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Properties

Label 2-273-91.51-c1-0-14
Degree $2$
Conductor $273$
Sign $0.633 + 0.773i$
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  + (1.34 + 0.779i)2-s + (−0.5 − 0.866i)3-s + (0.214 + 0.371i)4-s + (−1.85 − 1.06i)5-s − 1.55i·6-s + (1.34 − 2.27i)7-s − 2.44i·8-s + (−0.499 + 0.866i)9-s + (−1.66 − 2.88i)10-s + (4.76 − 2.75i)11-s + (0.214 − 0.371i)12-s + (−2.90 + 2.13i)13-s + (3.59 − 2.01i)14-s + 2.13i·15-s + (2.33 − 4.04i)16-s + (3.71 + 6.43i)17-s + ⋯
L(s)  = 1  + (0.954 + 0.550i)2-s + (−0.288 − 0.499i)3-s + (0.107 + 0.185i)4-s + (−0.828 − 0.478i)5-s − 0.636i·6-s + (0.510 − 0.860i)7-s − 0.865i·8-s + (−0.166 + 0.288i)9-s + (−0.526 − 0.912i)10-s + (1.43 − 0.829i)11-s + (0.0618 − 0.107i)12-s + (−0.805 + 0.592i)13-s + (0.960 − 0.539i)14-s + 0.552i·15-s + (0.584 − 1.01i)16-s + (0.901 + 1.56i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.633 + 0.773i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.633 + 0.773i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.633 + 0.773i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (142, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.633 + 0.773i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.49223 - 0.706523i\)
\(L(\frac12)\) \(\approx\) \(1.49223 - 0.706523i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.34 + 2.27i)T \)
13 \( 1 + (2.90 - 2.13i)T \)
good2 \( 1 + (-1.34 - 0.779i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (1.85 + 1.06i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-4.76 + 2.75i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-3.71 - 6.43i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.20 + 1.84i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.88 - 3.26i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 0.903T + 29T^{2} \)
31 \( 1 + (-2.62 + 1.51i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.824 + 0.476i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 4.86iT - 41T^{2} \)
43 \( 1 - 4.89T + 43T^{2} \)
47 \( 1 + (-4.55 - 2.63i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.49 - 11.2i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-7.30 + 4.21i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.81 + 3.14i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.52 - 0.882i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 11.7iT - 71T^{2} \)
73 \( 1 + (-6.43 + 3.71i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (7.64 - 13.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 10.4iT - 83T^{2} \)
89 \( 1 + (4.49 + 2.59i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 7.06iT - 97T^{2} \)
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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

−12.02543576121497409573858004965, −11.18180751025707066709673225561, −9.950562390533016824465943660158, −8.584819164399659897125275596819, −7.61269178582735606646684897020, −6.65908921087378207239509803213, −5.76188065701536529356342445784, −4.35906599339321936410893089119, −3.89221991645026908521545630920, −1.11902489759947870406925967392, 2.47634895917672510493592677971, 3.71813756802668303182377438433, 4.63869501291783598126423537445, 5.55496627370897911279801282252, 7.04078821232186785920624131197, 8.180972500754764448006264589501, 9.305973678692144903608678799330, 10.43087030891869382253561800467, 11.59483613854148637837315251292, 11.98318825500307667351999982892

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