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Properties

Label 2-273-91.20-c1-0-3
Degree $2$
Conductor $273$
Sign $0.240 - 0.970i$
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  + (0.604 − 0.161i)2-s + (−0.866 − 0.5i)3-s + (−1.39 + 0.804i)4-s + (−0.965 + 0.965i)5-s + (−0.604 − 0.161i)6-s + (2.62 − 0.305i)7-s + (−1.59 + 1.59i)8-s + (0.499 + 0.866i)9-s + (−0.427 + 0.739i)10-s + (1.26 + 4.72i)11-s + 1.60·12-s + (−1.35 + 3.34i)13-s + (1.53 − 0.610i)14-s + (1.31 − 0.353i)15-s + (0.902 − 1.56i)16-s + (2.72 + 4.72i)17-s + ⋯
L(s)  = 1  + (0.427 − 0.114i)2-s + (−0.499 − 0.288i)3-s + (−0.696 + 0.402i)4-s + (−0.431 + 0.431i)5-s + (−0.246 − 0.0660i)6-s + (0.993 − 0.115i)7-s + (−0.564 + 0.564i)8-s + (0.166 + 0.288i)9-s + (−0.135 + 0.233i)10-s + (0.382 + 1.42i)11-s + 0.464·12-s + (−0.374 + 0.927i)13-s + (0.411 − 0.163i)14-s + (0.340 − 0.0912i)15-s + (0.225 − 0.390i)16-s + (0.660 + 1.14i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.240 - 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.240 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.240 - 0.970i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (202, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.240 - 0.970i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.790341 + 0.618512i\)
\(L(\frac12)\) \(\approx\) \(0.790341 + 0.618512i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.62 + 0.305i)T \)
13 \( 1 + (1.35 - 3.34i)T \)
good2 \( 1 + (-0.604 + 0.161i)T + (1.73 - i)T^{2} \)
5 \( 1 + (0.965 - 0.965i)T - 5iT^{2} \)
11 \( 1 + (-1.26 - 4.72i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (-2.72 - 4.72i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.47 + 1.19i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (3.14 + 1.81i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.00 + 1.74i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-5.91 + 5.91i)T - 31iT^{2} \)
37 \( 1 + (2.84 + 10.6i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (-1.08 - 4.04i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (0.669 - 0.386i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.65 - 5.65i)T + 47iT^{2} \)
53 \( 1 + 6.72T + 53T^{2} \)
59 \( 1 + (3.87 - 14.4i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (0.210 - 0.121i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.55 + 1.48i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (0.711 - 2.65i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (2.17 + 2.17i)T + 73iT^{2} \)
79 \( 1 - 8.38T + 79T^{2} \)
83 \( 1 + (-11.2 + 11.2i)T - 83iT^{2} \)
89 \( 1 + (3.25 - 0.872i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (9.46 + 2.53i)T + (84.0 + 48.5i)T^{2} \)
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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

−12.17101524825335925375067253214, −11.44282684538498202260465897295, −10.39620192710372098406032404318, −9.225958824984286503135797558033, −8.041305201413776442069300641781, −7.30812249474417597535212494855, −6.01756647883795243181190798547, −4.56993722848678351723491266266, −4.12903472186746465539399275399, −2.07315747600632648968185035242, 0.76975440690701761641262780412, 3.45164092641861319722034616399, 4.71515623695806463517350123548, 5.32644972918506097444089853767, 6.37902074176313264600869703283, 8.094370619904603673399122206833, 8.662822619455382839952926866774, 9.928406721874903455557355460844, 10.79675270908968119247116678655, 11.87598526166422037812399640368

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