Properties

Label 2-273-91.6-c1-0-5
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: $\chi_{273} (97, \cdot )$
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

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

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