↑

Properties

Label 2-273-91.6-c1-0-9
Degree $2$
Conductor $273$
Sign $0.999 - 0.00540i$
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  + (0.0473 + 0.176i)2-s + (−0.866 − 0.5i)3-s + (1.70 − 0.983i)4-s + (2.80 + 2.80i)5-s + (0.0473 − 0.176i)6-s + (−0.467 − 2.60i)7-s + (0.513 + 0.513i)8-s + (0.499 + 0.866i)9-s + (−0.362 + 0.627i)10-s + (−2.53 + 0.679i)11-s − 1.96·12-s + (1.37 + 3.33i)13-s + (0.437 − 0.205i)14-s + (−1.02 − 3.82i)15-s + (1.90 − 3.29i)16-s + (−1.43 − 2.48i)17-s + ⋯
L(s)  = 1  + (0.0334 + 0.124i)2-s + (−0.499 − 0.288i)3-s + (0.851 − 0.491i)4-s + (1.25 + 1.25i)5-s + (0.0193 − 0.0721i)6-s + (−0.176 − 0.984i)7-s + (0.181 + 0.181i)8-s + (0.166 + 0.288i)9-s + (−0.114 + 0.198i)10-s + (−0.764 + 0.204i)11-s − 0.567·12-s + (0.380 + 0.924i)13-s + (0.117 − 0.0550i)14-s + (−0.264 − 0.987i)15-s + (0.475 − 0.822i)16-s + (−0.347 − 0.601i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.00540i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.00540i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.999 - 0.00540i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.999 - 0.00540i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.55037 + 0.00419139i\)
\(L(\frac12)\) \(\approx\) \(1.55037 + 0.00419139i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.467 + 2.60i)T \)
13 \( 1 + (-1.37 - 3.33i)T \)
good2 \( 1 + (-0.0473 - 0.176i)T + (-1.73 + i)T^{2} \)
5 \( 1 + (-2.80 - 2.80i)T + 5iT^{2} \)
11 \( 1 + (2.53 - 0.679i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (1.43 + 2.48i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.759 + 2.83i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-7.27 - 4.19i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.66 + 2.87i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (6.75 + 6.75i)T + 31iT^{2} \)
37 \( 1 + (6.77 - 1.81i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (2.79 - 0.747i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (2.43 - 1.40i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.85 - 4.85i)T - 47iT^{2} \)
53 \( 1 + 5.43T + 53T^{2} \)
59 \( 1 + (0.00666 + 0.00178i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-5.65 + 3.26i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.10 - 7.84i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (14.5 + 3.91i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (-0.321 + 0.321i)T - 73iT^{2} \)
79 \( 1 - 0.280T + 79T^{2} \)
83 \( 1 + (2.42 + 2.42i)T + 83iT^{2} \)
89 \( 1 + (-0.0536 - 0.200i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-0.197 + 0.736i)T + (-84.0 - 48.5i)T^{2} \)
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

−11.34213334141124735281674727028, −11.09641184931661417010751978581, −10.18250398761848916507529556151, −9.473332416415418969966733915605, −7.33466039191884260834465430848, −6.95038621502901646081820443077, −6.13740770942159434803705945743, −5.06454220199130085377892318517, −2.98802421535780030892117527889, −1.74336099782161340308973105049, 1.71420286944366368372770193976, 3.13975699990082171899387736367, 5.11224585800308918437566961066, 5.65266425166904027927764755142, 6.69125989108883082314353096419, 8.360846683320718013382705570916, 8.914197118755909177339282513536, 10.22675505367413257982276501764, 10.82372021295851859767672554953, 12.15547658261281310256609257668

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