.62i)T + (-5.64 + 96.8i)T^{2} \)
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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

−10.65795229996278987006624207402, −9.544615892547079100857035332967, −8.645102529220618484564256754217, −7.905388026894487888535578812516, −7.23240341419865954025932925950, −5.46671511583230642925673296929, −5.00809813149367326728010469077, −4.17388829366212532653414151648, −2.72349597666272633688659943408, −1.66259339929559840598405799377, 1.14282462103448127595297095064, 2.57942787182437347095882414864, 4.16361503700700928034007597134, 4.94213075669194995492254737609, 5.81130449196698595867822294569, 6.64364972009758406998538310872, 7.58822401350265633838165170564, 8.756438864149266988804320287260, 9.348737939752917283645954191613, 10.58341059247748032480933202499

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