Properties

Label 2-273-273.32-c1-0-6
Degree $2$
Conductor $273$
Sign $-0.646 - 0.762i$
Analytic cond. $2.17991$
Root an. cond. $1.47645$
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  + (1.61 + 1.61i)2-s + (−1.37 − 1.05i)3-s + 3.19i·4-s + (−0.511 + 1.90i)5-s + (−0.506 − 3.91i)6-s + (−1.48 + 2.19i)7-s + (−1.92 + 1.92i)8-s + (0.764 + 2.90i)9-s + (−3.90 + 2.25i)10-s + (1.51 + 0.404i)11-s + (3.38 − 4.38i)12-s + (−3.19 + 1.67i)13-s + (−5.92 + 1.14i)14-s + (2.71 − 2.07i)15-s + 0.173·16-s + 4.97·17-s + ⋯
L(s)  = 1  + (1.13 + 1.13i)2-s + (−0.792 − 0.610i)3-s + 1.59i·4-s + (−0.228 + 0.853i)5-s + (−0.206 − 1.59i)6-s + (−0.559 + 0.828i)7-s + (−0.682 + 0.682i)8-s + (0.254 + 0.967i)9-s + (−1.23 + 0.712i)10-s + (0.455 + 0.122i)11-s + (0.975 − 1.26i)12-s + (−0.884 + 0.465i)13-s + (−1.58 + 0.307i)14-s + (0.702 − 0.536i)15-s + 0.0434·16-s + 1.20·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.646 - 0.762i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.646 - 0.762i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.646 - 0.762i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.646 - 0.762i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.667055 + 1.43963i\)
\(L(\frac12)\) \(\approx\) \(0.667055 + 1.43963i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.37 + 1.05i)T \)
7 \( 1 + (1.48 - 2.19i)T \)
13 \( 1 + (3.19 - 1.67i)T \)
good2 \( 1 + (-1.61 - 1.61i)T + 2iT^{2} \)
5 \( 1 + (0.511 - 1.90i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (-1.51 - 0.404i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 - 4.97T + 17T^{2} \)
19 \( 1 + (0.803 + 2.99i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 - 0.470T + 23T^{2} \)
29 \( 1 + (3.57 + 2.06i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.68 + 6.28i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (-1.21 - 1.21i)T + 37iT^{2} \)
41 \( 1 + (-2.86 + 0.767i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-6.22 + 3.59i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-11.3 - 3.04i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-5.73 - 3.31i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.16 - 5.16i)T - 59iT^{2} \)
61 \( 1 + (3.50 - 6.06i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (15.1 + 4.07i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-1.32 + 4.94i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (3.37 - 0.904i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (2.83 + 4.91i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-10.6 - 10.6i)T + 83iT^{2} \)
89 \( 1 + (7.39 - 7.39i)T - 89iT^{2} \)
97 \( 1 + (-10.3 - 2.78i)T + (84.0 + 48.5i)T^{2} \)
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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

−12.29110846471368351188623537696, −11.84494980519861754064502919638, −10.60243390992686763180459212321, −9.273573693922278227962714365000, −7.58747337729276357515549200597, −7.15048809103728269819147042573, −6.15111620893374675058012552049, −5.53598429770508466851570947062, −4.25255159338576754446482442967, −2.71790014719886745903394388184, 1.02024720192773573938503038813, 3.30446453932887491782196649325, 4.16922333776625917085480904307, 5.06676959926116452347541367754, 5.95831006967552392768353267778, 7.47828311541927808522292870971, 9.190121008691176149259620341856, 10.17839690342584031790243241328, 10.67957319390527955854638885167, 11.83046729529394586506339798311

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