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Properties

Label 2-273-91.62-c2-0-6
Degree $2$
Conductor $273$
Sign $-0.124 - 0.992i$
Analytic cond. $7.43871$
Root an. cond. $2.72740$
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  + (2.24 − 1.29i)2-s + (−1.5 + 0.866i)3-s + (1.36 − 2.37i)4-s − 3.07·5-s + (−2.24 + 3.89i)6-s + (−4.26 + 5.54i)7-s + 3.27i·8-s + (1.5 − 2.59i)9-s + (−6.90 + 3.98i)10-s + (−6.01 + 3.47i)11-s + 4.74i·12-s + (−9.11 + 9.26i)13-s + (−2.38 + 18.0i)14-s + (4.60 − 2.66i)15-s + (9.72 + 16.8i)16-s + (3.62 + 2.09i)17-s + ⋯
L(s)  = 1  + (1.12 − 0.648i)2-s + (−0.5 + 0.288i)3-s + (0.342 − 0.592i)4-s − 0.614·5-s + (−0.374 + 0.648i)6-s + (−0.609 + 0.792i)7-s + 0.409i·8-s + (0.166 − 0.288i)9-s + (−0.690 + 0.398i)10-s + (−0.546 + 0.315i)11-s + 0.395i·12-s + (−0.701 + 0.712i)13-s + (−0.170 + 1.28i)14-s + (0.307 − 0.177i)15-s + (0.607 + 1.05i)16-s + (0.213 + 0.123i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.124 - 0.992i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.124 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.124 - 0.992i$
Analytic conductor: \(7.43871\)
Root analytic conductor: \(2.72740\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (244, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1),\ -0.124 - 0.992i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.803485 + 0.910927i\)
\(L(\frac12)\) \(\approx\) \(0.803485 + 0.910927i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.5 - 0.866i)T \)
7 \( 1 + (4.26 - 5.54i)T \)
13 \( 1 + (9.11 - 9.26i)T \)
good2 \( 1 + (-2.24 + 1.29i)T + (2 - 3.46i)T^{2} \)
5 \( 1 + 3.07T + 25T^{2} \)
11 \( 1 + (6.01 - 3.47i)T + (60.5 - 104. i)T^{2} \)
17 \( 1 + (-3.62 - 2.09i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-2.88 + 5.00i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (2.29 + 3.96i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (-10.2 - 17.7i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 - 10.8T + 961T^{2} \)
37 \( 1 + (-1.67 + 0.964i)T + (684.5 - 1.18e3i)T^{2} \)
41 \( 1 + (17.7 + 30.7i)T + (-840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (21.1 - 36.6i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 - 36.3T + 2.20e3T^{2} \)
53 \( 1 + 7.21T + 2.80e3T^{2} \)
59 \( 1 + (-23.6 + 40.9i)T + (-1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (32.2 + 18.5i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (48.5 - 28.0i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + (-100. - 57.8i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 - 6.27T + 5.32e3T^{2} \)
79 \( 1 - 47.7T + 6.24e3T^{2} \)
83 \( 1 - 24.0T + 6.88e3T^{2} \)
89 \( 1 + (-61.6 - 106. i)T + (-3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (26.5 - 45.9i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−12.09325099407510144075057471905, −11.38132980479473527055515483185, −10.34079569968526608830963402313, −9.295547842950204868247035979113, −8.046778591175777998112660505832, −6.68292595337409239803712785347, −5.48867802394422382493635979821, −4.66217086310229195580410428277, −3.56166863873230334394596276993, −2.35822872184818076207821410155, 0.43718934223695743997508405746, 3.17659313020711425189727113235, 4.28182998214944525730480097335, 5.34212546159797591995890099691, 6.27946112737827686263644941666, 7.29974279956187059610392164834, 7.934238984995642718345902342359, 9.783352158871361271222860798257, 10.52808878180687860743004028313, 11.82072689298269988642166115593

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