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Properties

Label 2-273-91.4-c1-0-7
Degree $2$
Conductor $273$
Sign $0.233 - 0.972i$
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  + 0.871i·2-s + (0.5 + 0.866i)3-s + 1.24·4-s + (1.34 − 0.773i)5-s + (−0.754 + 0.435i)6-s + (−2.02 + 1.70i)7-s + 2.82i·8-s + (−0.499 + 0.866i)9-s + (0.674 + 1.16i)10-s + (3.13 − 1.80i)11-s + (0.620 + 1.07i)12-s + (−3.37 − 1.26i)13-s + (−1.48 − 1.76i)14-s + (1.34 + 0.773i)15-s + 0.0187·16-s + 4.63·17-s + ⋯
L(s)  = 1  + 0.616i·2-s + (0.288 + 0.499i)3-s + 0.620·4-s + (0.599 − 0.345i)5-s + (−0.308 + 0.177i)6-s + (−0.766 + 0.642i)7-s + 0.998i·8-s + (−0.166 + 0.288i)9-s + (0.213 + 0.369i)10-s + (0.944 − 0.545i)11-s + (0.179 + 0.310i)12-s + (−0.936 − 0.351i)13-s + (−0.396 − 0.472i)14-s + (0.345 + 0.199i)15-s + 0.00467·16-s + 1.12·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.233 - 0.972i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.233 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.233 - 0.972i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.233 - 0.972i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.31305 + 1.03494i\)
\(L(\frac12)\) \(\approx\) \(1.31305 + 1.03494i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (2.02 - 1.70i)T \)
13 \( 1 + (3.37 + 1.26i)T \)
good2 \( 1 - 0.871iT - 2T^{2} \)
5 \( 1 + (-1.34 + 0.773i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-3.13 + 1.80i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 - 4.63T + 17T^{2} \)
19 \( 1 + (0.508 + 0.293i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 2.28T + 23T^{2} \)
29 \( 1 + (-1.07 + 1.86i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (8.99 + 5.19i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 5.80iT - 37T^{2} \)
41 \( 1 + (1.60 + 0.926i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.93 - 6.81i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.10 + 1.21i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6.49 + 11.2i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 5.41iT - 59T^{2} \)
61 \( 1 + (-4.85 + 8.40i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.66 - 0.960i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (9.14 - 5.28i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (1.96 + 1.13i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.78 - 10.0i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 9.45iT - 83T^{2} \)
89 \( 1 + 12.6iT - 89T^{2} \)
97 \( 1 + (14.3 - 8.30i)T + (48.5 - 84.0i)T^{2} \)
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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

−12.10597485413885914193797582878, −11.19049218923742806964429434123, −9.899467530672153538976316690044, −9.332517499137335938003073259628, −8.243929567332677747666447548754, −7.14791412763927551400083945759, −5.92971418178355713764184647018, −5.41465625472794761903784624943, −3.56492119873196176073300555031, −2.23129486069410932403323326866, 1.54417878719258237089060939588, 2.80541059258565915072751149659, 3.95932465338180858340040832612, 5.93379263595658498294162142865, 6.92237888762944166759920544890, 7.41511178529872430028839708041, 9.173269652718831353798865358834, 9.998724773186742263115762018307, 10.58129088059041878380856878284, 12.15213767116255428579301121933

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