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Properties

Label 2-273-13.9-c1-0-0
Degree $2$
Conductor $273$
Sign $-0.833 + 0.552i$
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  + (−0.400 + 0.694i)2-s + (−0.5 + 0.866i)3-s + (0.678 + 1.17i)4-s − 3.04·5-s + (−0.400 − 0.694i)6-s + (−0.5 − 0.866i)7-s − 2.69·8-s + (−0.499 − 0.866i)9-s + (1.22 − 2.11i)10-s + (−1.07 + 1.86i)11-s − 1.35·12-s + (−1.16 − 3.41i)13-s + 0.801·14-s + (1.52 − 2.64i)15-s + (−0.277 + 0.480i)16-s + (−0.400 − 0.694i)17-s + ⋯
L(s)  = 1  + (−0.283 + 0.491i)2-s + (−0.288 + 0.499i)3-s + (0.339 + 0.587i)4-s − 1.36·5-s + (−0.163 − 0.283i)6-s + (−0.188 − 0.327i)7-s − 0.951·8-s + (−0.166 − 0.288i)9-s + (0.386 − 0.669i)10-s + (−0.325 + 0.563i)11-s − 0.391·12-s + (−0.323 − 0.946i)13-s + 0.214·14-s + (0.393 − 0.681i)15-s + (−0.0693 + 0.120i)16-s + (−0.0972 − 0.168i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.833 + 0.552i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.833 + 0.552i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.833 + 0.552i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.833 + 0.552i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0850884 - 0.282477i\)
\(L(\frac12)\) \(\approx\) \(0.0850884 - 0.282477i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (1.16 + 3.41i)T \)
good2 \( 1 + (0.400 - 0.694i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + 3.04T + 5T^{2} \)
11 \( 1 + (1.07 - 1.86i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (0.400 + 0.694i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.698 - 1.20i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.51 - 2.62i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.23 - 2.14i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 6.26T + 31T^{2} \)
37 \( 1 + (5.37 - 9.31i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.60 + 4.51i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.31 - 7.47i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 9.15T + 47T^{2} \)
53 \( 1 - 1.21T + 53T^{2} \)
59 \( 1 + (-4.31 - 7.47i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.04 - 8.73i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.39 + 11.0i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-7.06 - 12.2i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 10.1T + 73T^{2} \)
79 \( 1 - 1.33T + 79T^{2} \)
83 \( 1 - 8.92T + 83T^{2} \)
89 \( 1 + (-1.25 + 2.17i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.16 + 2.01i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.26676757659650119029896363102, −11.57664963669789372783854179633, −10.66448512985586644043215759147, −9.568276919789339437309833628544, −8.320202906480998064202562243113, −7.63687728655999498310767920654, −6.87957905467071308981057224390, −5.40196261826481122380228474296, −4.04940160906299348713785525590, −3.11595964653703821154933835006, 0.24029030538118187778069575354, 2.19543889338390468228331374061, 3.71262508754525499584550170812, 5.26049245319355174005588041605, 6.45266526660306856196917031163, 7.38116576812073827756351088148, 8.490758939951911546100297747890, 9.459791707815984653704961559727, 10.76977124458491603257235500561, 11.35195945808289658362856728402

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