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Properties

Label 2-270-45.13-c2-0-2
Degree $2$
Conductor $270$
Sign $-0.900 - 0.433i$
Analytic cond. $7.35696$
Root an. cond. $2.71237$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.36 + 0.366i)2-s + (1.73 + i)4-s + (−4.54 + 2.07i)5-s + (−9.78 − 2.62i)7-s + (1.99 + 2i)8-s + (−6.97 + 1.16i)10-s + (8.11 + 14.0i)11-s + (−20.2 + 5.41i)13-s + (−12.4 − 7.16i)14-s + (1.99 + 3.46i)16-s + (−7.49 + 7.49i)17-s − 1.15i·19-s + (−9.95 − 0.954i)20-s + (5.93 + 22.1i)22-s + (−19.1 + 5.14i)23-s + ⋯
L(s)  = 1  + (0.683 + 0.183i)2-s + (0.433 + 0.250i)4-s + (−0.909 + 0.415i)5-s + (−1.39 − 0.374i)7-s + (0.249 + 0.250i)8-s + (−0.697 + 0.116i)10-s + (0.737 + 1.27i)11-s + (−1.55 + 0.416i)13-s + (−0.886 − 0.511i)14-s + (0.124 + 0.216i)16-s + (−0.441 + 0.441i)17-s − 0.0607i·19-s + (−0.497 − 0.0477i)20-s + (0.269 + 1.00i)22-s + (−0.834 + 0.223i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.900 - 0.433i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.900 - 0.433i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(270\)    =    \(2 \cdot 3^{3} \cdot 5\)
Sign: $-0.900 - 0.433i$
Analytic conductor: \(7.35696\)
Root analytic conductor: \(2.71237\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{270} (253, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 270,\ (\ :1),\ -0.900 - 0.433i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.189969 + 0.832269i\)
\(L(\frac12)\) \(\approx\) \(0.189969 + 0.832269i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.36 - 0.366i)T \)
3 \( 1 \)
5 \( 1 + (4.54 - 2.07i)T \)
good7 \( 1 + (9.78 + 2.62i)T + (42.4 + 24.5i)T^{2} \)
11 \( 1 + (-8.11 - 14.0i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (20.2 - 5.41i)T + (146. - 84.5i)T^{2} \)
17 \( 1 + (7.49 - 7.49i)T - 289iT^{2} \)
19 \( 1 + 1.15iT - 361T^{2} \)
23 \( 1 + (19.1 - 5.14i)T + (458. - 264.5i)T^{2} \)
29 \( 1 + (-19.9 + 11.5i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (-2.50 + 4.34i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (-33.0 + 33.0i)T - 1.36e3iT^{2} \)
41 \( 1 + (26.2 - 45.4i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (8.94 - 33.3i)T + (-1.60e3 - 924.5i)T^{2} \)
47 \( 1 + (15.5 + 4.17i)T + (1.91e3 + 1.10e3i)T^{2} \)
53 \( 1 + (-30.1 - 30.1i)T + 2.80e3iT^{2} \)
59 \( 1 + (-0.730 - 0.421i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (15.8 + 27.5i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (9.02 + 33.6i)T + (-3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 + 1.68T + 5.04e3T^{2} \)
73 \( 1 + (-100. - 100. i)T + 5.32e3iT^{2} \)
79 \( 1 + (9.01 - 5.20i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (6.02 - 22.4i)T + (-5.96e3 - 3.44e3i)T^{2} \)
89 \( 1 - 102. iT - 7.92e3T^{2} \)
97 \( 1 + (21.8 + 5.85i)T + (8.14e3 + 4.70e3i)T^{2} \)
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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

−12.28988838025930420473715626979, −11.46319102363113564281520916679, −10.12083535147547913989649454072, −9.505437305079726069315476381636, −7.85256204569371295992875881715, −6.96471599725986176357750718078, −6.41621148623826071800359046283, −4.61190825078706272500793570273, −3.86241585160708553465462308696, −2.53170403077611751253859308608, 0.32539369701680413845902081507, 2.81362877975907162664079370679, 3.74091021313876944661419780509, 5.01068231733160198636148705975, 6.20513064143419320831689284074, 7.13167491622910228103454931999, 8.429043113949942195959399961161, 9.428951519551830840070332424365, 10.43486309191310895634421877723, 11.74077573756493056084059913390

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