↑

Properties

Label 2-273-91.9-c1-0-10
Degree $2$
Conductor $273$
Sign $-0.659 + 0.751i$
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  + (−1.31 − 2.27i)2-s + 3-s + (−2.44 + 4.23i)4-s + (0.734 − 1.27i)5-s + (−1.31 − 2.27i)6-s + (2.40 − 1.09i)7-s + 7.59·8-s + 9-s − 3.85·10-s + 4.48·11-s + (−2.44 + 4.23i)12-s + (−3.58 + 0.374i)13-s + (−5.65 − 4.03i)14-s + (0.734 − 1.27i)15-s + (−5.08 − 8.80i)16-s + (1.81 − 3.14i)17-s + ⋯
L(s)  = 1  + (−0.928 − 1.60i)2-s + 0.577·3-s + (−1.22 + 2.11i)4-s + (0.328 − 0.569i)5-s + (−0.535 − 0.928i)6-s + (0.910 − 0.414i)7-s + 2.68·8-s + 0.333·9-s − 1.21·10-s + 1.35·11-s + (−0.706 + 1.22i)12-s + (−0.994 + 0.103i)13-s + (−1.51 − 1.07i)14-s + (0.189 − 0.328i)15-s + (−1.27 − 2.20i)16-s + (0.440 − 0.762i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.659 + 0.751i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.659 + 0.751i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.659 + 0.751i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (100, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.659 + 0.751i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.431331 - 0.952975i\)
\(L(\frac12)\) \(\approx\) \(0.431331 - 0.952975i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
7 \( 1 + (-2.40 + 1.09i)T \)
13 \( 1 + (3.58 - 0.374i)T \)
good2 \( 1 + (1.31 + 2.27i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-0.734 + 1.27i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 - 4.48T + 11T^{2} \)
17 \( 1 + (-1.81 + 3.14i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + 3.01T + 19T^{2} \)
23 \( 1 + (3.87 + 6.71i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.98 - 6.90i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.552 + 0.957i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.57 - 2.72i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.11 + 1.92i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.54 - 4.41i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.69 + 2.94i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.25 + 7.37i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (7.22 - 12.5i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 - 7.62T + 61T^{2} \)
67 \( 1 - 4.67T + 67T^{2} \)
71 \( 1 + (-1.24 - 2.15i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-3.03 - 5.24i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.59 - 6.22i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 9.30T + 83T^{2} \)
89 \( 1 + (3.18 + 5.50i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (4.02 + 6.96i)T + (-48.5 + 84.0i)T^{2} \)
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

−11.50334303458542488245821225282, −10.54516203879631099721309059000, −9.620525352049623475401715794013, −8.978964478867718458537208373022, −8.198381684453272470334225321466, −7.11836519443937516678284389425, −4.79847050372398418286204034097, −3.85962491786833665956446709060, −2.33943966874041079875582125517, −1.22229014842293744782410772822, 1.83989964732314826392000743735, 4.23902982354403949997243311507, 5.63492089424126354326352686698, 6.49019414554053376084814797548, 7.57111193813465832071555234211, 8.177223049719286695411838408453, 9.250700631372466240807798751156, 9.788130341060995088150145377700, 10.93466044570791947453319683065, 12.25822249475599349359559766642

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