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Properties

Label 2-273-91.76-c1-0-6
Degree $2$
Conductor $273$
Sign $0.323 - 0.946i$
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  + (−0.500 + 1.86i)2-s + (−0.866 + 0.5i)3-s + (−1.50 − 0.870i)4-s + (2.44 − 2.44i)5-s + (−0.500 − 1.86i)6-s + (2.02 − 1.70i)7-s + (−0.353 + 0.353i)8-s + (0.499 − 0.866i)9-s + (3.33 + 5.78i)10-s + (2.85 + 0.764i)11-s + 1.74·12-s + (3.60 + 0.0697i)13-s + (2.16 + 4.63i)14-s + (−0.893 + 3.33i)15-s + (−2.22 − 3.85i)16-s + (−0.667 + 1.15i)17-s + ⋯
L(s)  = 1  + (−0.354 + 1.32i)2-s + (−0.499 + 0.288i)3-s + (−0.754 − 0.435i)4-s + (1.09 − 1.09i)5-s + (−0.204 − 0.762i)6-s + (0.765 − 0.643i)7-s + (−0.124 + 0.124i)8-s + (0.166 − 0.288i)9-s + (1.05 + 1.82i)10-s + (0.860 + 0.230i)11-s + 0.502·12-s + (0.999 + 0.0193i)13-s + (0.579 + 1.23i)14-s + (−0.230 + 0.860i)15-s + (−0.556 − 0.963i)16-s + (−0.161 + 0.280i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.323 - 0.946i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.323 - 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.323 - 0.946i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (76, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.323 - 0.946i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.967455 + 0.692046i\)
\(L(\frac12)\) \(\approx\) \(0.967455 + 0.692046i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.02 + 1.70i)T \)
13 \( 1 + (-3.60 - 0.0697i)T \)
good2 \( 1 + (0.500 - 1.86i)T + (-1.73 - i)T^{2} \)
5 \( 1 + (-2.44 + 2.44i)T - 5iT^{2} \)
11 \( 1 + (-2.85 - 0.764i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (0.667 - 1.15i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.32 + 4.94i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (7.61 - 4.39i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.97 - 6.89i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.04 - 2.04i)T - 31iT^{2} \)
37 \( 1 + (-4.73 - 1.26i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (1.45 + 0.390i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (0.212 + 0.122i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.13 + 1.13i)T + 47iT^{2} \)
53 \( 1 - 2.62T + 53T^{2} \)
59 \( 1 + (-3.96 + 1.06i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (7.82 + 4.52i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.97 - 11.1i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (11.0 - 2.96i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (-1.06 - 1.06i)T + 73iT^{2} \)
79 \( 1 - 7.65T + 79T^{2} \)
83 \( 1 + (10.1 - 10.1i)T - 83iT^{2} \)
89 \( 1 + (-4.11 + 15.3i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-3.28 - 12.2i)T + (-84.0 + 48.5i)T^{2} \)
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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

−12.05878202681223027523683585354, −11.07732080012487057181742632806, −9.880456599001151484064584039265, −8.969798782773838677012424838635, −8.347603856745362684008767112193, −7.01188609228087480116398380848, −6.10895488391369890595578467303, −5.26981563558707377393452692993, −4.29188805021826677899014435100, −1.42691865436131408675204426465, 1.60921308752296329648048158585, 2.50751357076710404442154129680, 4.04088265695655416821054782250, 6.07413045785750674091387168827, 6.26554506983280728724743681202, 8.092978644256118910861240289378, 9.217666397235648776179432222197, 10.18721933496009856311065550292, 10.77116980135807264235101062446, 11.64132631697586308696118859359

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