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Properties

Label 2-273-91.45-c1-0-18
Degree $2$
Conductor $273$
Sign $-0.240 + 0.970i$
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  + (1.74 − 1.74i)2-s + (0.866 − 0.5i)3-s − 4.08i·4-s + (0.130 − 0.488i)5-s + (0.638 − 2.38i)6-s + (1.09 + 2.40i)7-s + (−3.63 − 3.63i)8-s + (0.499 − 0.866i)9-s + (−0.623 − 1.08i)10-s + (−1.54 + 5.75i)11-s + (−2.04 − 3.53i)12-s + (−3.50 − 0.833i)13-s + (6.11 + 2.29i)14-s + (−0.130 − 0.488i)15-s − 4.51·16-s + 0.933·17-s + ⋯
L(s)  = 1  + (1.23 − 1.23i)2-s + (0.499 − 0.288i)3-s − 2.04i·4-s + (0.0585 − 0.218i)5-s + (0.260 − 0.972i)6-s + (0.413 + 0.910i)7-s + (−1.28 − 1.28i)8-s + (0.166 − 0.288i)9-s + (−0.197 − 0.341i)10-s + (−0.464 + 1.73i)11-s + (−0.589 − 1.02i)12-s + (−0.972 − 0.231i)13-s + (1.63 + 0.613i)14-s + (−0.0337 − 0.126i)15-s − 1.12·16-s + 0.226·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.240 + 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.240 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.240 + 0.970i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (136, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.240 + 0.970i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.61472 - 2.06312i\)
\(L(\frac12)\) \(\approx\) \(1.61472 - 2.06312i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.866 + 0.5i)T \)
7 \( 1 + (-1.09 - 2.40i)T \)
13 \( 1 + (3.50 + 0.833i)T \)
good2 \( 1 + (-1.74 + 1.74i)T - 2iT^{2} \)
5 \( 1 + (-0.130 + 0.488i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (1.54 - 5.75i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 - 0.933T + 17T^{2} \)
19 \( 1 + (7.66 - 2.05i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + 8.12iT - 23T^{2} \)
29 \( 1 + (-1.96 + 3.40i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.37 - 0.636i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (-0.859 - 0.859i)T + 37iT^{2} \)
41 \( 1 + (-7.84 + 2.10i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (0.152 - 0.0881i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.65 - 0.444i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-0.750 + 1.30i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.03 - 3.03i)T - 59iT^{2} \)
61 \( 1 + (6.74 + 3.89i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.37 + 1.97i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-6.54 - 1.75i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (3.27 + 12.2i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-4.64 - 8.03i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.66 - 1.66i)T + 83iT^{2} \)
89 \( 1 + (-3.02 + 3.02i)T - 89iT^{2} \)
97 \( 1 + (0.856 - 3.19i)T + (-84.0 - 48.5i)T^{2} \)
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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

−12.27329083285209411522204308282, −10.79888534714445479586509090745, −10.06958849587780270334993204554, −9.012408988518330521939101365284, −7.75559680255824639332437177643, −6.31253997271405055370129686431, −4.96861063931510702215245314954, −4.37234137437714351853570846022, −2.59523392982352595993081923159, −2.03411889615992003078403330498, 2.96765826466375887194223262718, 4.04947795358044632093590619363, 5.01683552888087636399535798911, 6.11799466443866347426318574214, 7.20529542245460047665612217243, 7.956822076413556367354376270147, 8.896904732847407968689786498112, 10.45536601544084226938690893870, 11.29295506393415527239118023856, 12.68457875043428256798394546252

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