Properties

Label 2-273-91.23-c1-0-6
Degree $2$
Conductor $273$
Sign $-0.391 - 0.920i$
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  + 2.44i·2-s + (0.5 − 0.866i)3-s − 3.97·4-s + (3.36 + 1.94i)5-s + (2.11 + 1.22i)6-s + (1.82 − 1.91i)7-s − 4.82i·8-s + (−0.499 − 0.866i)9-s + (−4.75 + 8.23i)10-s + (3.91 + 2.26i)11-s + (−1.98 + 3.44i)12-s + (−2.00 − 2.99i)13-s + (4.68 + 4.46i)14-s + (3.36 − 1.94i)15-s + 3.84·16-s − 6.32·17-s + ⋯
L(s)  = 1  + 1.72i·2-s + (0.288 − 0.499i)3-s − 1.98·4-s + (1.50 + 0.869i)5-s + (0.864 + 0.498i)6-s + (0.689 − 0.723i)7-s − 1.70i·8-s + (−0.166 − 0.288i)9-s + (−1.50 + 2.60i)10-s + (1.18 + 0.681i)11-s + (−0.573 + 0.993i)12-s + (−0.555 − 0.831i)13-s + (1.25 + 1.19i)14-s + (0.869 − 0.502i)15-s + 0.960·16-s − 1.53·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.391 - 0.920i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.391 - 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.391 - 0.920i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (205, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.391 - 0.920i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.894152 + 1.35201i\)
\(L(\frac12)\) \(\approx\) \(0.894152 + 1.35201i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.82 + 1.91i)T \)
13 \( 1 + (2.00 + 2.99i)T \)
good2 \( 1 - 2.44iT - 2T^{2} \)
5 \( 1 + (-3.36 - 1.94i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-3.91 - 2.26i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + 6.32T + 17T^{2} \)
19 \( 1 + (1.68 - 0.971i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 2.66T + 23T^{2} \)
29 \( 1 + (1.99 + 3.45i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.77 - 1.02i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 3.00iT - 37T^{2} \)
41 \( 1 + (-4.70 + 2.71i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.41 + 2.45i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.60 + 0.926i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.173 + 0.300i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 6.06iT - 59T^{2} \)
61 \( 1 + (5.49 + 9.50i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.214 - 0.123i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-9.28 - 5.36i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-6.89 + 3.98i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.35 - 11.0i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 0.782iT - 83T^{2} \)
89 \( 1 - 12.3iT - 89T^{2} \)
97 \( 1 + (11.4 + 6.59i)T + (48.5 + 84.0i)T^{2} \)
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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

−12.70964057334819946154293317375, −11.06771859338249851898053510780, −9.925292468722388482374814432885, −9.142666748352949412490603267139, −8.013552099354735354933811611846, −6.99094388777731238979722874421, −6.56837034847034652111065694424, −5.55019013524959180880299104506, −4.25785755053486736987938311886, −2.06676193994336237756742339771, 1.65300450097064279518647530931, 2.41465269933977861379325379294, 4.17780597623246074069005746448, 4.97243575335375806369882268770, 6.22989281001793930404523286309, 8.662248877389155824570924626188, 9.125430090588972763606791493474, 9.560334207084747726149706608476, 10.79517502535713435154408311570, 11.49318606203807911269814861644

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