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Properties

Label 2-273-273.185-c1-0-10
Degree $2$
Conductor $273$
Sign $0.0699 + 0.997i$
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  + (−0.913 − 0.527i)2-s + (−1.53 + 0.808i)3-s + (−0.443 − 0.768i)4-s + (−0.000964 − 0.00167i)5-s + (1.82 + 0.0692i)6-s + (1.27 + 2.31i)7-s + 3.04i·8-s + (1.69 − 2.47i)9-s + 0.00203i·10-s − 2.49i·11-s + (1.30 + 0.818i)12-s + (−0.922 − 3.48i)13-s + (0.0616 − 2.78i)14-s + (0.00282 + 0.00177i)15-s + (0.717 − 1.24i)16-s + (−2.08 − 3.61i)17-s + ⋯
L(s)  = 1  + (−0.645 − 0.372i)2-s + (−0.884 + 0.466i)3-s + (−0.221 − 0.384i)4-s + (−0.000431 − 0.000747i)5-s + (0.745 + 0.0282i)6-s + (0.480 + 0.876i)7-s + 1.07i·8-s + (0.564 − 0.825i)9-s + 0.000643i·10-s − 0.752i·11-s + (0.375 + 0.236i)12-s + (−0.255 − 0.966i)13-s + (0.0164 − 0.745i)14-s + (0.000730 + 0.000459i)15-s + (0.179 − 0.310i)16-s + (−0.506 − 0.876i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0699 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0699 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.0699 + 0.997i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (185, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.0699 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.408127 - 0.380510i\)
\(L(\frac12)\) \(\approx\) \(0.408127 - 0.380510i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.53 - 0.808i)T \)
7 \( 1 + (-1.27 - 2.31i)T \)
13 \( 1 + (0.922 + 3.48i)T \)
good2 \( 1 + (0.913 + 0.527i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (0.000964 + 0.00167i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + 2.49iT - 11T^{2} \)
17 \( 1 + (2.08 + 3.61i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + 5.36iT - 19T^{2} \)
23 \( 1 + (-4.21 - 2.43i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-7.57 + 4.37i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (6.57 + 3.79i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.82 - 3.16i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.83 - 3.17i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.32 + 2.30i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.79 - 3.10i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (12.3 + 7.14i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.61 - 4.53i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + 7.57iT - 61T^{2} \)
67 \( 1 - 14.2T + 67T^{2} \)
71 \( 1 + (1.24 + 0.716i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-1.61 - 0.929i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.64 - 6.31i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 10.4T + 83T^{2} \)
89 \( 1 + (-2.70 + 4.67i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.07 + 1.19i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.20413847316022970798648047864, −10.99264141083584638159295869018, −9.760064614368240254357379649876, −9.087977874450718065085592085123, −8.121684155932047453061084646979, −6.51288440895331140023272836381, −5.37483374222330328816233383663, −4.80862883145424210091067134733, −2.71415038968547202156504200414, −0.65123597112070410503291738553, 1.48279006863789207919025445986, 3.98172731398836371655865398538, 4.96258012413120348058663970546, 6.64597287703719375887182661368, 7.15360559286636061455601121536, 8.083887496980624745323231278018, 9.179576421449984887769952061923, 10.35743320551162614732659792374, 10.99702435182621594256453455121, 12.35025244615818099821661141076

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