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Properties

Label 2-273-91.51-c1-0-17
Degree $2$
Conductor $273$
Sign $-0.994 - 0.101i$
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  + (0.161 + 0.0930i)2-s + (−0.5 − 0.866i)3-s + (−0.982 − 1.70i)4-s + (−2.28 − 1.31i)5-s − 0.186i·6-s + (0.161 + 2.64i)7-s − 0.738i·8-s + (−0.499 + 0.866i)9-s + (−0.245 − 0.425i)10-s + (−1.56 + 0.904i)11-s + (−0.982 + 1.70i)12-s + (−2.45 + 2.63i)13-s + (−0.219 + 0.440i)14-s + 2.63i·15-s + (−1.89 + 3.28i)16-s + (−3.13 − 5.42i)17-s + ⋯
L(s)  = 1  + (0.113 + 0.0658i)2-s + (−0.288 − 0.499i)3-s + (−0.491 − 0.851i)4-s + (−1.02 − 0.590i)5-s − 0.0759i·6-s + (0.0609 + 0.998i)7-s − 0.260i·8-s + (−0.166 + 0.288i)9-s + (−0.0776 − 0.134i)10-s + (−0.472 + 0.272i)11-s + (−0.283 + 0.491i)12-s + (−0.681 + 0.731i)13-s + (−0.0587 + 0.117i)14-s + 0.681i·15-s + (−0.474 + 0.821i)16-s + (−0.760 − 1.31i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 - 0.101i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 - 0.101i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.994 - 0.101i$
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: \((2,\ 273,\ (\ :1/2),\ -0.994 - 0.101i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0151703 + 0.298256i\)
\(L(\frac12)\) \(\approx\) \(0.0151703 + 0.298256i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.161 - 2.64i)T \)
13 \( 1 + (2.45 - 2.63i)T \)
good2 \( 1 + (-0.161 - 0.0930i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (2.28 + 1.31i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.56 - 0.904i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (3.13 + 5.42i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.44 + 1.41i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.12 + 3.68i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 0.456T + 29T^{2} \)
31 \( 1 + (-2.76 + 1.59i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.00 + 1.73i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 7.18iT - 41T^{2} \)
43 \( 1 - 7.23T + 43T^{2} \)
47 \( 1 + (10.3 + 5.95i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.23 - 3.87i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (8.83 - 5.09i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.59 - 7.95i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.40 - 0.810i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 1.85iT - 71T^{2} \)
73 \( 1 + (-12.3 + 7.15i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.86 + 10.1i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 11.7iT - 83T^{2} \)
89 \( 1 + (0.0791 + 0.0457i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 1.62iT - 97T^{2} \)
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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

−11.65725742141290686420162601519, −10.59937563150778394200965801332, −9.248754321659854059421290380425, −8.695876703629953123179387993140, −7.43517409898615306884637938185, −6.36761818870154946593675498890, −5.05982502072571631395988108568, −4.52027136567430374991273778869, −2.31054175877019514651991694868, −0.22163788301526042071666887973, 3.16392909525715900287270507496, 3.96868132989826232850218258348, 4.90956532580970360417078319182, 6.60820397104374629022543287485, 7.77271458959824161315704207057, 8.222453744395989572047177681238, 9.688372205731021124120054272233, 10.77403636021559299893408787719, 11.23736484546747680214167426007, 12.43705358685005767849642377094

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