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Properties

Label 2-3e6-243.103-c1-0-0
Degree $2$
Conductor $729$
Sign $-0.912 - 0.408i$
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  + (0.626 + 0.174i)2-s + (−1.34 − 0.814i)4-s + (−0.644 + 0.0250i)5-s + (−0.504 − 0.0789i)7-s + (−1.59 − 1.69i)8-s + (−0.408 − 0.0967i)10-s + (0.486 + 3.56i)11-s + (0.0982 − 0.143i)13-s + (−0.302 − 0.137i)14-s + (0.763 + 1.44i)16-s + (−6.29 + 3.16i)17-s + (−0.433 − 7.43i)19-s + (0.890 + 0.491i)20-s + (−0.316 + 2.31i)22-s + (−2.32 + 6.02i)23-s + ⋯
L(s)  = 1  + (0.443 + 0.123i)2-s + (−0.674 − 0.407i)4-s + (−0.288 + 0.0111i)5-s + (−0.190 − 0.0298i)7-s + (−0.564 − 0.598i)8-s + (−0.129 − 0.0305i)10-s + (0.146 + 1.07i)11-s + (0.0272 − 0.0397i)13-s + (−0.0808 − 0.0367i)14-s + (0.190 + 0.362i)16-s + (−1.52 + 0.766i)17-s + (−0.0993 − 1.70i)19-s + (0.199 + 0.109i)20-s + (−0.0675 + 0.494i)22-s + (−0.485 + 1.25i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.912 - 0.408i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.912 - 0.408i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(729\)    =    \(3^{6}\)
Sign: $-0.912 - 0.408i$
Analytic conductor: \(5.82109\)
Root analytic conductor: \(2.41269\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{729} (685, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 729,\ (\ :1/2),\ -0.912 - 0.408i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0533320 + 0.249591i\)
\(L(\frac12)\) \(\approx\) \(0.0533320 + 0.249591i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.626 - 0.174i)T + (1.71 + 1.03i)T^{2} \)
5 \( 1 + (0.644 - 0.0250i)T + (4.98 - 0.387i)T^{2} \)
7 \( 1 + (0.504 + 0.0789i)T + (6.66 + 2.13i)T^{2} \)
11 \( 1 + (-0.486 - 3.56i)T + (-10.5 + 2.94i)T^{2} \)
13 \( 1 + (-0.0982 + 0.143i)T + (-4.68 - 12.1i)T^{2} \)
17 \( 1 + (6.29 - 3.16i)T + (10.1 - 13.6i)T^{2} \)
19 \( 1 + (0.433 + 7.43i)T + (-18.8 + 2.20i)T^{2} \)
23 \( 1 + (2.32 - 6.02i)T + (-17.0 - 15.4i)T^{2} \)
29 \( 1 + (3.25 + 2.32i)T + (9.38 + 27.4i)T^{2} \)
31 \( 1 + (5.44 - 6.23i)T + (-4.19 - 30.7i)T^{2} \)
37 \( 1 + (-3.71 - 4.98i)T + (-10.6 + 35.4i)T^{2} \)
41 \( 1 + (0.706 - 2.74i)T + (-35.8 - 19.8i)T^{2} \)
43 \( 1 + (5.41 + 4.90i)T + (4.16 + 42.7i)T^{2} \)
47 \( 1 + (-5.10 - 5.84i)T + (-6.36 + 46.5i)T^{2} \)
53 \( 1 + (-1.26 + 7.14i)T + (-49.8 - 18.1i)T^{2} \)
59 \( 1 + (0.549 - 0.224i)T + (42.1 - 41.3i)T^{2} \)
61 \( 1 + (-3.13 + 1.89i)T + (28.4 - 53.9i)T^{2} \)
67 \( 1 + (-5.50 + 3.93i)T + (21.6 - 63.3i)T^{2} \)
71 \( 1 + (3.45 + 11.5i)T + (-59.3 + 39.0i)T^{2} \)
73 \( 1 + (-8.73 + 2.06i)T + (65.2 - 32.7i)T^{2} \)
79 \( 1 + (-0.340 + 0.333i)T + (1.53 - 78.9i)T^{2} \)
83 \( 1 + (-0.138 - 0.536i)T + (-72.6 + 40.0i)T^{2} \)
89 \( 1 + (3.77 - 12.6i)T + (-74.3 - 48.9i)T^{2} \)
97 \( 1 + (6.14 + 0.238i)T + (96.7 + 7.51i)T^{2} \)
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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

−10.79576356849900422183126607767, −9.685417644184767542135813537966, −9.254459846057200006285561599145, −8.248634453067806567861192326967, −7.07886493334577602791454488335, −6.37017207593784699252076613421, −5.19832821626837996987389488160, −4.43736691781359599814301035766, −3.57542013082488548026686159724, −1.91622406007387393635620066521, 0.11028815781348492967772595371, 2.38036373936435666923225054329, 3.68944521889598781386244621502, 4.22836090567318874995035612437, 5.50228781998052158196020503264, 6.26000772805638777403955891730, 7.55997156718094570259106965920, 8.399905604726048084191497580918, 9.027265809385035310753350918648, 9.960703888948800896845548313302

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