↑

Properties

Label 2-273-273.272-c1-0-1
Degree $2$
Conductor $273$
Sign $-0.996 + 0.0824i$
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  − 1.10·2-s + (0.669 + 1.59i)3-s − 0.773·4-s − 1.62i·5-s + (−0.741 − 1.76i)6-s + (−2.53 + 0.762i)7-s + 3.07·8-s + (−2.10 + 2.13i)9-s + 1.80i·10-s − 2.37·11-s + (−0.517 − 1.23i)12-s + (−2.05 + 2.96i)13-s + (2.80 − 0.844i)14-s + (2.59 − 1.08i)15-s − 1.85·16-s − 5.91·17-s + ⋯
L(s)  = 1  − 0.783·2-s + (0.386 + 0.922i)3-s − 0.386·4-s − 0.727i·5-s + (−0.302 − 0.722i)6-s + (−0.957 + 0.288i)7-s + 1.08·8-s + (−0.701 + 0.712i)9-s + 0.569i·10-s − 0.715·11-s + (−0.149 − 0.356i)12-s + (−0.570 + 0.821i)13-s + (0.749 − 0.225i)14-s + (0.670 − 0.281i)15-s − 0.463·16-s − 1.43·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0824i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.996 + 0.0824i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.996 + 0.0824i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (272, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.996 + 0.0824i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00768355 - 0.186087i\)
\(L(\frac12)\) \(\approx\) \(0.00768355 - 0.186087i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.669 - 1.59i)T \)
7 \( 1 + (2.53 - 0.762i)T \)
13 \( 1 + (2.05 - 2.96i)T \)
good2 \( 1 + 1.10T + 2T^{2} \)
5 \( 1 + 1.62iT - 5T^{2} \)
11 \( 1 + 2.37T + 11T^{2} \)
17 \( 1 + 5.91T + 17T^{2} \)
19 \( 1 + 2.16T + 19T^{2} \)
23 \( 1 + 7.30iT - 23T^{2} \)
29 \( 1 + 1.65iT - 29T^{2} \)
31 \( 1 + 3.64T + 31T^{2} \)
37 \( 1 - 11.3iT - 37T^{2} \)
41 \( 1 - 2.27iT - 41T^{2} \)
43 \( 1 - 2.85T + 43T^{2} \)
47 \( 1 - 9.20iT - 47T^{2} \)
53 \( 1 - 12.3iT - 53T^{2} \)
59 \( 1 + 9.86iT - 59T^{2} \)
61 \( 1 - 2.47iT - 61T^{2} \)
67 \( 1 + 5.91iT - 67T^{2} \)
71 \( 1 - 8.43T + 71T^{2} \)
73 \( 1 - 7.77T + 73T^{2} \)
79 \( 1 + 12.3T + 79T^{2} \)
83 \( 1 + 5.44iT - 83T^{2} \)
89 \( 1 - 10.4iT - 89T^{2} \)
97 \( 1 - 4.17T + 97T^{2} \)
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

−12.52969822636898711964410299381, −11.02217717792594870300044725742, −10.22520906008663467776484977873, −9.317534613364005027297252747598, −8.889732705946348408885579242146, −8.039594577775175426360272278849, −6.53644297867830778113251497784, −4.91237298205317992859006502949, −4.30051709492357910964521416103, −2.52583318898526561915578395266, 0.16632172656679928212832636676, 2.33318001580895647087079642154, 3.66515087891213548469964712073, 5.52738596531717391242354271447, 6.95669597590931885106909736137, 7.41801212168163458932951127084, 8.550799393050406912602275245004, 9.399011865863561663744675298875, 10.36055961103198981375164084285, 11.15002246529830437002908118672

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