Properties

Label 2-273-7.2-c1-0-1
Degree $2$
Conductor $273$
Sign $-0.725 - 0.688i$
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  + (−0.410 + 0.711i)2-s + (−0.5 − 0.866i)3-s + (0.662 + 1.14i)4-s + (−0.910 + 1.57i)5-s + 0.821·6-s + (0.589 + 2.57i)7-s − 2.73·8-s + (−0.499 + 0.866i)9-s + (−0.748 − 1.29i)10-s + (−2.57 − 4.45i)11-s + (0.662 − 1.14i)12-s − 13-s + (−2.07 − 0.640i)14-s + 1.82·15-s + (−0.203 + 0.351i)16-s + (2.43 + 4.22i)17-s + ⋯
L(s)  = 1  + (−0.290 + 0.503i)2-s + (−0.288 − 0.499i)3-s + (0.331 + 0.573i)4-s + (−0.407 + 0.705i)5-s + 0.335·6-s + (0.222 + 0.974i)7-s − 0.965·8-s + (−0.166 + 0.288i)9-s + (−0.236 − 0.409i)10-s + (−0.775 − 1.34i)11-s + (0.191 − 0.331i)12-s − 0.277·13-s + (−0.555 − 0.171i)14-s + 0.470·15-s + (−0.0507 + 0.0879i)16-s + (0.591 + 1.02i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.725 - 0.688i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.725 - 0.688i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.725 - 0.688i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.725 - 0.688i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.287969 + 0.721800i\)
\(L(\frac12)\) \(\approx\) \(0.287969 + 0.721800i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.589 - 2.57i)T \)
13 \( 1 + T \)
good2 \( 1 + (0.410 - 0.711i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (0.910 - 1.57i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.57 + 4.45i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-2.43 - 4.22i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.82 - 6.62i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.11 - 1.93i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 1.82T + 29T^{2} \)
31 \( 1 + (0.865 + 1.49i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.53 + 9.59i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 5.37T + 41T^{2} \)
43 \( 1 - 11.2T + 43T^{2} \)
47 \( 1 + (4.09 - 7.09i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.25 - 5.63i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.18 + 7.25i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.26 + 2.18i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.248 + 0.429i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 2.27T + 71T^{2} \)
73 \( 1 + (-1.68 - 2.92i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.01 + 6.94i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 16.1T + 83T^{2} \)
89 \( 1 + (6.52 - 11.3i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 13.0T + 97T^{2} \)
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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

−12.36448038715067604678630732534, −11.27830464414153673517194645623, −10.70074560125465341663669429970, −9.068559998499333158672359251118, −7.948773817568249879957825668016, −7.75235940450373091593128129108, −6.15042768016344330815635561786, −5.81542304376599674915628679041, −3.61527304182223182837370810943, −2.44389616804380835331718284772, 0.65508489841941272608927571168, 2.56917808053423526114299884730, 4.48113948720563578738879066307, 5.02280740084177571744647895052, 6.64676668954877523019066424630, 7.63281088736469043367834290579, 8.971984996916881599035691186118, 9.927539355697379494915072269085, 10.49090982450438950772144295276, 11.41315913106996954017826817509

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