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Properties

Label 2-273-91.81-c1-0-7
Degree $2$
Conductor $273$
Sign $0.982 - 0.188i$
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  + (−1.02 + 1.77i)2-s − 3-s + (−1.11 − 1.92i)4-s + (−0.274 − 0.475i)5-s + (1.02 − 1.77i)6-s + (−0.839 − 2.50i)7-s + 0.456·8-s + 9-s + 1.12·10-s + 4.69·11-s + (1.11 + 1.92i)12-s + (−0.663 − 3.54i)13-s + (5.32 + 1.08i)14-s + (0.274 + 0.475i)15-s + (1.75 − 3.03i)16-s + (0.301 + 0.522i)17-s + ⋯
L(s)  = 1  + (−0.726 + 1.25i)2-s − 0.577·3-s + (−0.555 − 0.962i)4-s + (−0.122 − 0.212i)5-s + (0.419 − 0.726i)6-s + (−0.317 − 0.948i)7-s + 0.161·8-s + 0.333·9-s + 0.356·10-s + 1.41·11-s + (0.320 + 0.555i)12-s + (−0.184 − 0.982i)13-s + (1.42 + 0.289i)14-s + (0.0709 + 0.122i)15-s + (0.438 − 0.759i)16-s + (0.0731 + 0.126i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 - 0.188i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.982 - 0.188i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.982 - 0.188i$
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: \((2,\ 273,\ (\ :1/2),\ 0.982 - 0.188i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.630908 + 0.0599350i\)
\(L(\frac12)\) \(\approx\) \(0.630908 + 0.0599350i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.839 + 2.50i)T \)
13 \( 1 + (0.663 + 3.54i)T \)
good2 \( 1 + (1.02 - 1.77i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (0.274 + 0.475i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 - 4.69T + 11T^{2} \)
17 \( 1 + (-0.301 - 0.522i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 - 0.561T + 19T^{2} \)
23 \( 1 + (0.188 - 0.326i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.09 - 3.62i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.577 - 0.999i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.40 + 7.62i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.96 + 6.85i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.747 - 1.29i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.09 + 1.90i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.52 + 7.83i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.26 + 7.39i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 - 7.42T + 61T^{2} \)
67 \( 1 + 9.59T + 67T^{2} \)
71 \( 1 + (-2.88 + 5.00i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-7.24 + 12.5i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-7.31 - 12.6i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 14.8T + 83T^{2} \)
89 \( 1 + (4.59 - 7.95i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.15 + 5.45i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.96407603723368187825821301562, −10.72719505666332041095900261006, −9.856204368463384322600773462028, −8.920620071784883412365309138982, −7.88172306793454553213183837920, −6.95508806643046807821062647616, −6.30658164792881762200545796294, −5.11736532542936516305146223307, −3.69978478796663003607038457666, −0.73737886392543369032912679653, 1.48506124017937655550754541122, 2.94292541003811237086543067578, 4.30520168099055483826575797397, 5.95672201312471905339648487846, 6.86129408497214886545980836989, 8.501872310964955824792969009905, 9.352571753308176791549635884622, 9.904861911343672001946650164626, 11.18911784123837814001879081322, 11.73884120099688853812059414302

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