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Properties

Label 2-273-91.30-c1-0-3
Degree $2$
Conductor $273$
Sign $-0.313 - 0.949i$
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  + (−0.395 + 0.228i)2-s + 3-s + (−0.895 + 1.55i)4-s + (−1.5 − 0.866i)5-s + (−0.395 + 0.228i)6-s + 2.64i·7-s − 1.73i·8-s + 9-s + 0.791·10-s + 3.46i·11-s + (−0.895 + 1.55i)12-s + (−1 + 3.46i)13-s + (−0.604 − 1.04i)14-s + (−1.5 − 0.866i)15-s + (−1.39 − 2.41i)16-s + (−0.5 + 0.866i)17-s + ⋯
L(s)  = 1  + (−0.279 + 0.161i)2-s + 0.577·3-s + (−0.447 + 0.775i)4-s + (−0.670 − 0.387i)5-s + (−0.161 + 0.0932i)6-s + 0.999i·7-s − 0.612i·8-s + 0.333·9-s + 0.250·10-s + 1.04i·11-s + (−0.258 + 0.447i)12-s + (−0.277 + 0.960i)13-s + (−0.161 − 0.279i)14-s + (−0.387 − 0.223i)15-s + (−0.348 − 0.604i)16-s + (−0.121 + 0.210i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.313 - 0.949i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.313 - 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.313 - 0.949i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.313 - 0.949i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.544920 + 0.753660i\)
\(L(\frac12)\) \(\approx\) \(0.544920 + 0.753660i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
7 \( 1 - 2.64iT \)
13 \( 1 + (1 - 3.46i)T \)
good2 \( 1 + (0.395 - 0.228i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + (1.5 + 0.866i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 - 3.46iT - 11T^{2} \)
17 \( 1 + (0.5 - 0.866i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 - 5.29iT - 19T^{2} \)
23 \( 1 + (0.291 + 0.504i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.5 + 6.06i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.708 - 0.409i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.08 + 1.77i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-6.08 - 3.51i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.29 - 3.96i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-5.29 - 3.05i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (6.08 + 10.5i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (8.29 + 4.78i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 - 13.1T + 61T^{2} \)
67 \( 1 - 7.11iT - 67T^{2} \)
71 \( 1 + (-9.87 + 5.70i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-7.5 + 4.33i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.29 + 9.16i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 3.46iT - 83T^{2} \)
89 \( 1 + (13.5 - 7.79i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.0825 - 0.0476i)T + (48.5 - 84.0i)T^{2} \)
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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

−12.34849049947856212182169534787, −11.57787410021481488917627015908, −9.818390540351721501623747849826, −9.254025542471679652760007981909, −8.212333687374287052917123440502, −7.75549703156030026352355146874, −6.44464104865362991986239792935, −4.68755313684334439612235489684, −3.88771264048786666034809925860, −2.30657838384625996078069108282, 0.76249376074704175307859539257, 2.93837800080755273840951834984, 4.15888800838187748054410391102, 5.42127565260545621576359315806, 6.87913316312645328137726413251, 7.84080404730479738269794113401, 8.774886395774201195497950970186, 9.736856686241309379605988488294, 10.82495371950902107498224065973, 11.09221195693291006099442353488

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