Properties

Label 2-273-273.38-c1-0-16
Degree $2$
Conductor $273$
Sign $0.983 - 0.178i$
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  + (1.5 + 0.866i)3-s + (1 − 1.73i)4-s + (−0.5 + 2.59i)7-s + (1.5 + 2.59i)9-s + (3 − 1.73i)12-s + (3.5 − 0.866i)13-s + (−1.99 − 3.46i)16-s + (−3.5 − 6.06i)19-s + (−3 + 3.46i)21-s + (−2.5 + 4.33i)25-s + 5.19i·27-s + (4 + 3.46i)28-s + (3.5 − 6.06i)31-s + 6·36-s + (−10.5 + 6.06i)37-s + ⋯
L(s)  = 1  + (0.866 + 0.499i)3-s + (0.5 − 0.866i)4-s + (−0.188 + 0.981i)7-s + (0.5 + 0.866i)9-s + (0.866 − 0.499i)12-s + (0.970 − 0.240i)13-s + (−0.499 − 0.866i)16-s + (−0.802 − 1.39i)19-s + (−0.654 + 0.755i)21-s + (−0.5 + 0.866i)25-s + 0.999i·27-s + (0.755 + 0.654i)28-s + (0.628 − 1.08i)31-s + 36-s + (−1.72 + 0.996i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 - 0.178i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.983 - 0.178i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.983 - 0.178i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (38, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.983 - 0.178i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.79610 + 0.161353i\)
\(L(\frac12)\) \(\approx\) \(1.79610 + 0.161353i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.5 - 0.866i)T \)
7 \( 1 + (0.5 - 2.59i)T \)
13 \( 1 + (-3.5 + 0.866i)T \)
good2 \( 1 + (-1 + 1.73i)T^{2} \)
5 \( 1 + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.5 + 6.06i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + (-3.5 + 6.06i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (10.5 - 6.06i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 5T + 43T^{2} \)
47 \( 1 + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (6 - 3.46i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-10.5 - 6.06i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (-3.5 + 6.06i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.5 + 11.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 14T + 97T^{2} \)
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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

−11.69669932473786528121947326696, −10.89788181295720255007777603854, −9.959398003452045865289801423866, −9.097450989864690586399106326145, −8.349732452444268304290543035431, −6.93883192950769298150807432880, −5.85499557552074227026409362878, −4.77241622397165056600171208375, −3.17114009101357850970073437250, −1.97989025242404553687679101681, 1.80058772273606642429886052316, 3.38298700081458132887613400183, 4.07425742122661719516366678293, 6.32528579230904434403694926178, 7.04112578121127861852040703844, 8.087841502792574714796864995698, 8.608829076573793060916093381791, 10.01809902762986195908210035193, 10.92550329199480167008489690258, 12.17086717762798092021319711217

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