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Properties

Label 2-273-273.257-c1-0-11
Degree $2$
Conductor $273$
Sign $0.541 - 0.840i$
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  + (1.5 + 0.866i)3-s − 2·4-s + (2 + 1.73i)7-s + (1.5 + 2.59i)9-s + (−3 − 1.73i)12-s + (1 + 3.46i)13-s + 4·16-s + (0.5 + 0.866i)19-s + (1.50 + 4.33i)21-s + (−2.5 − 4.33i)25-s + 5.19i·27-s + (−4 − 3.46i)28-s + (−3.5 − 6.06i)31-s + (−3 − 5.19i)36-s − 5.19i·37-s + ⋯
L(s)  = 1  + (0.866 + 0.499i)3-s − 4-s + (0.755 + 0.654i)7-s + (0.5 + 0.866i)9-s + (−0.866 − 0.499i)12-s + (0.277 + 0.960i)13-s + 16-s + (0.114 + 0.198i)19-s + (0.327 + 0.944i)21-s + (−0.5 − 0.866i)25-s + 0.999i·27-s + (−0.755 − 0.654i)28-s + (−0.628 − 1.08i)31-s + (−0.5 − 0.866i)36-s − 0.854i·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.541 - 0.840i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.541 - 0.840i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.541 - 0.840i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.541 - 0.840i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.26910 + 0.692107i\)
\(L(\frac12)\) \(\approx\) \(1.26910 + 0.692107i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.5 - 0.866i)T \)
7 \( 1 + (-2 - 1.73i)T \)
13 \( 1 + (-1 - 3.46i)T \)
good2 \( 1 + 2T^{2} \)
5 \( 1 + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.5 + 6.06i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 5.19iT - 37T^{2} \)
41 \( 1 + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4 + 6.92i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (26.5 - 45.8i)T^{2} \)
59 \( 1 - 59T^{2} \)
61 \( 1 + (-13.5 + 7.79i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (10.5 + 6.06i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (5 + 8.66i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.5 - 11.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 + (-9.5 + 16.4i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.11990515335903739985469052413, −11.01099641032388266832006557808, −9.876686972744191120894742608387, −9.087398315469244384431776791870, −8.462035230188345410544844106201, −7.54439321286363086520148134814, −5.75661948492866863806523905206, −4.63278912814543631048541880473, −3.79521017015359293978764829044, −2.09341921989679480560233999901, 1.24027636210832420307063631380, 3.22253338832772130347604381603, 4.31885719901572852730971147120, 5.55287418224715219080595583184, 7.15906306963011642793162970074, 8.012077359490792926095144984368, 8.704746423070926815940792846379, 9.715309858720150862828951638853, 10.64839563086742168272106466912, 11.92589734366127215568968297984

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