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Properties

Label 2-273-273.62-c1-0-28
Degree $2$
Conductor $273$
Sign $0.407 + 0.913i$
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  + (1.5 − 0.866i)3-s + (1 − 1.73i)4-s + (−2.5 − 0.866i)7-s + (1.5 − 2.59i)9-s − 3.46i·12-s + (2.5 + 2.59i)13-s + (−1.99 − 3.46i)16-s + (−4 + 6.92i)19-s + (−4.5 + 0.866i)21-s + 5·25-s − 5.19i·27-s + (−4 + 3.46i)28-s + 7·31-s + (−3 − 5.19i)36-s + (6 − 3.46i)37-s + ⋯
L(s)  = 1  + (0.866 − 0.499i)3-s + (0.5 − 0.866i)4-s + (−0.944 − 0.327i)7-s + (0.5 − 0.866i)9-s − 0.999i·12-s + (0.693 + 0.720i)13-s + (−0.499 − 0.866i)16-s + (−0.917 + 1.58i)19-s + (−0.981 + 0.188i)21-s + 25-s − 0.999i·27-s + (−0.755 + 0.654i)28-s + 1.25·31-s + (−0.5 − 0.866i)36-s + (0.986 − 0.569i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.407 + 0.913i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.407 + 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.407 + 0.913i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (62, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.407 + 0.913i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.42669 - 0.925507i\)
\(L(\frac12)\) \(\approx\) \(1.42669 - 0.925507i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.5 + 0.866i)T \)
7 \( 1 + (2.5 + 0.866i)T \)
13 \( 1 + (-2.5 - 2.59i)T \)
good2 \( 1 + (-1 + 1.73i)T^{2} \)
5 \( 1 - 5T^{2} \)
11 \( 1 + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4 - 6.92i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 7T + 31T^{2} \)
37 \( 1 + (-6 + 3.46i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (6.5 - 11.2i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (7.5 + 4.33i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (10.5 - 6.06i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 17T + 73T^{2} \)
79 \( 1 - 13T + 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.5 - 4.33i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.81304630716341968080759031897, −10.58921493789206326309490353247, −9.833109972530714809126527562638, −8.937869644527916290778562486477, −7.79933088963558749217157555327, −6.57095022960282410074537069827, −6.17911972433700601838042301730, −4.23442401067337627477533254992, −2.89219752080197295347989655315, −1.42830943399969465503204095885, 2.58903831305641193565273178659, 3.31460287835113398316659963278, 4.58740299506504913870549686021, 6.30326179667536165592530001401, 7.26179450250040413496658364663, 8.467749300075130034196741846508, 8.940005621434105175667922144109, 10.19824282904115939800797122599, 11.00985004001999004616389462856, 12.20451991458521757531125864133

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