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Properties

Label 2-273-91.81-c1-0-16
Degree $2$
Conductor $273$
Sign $-0.467 + 0.884i$
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  + (1.35 − 2.35i)2-s + 3-s + (−2.68 − 4.65i)4-s + (1.94 + 3.36i)5-s + (1.35 − 2.35i)6-s + (0.587 − 2.57i)7-s − 9.16·8-s + 9-s + 10.5·10-s − 1.63·11-s + (−2.68 − 4.65i)12-s + (−3.59 + 0.202i)13-s + (−5.26 − 4.88i)14-s + (1.94 + 3.36i)15-s + (−7.06 + 12.2i)16-s + (2.09 + 3.63i)17-s + ⋯
L(s)  = 1  + (0.960 − 1.66i)2-s + 0.577·3-s + (−1.34 − 2.32i)4-s + (0.869 + 1.50i)5-s + (0.554 − 0.960i)6-s + (0.221 − 0.975i)7-s − 3.23·8-s + 0.333·9-s + 3.33·10-s − 0.491·11-s + (−0.775 − 1.34i)12-s + (−0.998 + 0.0562i)13-s + (−1.40 − 1.30i)14-s + (0.501 + 0.869i)15-s + (−1.76 + 3.05i)16-s + (0.508 + 0.880i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.467 + 0.884i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.467 + 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.467 + 0.884i$
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: \((2,\ 273,\ (\ :1/2),\ -0.467 + 0.884i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.22179 - 2.02774i\)
\(L(\frac12)\) \(\approx\) \(1.22179 - 2.02774i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.587 + 2.57i)T \)
13 \( 1 + (3.59 - 0.202i)T \)
good2 \( 1 + (-1.35 + 2.35i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (-1.94 - 3.36i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + 1.63T + 11T^{2} \)
17 \( 1 + (-2.09 - 3.63i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 - 1.69T + 19T^{2} \)
23 \( 1 + (0.395 - 0.685i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.242 + 0.419i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.915 - 1.58i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.344 - 0.596i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.96 - 5.14i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.79 + 4.83i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.292 + 0.506i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.04 + 5.28i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.13 + 7.16i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + 9.08T + 61T^{2} \)
67 \( 1 - 1.00T + 67T^{2} \)
71 \( 1 + (-7.93 + 13.7i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (2.92 - 5.06i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.643 - 1.11i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 6.18T + 83T^{2} \)
89 \( 1 + (-2.20 + 3.81i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5.08 + 8.81i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.46512591867982449722739708829, −10.53293981840280292886633379628, −10.21554684014014106923290272684, −9.434069032865662389713587898909, −7.59934297719779133503924016895, −6.33805424833727083133541641290, −5.09053288476536895500970416380, −3.74745726056237636322126896234, −2.87080539323744406561502597833, −1.83587245275733258189936941279, 2.67413685104353343404128901411, 4.50901684783034205920764220299, 5.25323736161498325572429240065, 5.85843179189135299895945690132, 7.36724996373362935893197626507, 8.184077203168998604290513607797, 9.058912337905726375807854226652, 9.586692284425390851083826903413, 12.10059380085386625976592495862, 12.52305171949881466981957092117

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