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Properties

Label 2-540-9.4-c3-0-6
Degree $2$
Conductor $540$
Sign $0.987 + 0.158i$
Analytic cond. $31.8610$
Root an. cond. $5.64455$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.5 − 4.33i)5-s + (6.86 + 11.8i)7-s + (10.8 + 18.7i)11-s + (39.8 − 68.9i)13-s + 64.3·17-s − 144.·19-s + (−13.4 + 23.3i)23-s + (−12.5 − 21.6i)25-s + (148. + 257. i)29-s + (48.3 − 83.7i)31-s + 68.6·35-s + 401.·37-s + (−5.84 + 10.1i)41-s + (−146. − 253. i)43-s + (87.6 + 151. i)47-s + ⋯
L(s)  = 1  + (0.223 − 0.387i)5-s + (0.370 + 0.642i)7-s + (0.296 + 0.513i)11-s + (0.849 − 1.47i)13-s + 0.918·17-s − 1.74·19-s + (−0.122 + 0.211i)23-s + (−0.100 − 0.173i)25-s + (0.950 + 1.64i)29-s + (0.279 − 0.484i)31-s + 0.331·35-s + 1.78·37-s + (−0.0222 + 0.0385i)41-s + (−0.518 − 0.897i)43-s + (0.272 + 0.471i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 540 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.158i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 540 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.987 + 0.158i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(540\)    =    \(2^{2} \cdot 3^{3} \cdot 5\)
Sign: $0.987 + 0.158i$
Analytic conductor: \(31.8610\)
Root analytic conductor: \(5.64455\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{540} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 540,\ (\ :3/2),\ 0.987 + 0.158i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.275574503\)
\(L(\frac12)\) \(\approx\) \(2.275574503\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + (-2.5 + 4.33i)T \)
good7 \( 1 + (-6.86 - 11.8i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (-10.8 - 18.7i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-39.8 + 68.9i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 64.3T + 4.91e3T^{2} \)
19 \( 1 + 144.T + 6.85e3T^{2} \)
23 \( 1 + (13.4 - 23.3i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-148. - 257. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-48.3 + 83.7i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 401.T + 5.06e4T^{2} \)
41 \( 1 + (5.84 - 10.1i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (146. + 253. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-87.6 - 151. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 155.T + 1.48e5T^{2} \)
59 \( 1 + (-297. + 515. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-31.0 - 53.7i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-80.3 + 139. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 959.T + 3.57e5T^{2} \)
73 \( 1 - 763.T + 3.89e5T^{2} \)
79 \( 1 + (31.9 + 55.3i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-458. - 794. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 1.31e3T + 7.04e5T^{2} \)
97 \( 1 + (-64.0 - 110. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−10.41866131137243138576587409667, −9.494959396113792967398453663823, −8.451706970583458333172718525466, −8.022646290443197224805206042112, −6.59894721812784087558263310331, −5.70866101194824584782449968282, −4.85576131193121705256522512982, −3.58455335321932603975814701041, −2.25714426330342337022822076856, −0.931570742329795017826123926136, 1.00386672174661341341778171461, 2.32139843044102630314694774741, 3.82154501596779099185266056794, 4.53545077878795651697362341139, 6.15865346400346590677170431341, 6.54649309997530027686153775328, 7.83466901722604972357316159644, 8.594065509861454816022828442016, 9.616831100780768996117860692518, 10.51284056821798240568049431823

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