↑

Properties

Label 2-273-13.10-c1-0-8
Degree $2$
Conductor $273$
Sign $0.816 - 0.577i$
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  + (1.15 − 0.664i)2-s + (0.5 + 0.866i)3-s + (−0.117 + 0.204i)4-s + 1.55i·5-s + (1.15 + 0.664i)6-s + (−0.866 − 0.5i)7-s + 2.96i·8-s + (−0.499 + 0.866i)9-s + (1.03 + 1.79i)10-s + (3.66 − 2.11i)11-s − 0.235·12-s + (3.29 + 1.46i)13-s − 1.32·14-s + (−1.34 + 0.779i)15-s + (1.73 + 3.00i)16-s + (−0.391 + 0.678i)17-s + ⋯
L(s)  = 1  + (0.813 − 0.469i)2-s + (0.288 + 0.499i)3-s + (−0.0589 + 0.102i)4-s + 0.696i·5-s + (0.469 + 0.271i)6-s + (−0.327 − 0.188i)7-s + 1.04i·8-s + (−0.166 + 0.288i)9-s + (0.327 + 0.566i)10-s + (1.10 − 0.638i)11-s − 0.0680·12-s + (0.913 + 0.406i)13-s − 0.355·14-s + (−0.348 + 0.201i)15-s + (0.434 + 0.751i)16-s + (−0.0949 + 0.164i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.816 - 0.577i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.816 - 0.577i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.86063 + 0.591874i\)
\(L(\frac12)\) \(\approx\) \(1.86063 + 0.591874i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (0.866 + 0.5i)T \)
13 \( 1 + (-3.29 - 1.46i)T \)
good2 \( 1 + (-1.15 + 0.664i)T + (1 - 1.73i)T^{2} \)
5 \( 1 - 1.55iT - 5T^{2} \)
11 \( 1 + (-3.66 + 2.11i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (0.391 - 0.678i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (5.02 + 2.90i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.07 + 3.60i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (5.01 + 8.69i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 6.43iT - 31T^{2} \)
37 \( 1 + (-5.84 + 3.37i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.745 - 0.430i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.79 + 4.84i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 13.1iT - 47T^{2} \)
53 \( 1 + 2.25T + 53T^{2} \)
59 \( 1 + (-10.2 - 5.91i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.69 - 6.40i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (9.30 - 5.37i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (3.62 + 2.09i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 13.4iT - 73T^{2} \)
79 \( 1 - 13.8T + 79T^{2} \)
83 \( 1 + 0.989iT - 83T^{2} \)
89 \( 1 + (4.59 - 2.65i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.0673 + 0.0388i)T + (48.5 + 84.0i)T^{2} \)
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

−11.90338300158597864741429226368, −11.14634432103387280006489120535, −10.44100051995460162101006858419, −9.027209731444550750769513742995, −8.419797812319177666250749799863, −6.84202138956781449548764527247, −5.87377356677357845832445449370, −4.24437497501055975431836426635, −3.71000979585263535769402339548, −2.45615488697700526304118161982, 1.42153645387276627944332598064, 3.58010071384073912360046712057, 4.58593608058804414311283627225, 5.90192237844298104669733972937, 6.53554534772339379363320026055, 7.77863727959807163588435364069, 8.997340731324472910280546407083, 9.617278265661601316698468956219, 11.02597574112116052404646120514, 12.34842708878872872537921412624

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