↑

Properties

Label 2-273-91.20-c1-0-0
Degree $2$
Conductor $273$
Sign $-0.330 - 0.943i$
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  + (1.34 − 0.359i)2-s + (−0.866 − 0.5i)3-s + (−0.0625 + 0.0361i)4-s + (−2.52 + 2.52i)5-s + (−1.34 − 0.359i)6-s + (−1.59 + 2.11i)7-s + (−2.03 + 2.03i)8-s + (0.499 + 0.866i)9-s + (−2.48 + 4.29i)10-s + (0.0529 + 0.197i)11-s + 0.0722·12-s + (2.07 − 2.94i)13-s + (−1.37 + 3.40i)14-s + (3.45 − 0.925i)15-s + (−1.92 + 3.33i)16-s + (−1.13 − 1.97i)17-s + ⋯
L(s)  = 1  + (0.948 − 0.254i)2-s + (−0.499 − 0.288i)3-s + (−0.0312 + 0.0180i)4-s + (−1.13 + 1.13i)5-s + (−0.547 − 0.146i)6-s + (−0.602 + 0.798i)7-s + (−0.719 + 0.719i)8-s + (0.166 + 0.288i)9-s + (−0.785 + 1.35i)10-s + (0.0159 + 0.0596i)11-s + 0.0208·12-s + (0.575 − 0.817i)13-s + (−0.368 + 0.910i)14-s + (0.891 − 0.238i)15-s + (−0.481 + 0.833i)16-s + (−0.275 − 0.477i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.330 - 0.943i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.330 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.330 - 0.943i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (202, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.330 - 0.943i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.498473 + 0.702719i\)
\(L(\frac12)\) \(\approx\) \(0.498473 + 0.702719i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.59 - 2.11i)T \)
13 \( 1 + (-2.07 + 2.94i)T \)
good2 \( 1 + (-1.34 + 0.359i)T + (1.73 - i)T^{2} \)
5 \( 1 + (2.52 - 2.52i)T - 5iT^{2} \)
11 \( 1 + (-0.0529 - 0.197i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (1.13 + 1.97i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.50 - 0.402i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-2.59 - 1.49i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.75 - 8.23i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.75 - 2.75i)T - 31iT^{2} \)
37 \( 1 + (-1.17 - 4.39i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (-1.36 - 5.07i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-1.76 + 1.01i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-9.42 - 9.42i)T + 47iT^{2} \)
53 \( 1 + 12.6T + 53T^{2} \)
59 \( 1 + (-0.510 + 1.90i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-0.850 + 0.491i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-15.1 + 4.06i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-0.00355 + 0.0132i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (2.24 + 2.24i)T + 73iT^{2} \)
79 \( 1 - 12.7T + 79T^{2} \)
83 \( 1 + (3.37 - 3.37i)T - 83iT^{2} \)
89 \( 1 + (11.8 - 3.17i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (-7.85 - 2.10i)T + (84.0 + 48.5i)T^{2} \)
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

−12.37158667757070821688849730413, −11.29268153577046898296574187111, −10.94151790687541220442143217920, −9.358764908727157233874429598549, −8.160928014668061970972329938684, −7.09273975688366563590946674230, −6.06115263911148989686261242781, −5.02871186809456000153516289548, −3.56470268188976798882362559265, −2.89788007120400461275293753603, 0.52465750620084538869660427143, 3.93049015493361235836712832995, 4.07731639013414369219748770332, 5.29655571434940651302392343467, 6.38472117600023841518144199884, 7.49927153574450587758637366059, 8.846421209497088014443932472180, 9.607093019314834971573880969541, 10.97066275383581259769828617762, 11.81811026845151014279390558371

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