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Properties

Label 2-273-91.81-c1-0-11
Degree $2$
Conductor $273$
Sign $0.982 - 0.183i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.613 + 1.06i)2-s + 3-s + (0.247 + 0.429i)4-s + (−2.10 − 3.64i)5-s + (−0.613 + 1.06i)6-s + (2.23 − 1.41i)7-s − 3.06·8-s + 9-s + 5.16·10-s + 5.52·11-s + (0.247 + 0.429i)12-s + (3.59 + 0.226i)13-s + (0.139 + 3.24i)14-s + (−2.10 − 3.64i)15-s + (1.38 − 2.39i)16-s + (−0.0891 − 0.154i)17-s + ⋯
L(s)  = 1  + (−0.433 + 0.751i)2-s + 0.577·3-s + (0.123 + 0.214i)4-s + (−0.942 − 1.63i)5-s + (−0.250 + 0.433i)6-s + (0.843 − 0.536i)7-s − 1.08·8-s + 0.333·9-s + 1.63·10-s + 1.66·11-s + (0.0715 + 0.123i)12-s + (0.998 + 0.0627i)13-s + (0.0372 + 0.866i)14-s + (−0.543 − 0.942i)15-s + (0.345 − 0.598i)16-s + (−0.0216 − 0.0374i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 - 0.183i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.982 - 0.183i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.982 - 0.183i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (172, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.982 - 0.183i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.25143 + 0.116095i\)
\(L(\frac12)\) \(\approx\) \(1.25143 + 0.116095i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
7 \( 1 + (-2.23 + 1.41i)T \)
13 \( 1 + (-3.59 - 0.226i)T \)
good2 \( 1 + (0.613 - 1.06i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (2.10 + 3.64i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 - 5.52T + 11T^{2} \)
17 \( 1 + (0.0891 + 0.154i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + 4.51T + 19T^{2} \)
23 \( 1 + (0.543 - 0.941i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.0731 + 0.126i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.19 + 7.26i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.15 - 3.73i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.782 - 1.35i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.66 - 2.88i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.636 - 1.10i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.93 - 6.82i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.01 - 1.75i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + 3.76T + 61T^{2} \)
67 \( 1 + 0.307T + 67T^{2} \)
71 \( 1 + (1.62 - 2.82i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (3.53 - 6.11i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.30 + 3.99i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 8.85T + 83T^{2} \)
89 \( 1 + (6.59 - 11.4i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.17 + 2.03i)T + (-48.5 - 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.87624740566773253148987089273, −11.31129554090089753534151449298, −9.461974453527608186168153958975, −8.607528146684777434517997648922, −8.302988401372126269590144192200, −7.36816411265250573399691343374, −6.15546047892485104048789334065, −4.44844956686828404759991312706, −3.82229366897289201960633030166, −1.27975158432179767045686574033, 1.79727083683669845454963174553, 3.09751984824603103040968409916, 4.05688292190028729707579040339, 6.21128983638838104305515465233, 6.91629697796176645813457376763, 8.307175235249411823043356075961, 8.957970132498888324321070642767, 10.27957640187194709467388994767, 10.97282099454650590207986818185, 11.59342770516434941923286366316

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