↑

Properties

Label 2-273-91.54-c1-0-15
Degree $2$
Conductor $273$
Sign $-0.0871 + 0.996i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.55 − 1.55i)2-s + (−0.866 − 0.5i)3-s − 2.84i·4-s + (3.12 − 0.837i)5-s + (−2.12 + 0.569i)6-s + (1.93 + 1.79i)7-s + (−1.31 − 1.31i)8-s + (0.499 + 0.866i)9-s + (3.56 − 6.16i)10-s + (−3.89 + 1.04i)11-s + (−1.42 + 2.46i)12-s + (−2.61 − 2.47i)13-s + (5.81 − 0.218i)14-s + (−3.12 − 0.837i)15-s + 1.59·16-s − 7.61·17-s + ⋯
L(s)  = 1  + (1.10 − 1.10i)2-s + (−0.499 − 0.288i)3-s − 1.42i·4-s + (1.39 − 0.374i)5-s + (−0.868 + 0.232i)6-s + (0.733 + 0.680i)7-s + (−0.465 − 0.465i)8-s + (0.166 + 0.288i)9-s + (1.12 − 1.95i)10-s + (−1.17 + 0.314i)11-s + (−0.410 + 0.711i)12-s + (−0.726 − 0.687i)13-s + (1.55 − 0.0584i)14-s + (−0.806 − 0.216i)15-s + 0.398·16-s − 1.84·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0871 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0871 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.0871 + 0.996i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.0871 + 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.52111 - 1.65991i\)
\(L(\frac12)\) \(\approx\) \(1.52111 - 1.65991i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.93 - 1.79i)T \)
13 \( 1 + (2.61 + 2.47i)T \)
good2 \( 1 + (-1.55 + 1.55i)T - 2iT^{2} \)
5 \( 1 + (-3.12 + 0.837i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (3.89 - 1.04i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + 7.61T + 17T^{2} \)
19 \( 1 + (0.714 - 2.66i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 - 4.49iT - 23T^{2} \)
29 \( 1 + (-1.49 - 2.58i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.691 + 2.58i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (-1.20 - 1.20i)T + 37iT^{2} \)
41 \( 1 + (-2.95 + 11.0i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-6.30 - 3.64i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.666 + 2.48i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (3.91 + 6.77i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.78 - 5.78i)T - 59iT^{2} \)
61 \( 1 + (7.97 - 4.60i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.02 - 7.54i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (0.721 + 2.69i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (-1.69 - 0.453i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-4.31 + 7.46i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.63 - 1.63i)T + 83iT^{2} \)
89 \( 1 + (-12.7 + 12.7i)T - 89iT^{2} \)
97 \( 1 + (-12.7 + 3.41i)T + (84.0 - 48.5i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.84478682428061279532342383625, −10.83452682024810933680533888290, −10.24107716582647064070985091365, −9.094789779625579765413917879476, −7.70114609050012002851435083686, −5.99301883842116982489730802056, −5.32243508169693153912287134866, −4.63882601408465810212330895971, −2.53558255917226960085488806564, −1.87004550823240534654825464739, 2.45944833582099573200861971743, 4.51400504728841693993170219694, 4.95243444233460940168962059037, 6.19354294200300024466816352292, 6.74058263361422583283404683754, 7.88328819500375997416982935002, 9.305658577518567057444786789451, 10.49244167751950865461140036278, 11.06381897501546405703523854625, 12.57460752337446221831693697712

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