↑

Properties

Label 2-273-273.242-c1-0-8
Degree $2$
Conductor $273$
Sign $0.591 - 0.806i$
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  + (−2.42 − 0.649i)2-s + (−1.28 + 1.16i)3-s + (3.72 + 2.14i)4-s + (3.58 + 0.959i)5-s + (3.86 − 1.98i)6-s + (1.55 + 2.14i)7-s + (−4.07 − 4.07i)8-s + (0.301 − 2.98i)9-s + (−8.06 − 4.65i)10-s + (−2.52 + 0.677i)11-s + (−7.27 + 1.56i)12-s + (3.59 + 0.242i)13-s + (−2.37 − 6.20i)14-s + (−5.71 + 2.92i)15-s + (2.93 + 5.08i)16-s + (−0.358 + 0.621i)17-s + ⋯
L(s)  = 1  + (−1.71 − 0.459i)2-s + (−0.741 + 0.670i)3-s + (1.86 + 1.07i)4-s + (1.60 + 0.429i)5-s + (1.57 − 0.808i)6-s + (0.586 + 0.809i)7-s + (−1.44 − 1.44i)8-s + (0.100 − 0.994i)9-s + (−2.54 − 1.47i)10-s + (−0.761 + 0.204i)11-s + (−2.10 + 0.450i)12-s + (0.997 + 0.0671i)13-s + (−0.634 − 1.65i)14-s + (−1.47 + 0.755i)15-s + (0.733 + 1.27i)16-s + (−0.0870 + 0.150i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.591 - 0.806i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.591 - 0.806i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.591 - 0.806i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (242, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.591 - 0.806i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.561842 + 0.284601i\)
\(L(\frac12)\) \(\approx\) \(0.561842 + 0.284601i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.28 - 1.16i)T \)
7 \( 1 + (-1.55 - 2.14i)T \)
13 \( 1 + (-3.59 - 0.242i)T \)
good2 \( 1 + (2.42 + 0.649i)T + (1.73 + i)T^{2} \)
5 \( 1 + (-3.58 - 0.959i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (2.52 - 0.677i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (0.358 - 0.621i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.19 + 4.46i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-2.80 - 4.86i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 1.19iT - 29T^{2} \)
31 \( 1 + (3.93 - 1.05i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (5.34 + 1.43i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-3.34 + 3.34i)T - 41iT^{2} \)
43 \( 1 - 4.83iT - 43T^{2} \)
47 \( 1 + (0.235 - 0.879i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-2.40 - 1.38i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (8.51 - 2.28i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-5.31 - 9.21i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.15 - 0.844i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-7.32 + 7.32i)T - 71iT^{2} \)
73 \( 1 + (1.53 + 5.73i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (1.74 + 3.02i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.93 - 1.93i)T - 83iT^{2} \)
89 \( 1 + (2.74 - 10.2i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (13.6 + 13.6i)T + 97iT^{2} \)
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

−11.34611602423473075493503817143, −10.90862175465518641530689668015, −10.12583321948107880761802042561, −9.277925252701226716756995867620, −8.797106644060159566605624494033, −7.27392863062268436911875677008, −6.11768388880698619990680273583, −5.24432458687195771014761192973, −2.87123756626455486661186063325, −1.59493508446747760561766733364, 1.02374296834741558106126508374, 1.99101074722664212411649223369, 5.21677991248586977276135968161, 6.07054686551712435884560347512, 6.93616213559414101013010983250, 7.992375773441832515227827295290, 8.754850512381663491465107327456, 9.971624624193778769047225058338, 10.56851494781589896567055060765, 11.19171283923357722561571427728

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