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Properties

Label 2-273-91.33-c1-0-14
Degree $2$
Conductor $273$
Sign $-0.154 + 0.987i$
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  + (0.306 − 1.14i)2-s − i·3-s + (0.518 + 0.299i)4-s + (−0.424 − 1.58i)5-s + (−1.14 − 0.306i)6-s + (1.65 + 2.06i)7-s + (2.17 − 2.17i)8-s − 9-s − 1.94·10-s + (2.06 − 2.06i)11-s + (0.299 − 0.518i)12-s + (−2.68 − 2.40i)13-s + (2.86 − 1.26i)14-s + (−1.58 + 0.424i)15-s + (−1.22 − 2.11i)16-s + (−0.405 + 0.702i)17-s + ⋯
L(s)  = 1  + (0.216 − 0.808i)2-s − 0.577i·3-s + (0.259 + 0.149i)4-s + (−0.189 − 0.708i)5-s + (−0.466 − 0.125i)6-s + (0.626 + 0.779i)7-s + (0.769 − 0.769i)8-s − 0.333·9-s − 0.613·10-s + (0.621 − 0.621i)11-s + (0.0863 − 0.149i)12-s + (−0.743 − 0.668i)13-s + (0.765 − 0.338i)14-s + (−0.408 + 0.109i)15-s + (−0.305 − 0.529i)16-s + (−0.0983 + 0.170i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.154 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.154 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.154 + 0.987i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (124, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.154 + 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.05749 - 1.23604i\)
\(L(\frac12)\) \(\approx\) \(1.05749 - 1.23604i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + iT \)
7 \( 1 + (-1.65 - 2.06i)T \)
13 \( 1 + (2.68 + 2.40i)T \)
good2 \( 1 + (-0.306 + 1.14i)T + (-1.73 - i)T^{2} \)
5 \( 1 + (0.424 + 1.58i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (-2.06 + 2.06i)T - 11iT^{2} \)
17 \( 1 + (0.405 - 0.702i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.56 - 4.56i)T - 19iT^{2} \)
23 \( 1 + (1.58 - 0.917i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.42 - 4.20i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.78 - 1.01i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-3.95 - 1.06i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-0.392 - 1.46i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (2.14 - 1.23i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.21 - 0.325i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (1.12 + 1.95i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6.46 + 1.73i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + 0.516iT - 61T^{2} \)
67 \( 1 + (-9.11 - 9.11i)T + 67iT^{2} \)
71 \( 1 + (2.21 - 8.27i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (2.64 - 9.87i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (2.17 - 3.76i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-7.44 + 7.44i)T - 83iT^{2} \)
89 \( 1 + (2.18 - 8.16i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (16.5 + 4.44i)T + (84.0 + 48.5i)T^{2} \)
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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

−11.79286263896319963908832717133, −11.05930898966047989353364264289, −9.922102587770723035257482898458, −8.560672800425456596420595889299, −8.001647113691786240410438331248, −6.66494823983755535924821562506, −5.45355009052743865331803168433, −4.13728066395875009515129400447, −2.69020339267525755410053745956, −1.41881288350112374265033035369, 2.26294060357442235605623796440, 4.18269416169133631281487950868, 4.90825623025624136010202474320, 6.45769330818524105475404058807, 7.06340872851658786826999961151, 7.992553108439584806099548741306, 9.344328612768592264340450528925, 10.43189928070427216567531273513, 11.10817604971474299313732252653, 11.86829169031749723831528065779

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