Properties

Label 2-273-13.9-c1-0-8
Degree $2$
Conductor $273$
Sign $0.5 + 0.866i$
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  + (0.636 − 1.10i)2-s + (0.5 − 0.866i)3-s + (0.188 + 0.326i)4-s + 1.10·5-s + (−0.636 − 1.10i)6-s + (0.5 + 0.866i)7-s + 3.02·8-s + (−0.499 − 0.866i)9-s + (0.702 − 1.21i)10-s + (0.174 − 0.302i)11-s + 0.377·12-s + (−3.47 + 0.955i)13-s + 1.27·14-s + (0.551 − 0.955i)15-s + (1.55 − 2.68i)16-s + (−0.363 − 0.628i)17-s + ⋯
L(s)  = 1  + (0.450 − 0.780i)2-s + (0.288 − 0.499i)3-s + (0.0943 + 0.163i)4-s + 0.493·5-s + (−0.260 − 0.450i)6-s + (0.188 + 0.327i)7-s + 1.07·8-s + (−0.166 − 0.288i)9-s + (0.222 − 0.384i)10-s + (0.0525 − 0.0911i)11-s + 0.108·12-s + (−0.964 + 0.265i)13-s + 0.340·14-s + (0.142 − 0.246i)15-s + (0.387 − 0.671i)16-s + (−0.0880 − 0.152i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.5 + 0.866i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.5 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.5 + 0.866i$
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: \((2,\ 273,\ (\ :1/2),\ 0.5 + 0.866i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.71259 - 0.988768i\)
\(L(\frac12)\) \(\approx\) \(1.71259 - 0.988768i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.47 - 0.955i)T \)
good2 \( 1 + (-0.636 + 1.10i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 - 1.10T + 5T^{2} \)
11 \( 1 + (-0.174 + 0.302i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (0.363 + 0.628i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.15 + 1.99i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.0375 - 0.0649i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.240 - 0.416i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 6.85T + 31T^{2} \)
37 \( 1 + (-0.551 + 0.955i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.54 - 2.68i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.31 - 4.00i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 4.70T + 47T^{2} \)
53 \( 1 - 5.54T + 53T^{2} \)
59 \( 1 + (-1.96 - 3.39i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.77 + 4.80i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.06 - 7.03i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-4.76 - 8.25i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 10.4T + 73T^{2} \)
79 \( 1 - 17.1T + 79T^{2} \)
83 \( 1 + 11.9T + 83T^{2} \)
89 \( 1 + (-5.98 + 10.3i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.05 + 8.74i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.86643902846197667771481075133, −11.08881145907612941286132281497, −9.963270308964125646430202017787, −8.970955603047209334858397799928, −7.76894267451525893037169687176, −6.92004775534024443588714569134, −5.52257422488013204477100648040, −4.23993664862428089946512540646, −2.80810832438042863005868921978, −1.87450293777002994741027986699, 2.07348242146880424158847942121, 3.94604048598894512256666093342, 5.06811462902136490464263870260, 5.90183285358460665847405354335, 7.10904625384817948049368263650, 7.940220817347427605980481855576, 9.284681114837419404278266080908, 10.19859433030528479946308840367, 10.84378230548680461330973341941, 12.18028181935897057202222427052

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