Properties

Label 2-273-91.41-c1-0-0
Degree $2$
Conductor $273$
Sign $-0.258 - 0.966i$
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  + (−2.11 − 0.567i)2-s + (0.866 − 0.5i)3-s + (2.43 + 1.40i)4-s + (−3.00 − 3.00i)5-s + (−2.11 + 0.567i)6-s + (−2.60 + 0.481i)7-s + (−1.26 − 1.26i)8-s + (0.499 − 0.866i)9-s + (4.65 + 8.06i)10-s + (−0.698 + 2.60i)11-s + 2.81·12-s + (0.373 + 3.58i)13-s + (5.78 + 0.455i)14-s + (−4.10 − 1.09i)15-s + (−0.857 − 1.48i)16-s + (0.599 − 1.03i)17-s + ⋯
L(s)  = 1  + (−1.49 − 0.401i)2-s + (0.499 − 0.288i)3-s + (1.21 + 0.703i)4-s + (−1.34 − 1.34i)5-s + (−0.865 + 0.231i)6-s + (−0.983 + 0.182i)7-s + (−0.445 − 0.445i)8-s + (0.166 − 0.288i)9-s + (1.47 + 2.55i)10-s + (−0.210 + 0.785i)11-s + 0.811·12-s + (0.103 + 0.994i)13-s + (1.54 + 0.121i)14-s + (−1.05 − 0.283i)15-s + (−0.214 − 0.371i)16-s + (0.145 − 0.251i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 - 0.966i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.258 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.258 - 0.966i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.258 - 0.966i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0335413 + 0.0436885i\)
\(L(\frac12)\) \(\approx\) \(0.0335413 + 0.0436885i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.866 + 0.5i)T \)
7 \( 1 + (2.60 - 0.481i)T \)
13 \( 1 + (-0.373 - 3.58i)T \)
good2 \( 1 + (2.11 + 0.567i)T + (1.73 + i)T^{2} \)
5 \( 1 + (3.00 + 3.00i)T + 5iT^{2} \)
11 \( 1 + (0.698 - 2.60i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (-0.599 + 1.03i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.89 + 0.507i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (4.65 - 2.68i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.47 - 2.56i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.36 + 3.36i)T + 31iT^{2} \)
37 \( 1 + (1.03 - 3.87i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (1.42 - 5.32i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (9.78 + 5.64i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.97 - 2.97i)T - 47iT^{2} \)
53 \( 1 + 11.0T + 53T^{2} \)
59 \( 1 + (3.14 + 11.7i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (4.55 + 2.63i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.15 - 1.11i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-0.800 - 2.98i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (-4.78 + 4.78i)T - 73iT^{2} \)
79 \( 1 + 1.16T + 79T^{2} \)
83 \( 1 + (3.24 + 3.24i)T + 83iT^{2} \)
89 \( 1 + (-4.80 - 1.28i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (14.7 - 3.95i)T + (84.0 - 48.5i)T^{2} \)
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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

−12.03368606713219653299877320430, −11.31987857877943545564586032855, −9.736887294922945886634474221892, −9.374023971170281346464893863945, −8.422953312086387773286194342345, −7.75951365427626141808180297888, −6.85204571764907948631106511155, −4.80757307030454517469966233181, −3.45068292650456768786033192540, −1.64593857596867235887455264532, 0.06383885940719181590390638305, 2.94759501955463700125350226897, 3.79767900299934777449787050343, 6.16576897067946086720095481134, 7.10193889939850056158691350311, 7.910916043264778029060926114294, 8.490924914746106428431103435246, 9.813173811617002724412722291806, 10.46962689424252228845750364224, 11.05734437023978553658543499800

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