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Properties

Label 2-273-91.45-c1-0-15
Degree $2$
Conductor $273$
Sign $0.196 + 0.980i$
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  + (1.09 − 1.09i)2-s + (0.866 − 0.5i)3-s − 0.419i·4-s + (0.745 − 2.78i)5-s + (0.402 − 1.50i)6-s + (−1.80 + 1.93i)7-s + (1.73 + 1.73i)8-s + (0.499 − 0.866i)9-s + (−2.24 − 3.88i)10-s + (1.41 − 5.28i)11-s + (−0.209 − 0.363i)12-s + (−0.662 + 3.54i)13-s + (0.136 + 4.11i)14-s + (−0.745 − 2.78i)15-s + 4.66·16-s − 4.36·17-s + ⋯
L(s)  = 1  + (0.777 − 0.777i)2-s + (0.499 − 0.288i)3-s − 0.209i·4-s + (0.333 − 1.24i)5-s + (0.164 − 0.613i)6-s + (−0.683 + 0.730i)7-s + (0.614 + 0.614i)8-s + (0.166 − 0.288i)9-s + (−0.708 − 1.22i)10-s + (0.427 − 1.59i)11-s + (−0.0605 − 0.104i)12-s + (−0.183 + 0.982i)13-s + (0.0363 + 1.09i)14-s + (−0.192 − 0.718i)15-s + 1.16·16-s − 1.05·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.196 + 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.196 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.196 + 0.980i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (136, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.196 + 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.68057 - 1.37660i\)
\(L(\frac12)\) \(\approx\) \(1.68057 - 1.37660i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.866 + 0.5i)T \)
7 \( 1 + (1.80 - 1.93i)T \)
13 \( 1 + (0.662 - 3.54i)T \)
good2 \( 1 + (-1.09 + 1.09i)T - 2iT^{2} \)
5 \( 1 + (-0.745 + 2.78i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (-1.41 + 5.28i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + 4.36T + 17T^{2} \)
19 \( 1 + (1.39 - 0.373i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 - 8.37iT - 23T^{2} \)
29 \( 1 + (-0.882 + 1.52i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.770 + 0.206i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (-3.86 - 3.86i)T + 37iT^{2} \)
41 \( 1 + (-3.88 + 1.03i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (5.58 - 3.22i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (8.52 + 2.28i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-0.139 + 0.241i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.16 - 5.16i)T - 59iT^{2} \)
61 \( 1 + (-4.10 - 2.37i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.87 - 0.502i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-11.3 - 3.04i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (3.72 + 13.9i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (0.431 + 0.746i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.29 - 4.29i)T + 83iT^{2} \)
89 \( 1 + (-4.21 + 4.21i)T - 89iT^{2} \)
97 \( 1 + (0.575 - 2.14i)T + (-84.0 - 48.5i)T^{2} \)
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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

−11.82301763860584138374723146322, −11.27073096270851684074667090810, −9.583526756839088488832771517789, −8.896940528703479994633829473280, −8.156712481477833092494655654220, −6.45984351584906432870077226363, −5.38525567591627932796271852344, −4.17165030912158020399654363533, −3.04863803096614503935002553155, −1.68955445586921780468651965367, 2.51428354276924780275523039501, 3.90793054217630226161380902575, 4.84283048253555496601507014373, 6.57627122229354777980210017149, 6.71952112412161142084288964406, 7.82614088600813406601992325535, 9.506452522168898430235564270646, 10.27119307637134884408170851494, 10.77164042085305331382070753056, 12.63473581860937197337171462939

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