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Properties

Label 2-273-13.3-c1-0-8
Degree $2$
Conductor $273$
Sign $0.798 + 0.601i$
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  + (0.277 + 0.480i)2-s + (−0.5 − 0.866i)3-s + (0.846 − 1.46i)4-s + 1.35·5-s + (0.277 − 0.480i)6-s + (−0.5 + 0.866i)7-s + 2.04·8-s + (−0.499 + 0.866i)9-s + (0.376 + 0.652i)10-s + (−0.568 − 0.984i)11-s − 1.69·12-s + (1.37 − 3.33i)13-s − 0.554·14-s + (−0.678 − 1.17i)15-s + (−1.12 − 1.94i)16-s + (0.277 − 0.480i)17-s + ⋯
L(s)  = 1  + (0.196 + 0.339i)2-s + (−0.288 − 0.499i)3-s + (0.423 − 0.732i)4-s + 0.606·5-s + (0.113 − 0.196i)6-s + (−0.188 + 0.327i)7-s + 0.724·8-s + (−0.166 + 0.288i)9-s + (0.119 + 0.206i)10-s + (−0.171 − 0.296i)11-s − 0.488·12-s + (0.380 − 0.924i)13-s − 0.148·14-s + (−0.175 − 0.303i)15-s + (−0.280 − 0.486i)16-s + (0.0672 − 0.116i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.798 + 0.601i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.798 + 0.601i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.798 + 0.601i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.798 + 0.601i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.45854 - 0.487632i\)
\(L(\frac12)\) \(\approx\) \(1.45854 - 0.487632i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.37 + 3.33i)T \)
good2 \( 1 + (-0.277 - 0.480i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 - 1.35T + 5T^{2} \)
11 \( 1 + (0.568 + 0.984i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-0.277 + 0.480i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.05 + 3.55i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.39 - 5.88i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.51 - 7.82i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 8.63T + 31T^{2} \)
37 \( 1 + (-2.59 - 4.48i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.109 + 0.190i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.94 - 3.37i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 8.13T + 47T^{2} \)
53 \( 1 - 7.98T + 53T^{2} \)
59 \( 1 + (1.94 - 3.37i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.46 - 9.45i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.381 + 0.660i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (2.25 - 3.90i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 8.50T + 73T^{2} \)
79 \( 1 + 3.74T + 79T^{2} \)
83 \( 1 + 11.0T + 83T^{2} \)
89 \( 1 + (-4.30 - 7.45i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-7.47 + 12.9i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.68744744283685985769895757615, −10.91929240209832384427343611909, −10.01504824573184754782145670110, −8.987418820707281528716984392698, −7.62308068314782686095522675659, −6.71686136603606349520195912969, −5.69655368273857886351559213938, −5.17891205848106372091672582490, −2.96853371462764835439504628630, −1.37850878789793898381021285109, 2.06904521135131527387069625392, 3.56761603230707365875578860635, 4.55686149886469228788887581623, 6.00241497680514940489817306498, 6.98702754423067842029219936219, 8.147885658048367704975605338496, 9.347441016278754790971136151514, 10.25066908134901782601220248217, 11.09334266901204494466501771394, 11.94518292869754914340264152889

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