Properties

Label 2-273-91.19-c1-0-9
Degree $2$
Conductor $273$
Sign $0.925 + 0.379i$
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  + (0.251 + 0.0673i)2-s + i·3-s + (−1.67 − 0.966i)4-s + (3.35 − 0.900i)5-s + (−0.0673 + 0.251i)6-s + (0.422 − 2.61i)7-s + (−0.723 − 0.723i)8-s − 9-s + 0.905·10-s + (1.69 + 1.69i)11-s + (0.966 − 1.67i)12-s + (−1.60 − 3.23i)13-s + (0.282 − 0.628i)14-s + (0.900 + 3.35i)15-s + (1.79 + 3.11i)16-s + (1.67 − 2.89i)17-s + ⋯
L(s)  = 1  + (0.177 + 0.0476i)2-s + 0.577i·3-s + (−0.836 − 0.483i)4-s + (1.50 − 0.402i)5-s + (−0.0275 + 0.102i)6-s + (0.159 − 0.987i)7-s + (−0.255 − 0.255i)8-s − 0.333·9-s + 0.286·10-s + (0.512 + 0.512i)11-s + (0.278 − 0.483i)12-s + (−0.443 − 0.896i)13-s + (0.0754 − 0.167i)14-s + (0.232 + 0.867i)15-s + (0.449 + 0.779i)16-s + (0.405 − 0.702i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.925 + 0.379i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.925 + 0.379i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.925 + 0.379i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.925 + 0.379i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.44073 - 0.284073i\)
\(L(\frac12)\) \(\approx\) \(1.44073 - 0.284073i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - iT \)
7 \( 1 + (-0.422 + 2.61i)T \)
13 \( 1 + (1.60 + 3.23i)T \)
good2 \( 1 + (-0.251 - 0.0673i)T + (1.73 + i)T^{2} \)
5 \( 1 + (-3.35 + 0.900i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (-1.69 - 1.69i)T + 11iT^{2} \)
17 \( 1 + (-1.67 + 2.89i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.43 - 4.43i)T + 19iT^{2} \)
23 \( 1 + (-0.136 + 0.0787i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.530 + 0.919i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.37 - 5.14i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (0.635 - 2.37i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (11.0 - 2.95i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (6.74 - 3.89i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.03 + 7.60i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (3.19 + 5.54i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.931 - 3.47i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 - 12.4iT - 61T^{2} \)
67 \( 1 + (-3.01 + 3.01i)T - 67iT^{2} \)
71 \( 1 + (-3.09 - 0.829i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (-7.36 - 1.97i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-3.17 + 5.49i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-10.4 - 10.4i)T + 83iT^{2} \)
89 \( 1 + (13.8 + 3.71i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (1.40 - 5.24i)T + (-84.0 - 48.5i)T^{2} \)
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

−11.94431772182333844667339913126, −10.29586629062971487968408655598, −10.01914948835133797350454153892, −9.375786826381371988765171745700, −8.168531984399957448829217255152, −6.66990548474948283600768865171, −5.36475707663652294293587802416, −4.95173441838192395225655294835, −3.49550290246675059651875854300, −1.33534338630234995584471679562, 1.93426456566254373795400580430, 3.19703543233619307363693561307, 5.02535463203058171366740511418, 5.87195290664211285481078920047, 6.86454849167013547118805647542, 8.276146568644259343135524086385, 9.216421363762783030448446317527, 9.694454986028136836433725248439, 11.24522144073298707481154788794, 12.12605758408646743758441326175

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