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Properties

Label 2-273-273.251-c1-0-9
Degree $2$
Conductor $273$
Sign $-0.804 - 0.593i$
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  + (−0.230 + 0.398i)2-s + (−0.688 + 1.58i)3-s + (0.894 + 1.54i)4-s + 2.35i·5-s + (−0.474 − 0.639i)6-s + (1.77 + 1.96i)7-s − 1.74·8-s + (−2.05 − 2.18i)9-s + (−0.937 − 0.541i)10-s + (1.89 − 3.28i)11-s + (−3.07 + 0.354i)12-s + (−0.259 − 3.59i)13-s + (−1.19 + 0.255i)14-s + (−3.73 − 1.61i)15-s + (−1.38 + 2.40i)16-s + (−0.112 − 0.194i)17-s + ⋯
L(s)  = 1  + (−0.162 + 0.281i)2-s + (−0.397 + 0.917i)3-s + (0.447 + 0.774i)4-s + 1.05i·5-s + (−0.193 − 0.261i)6-s + (0.670 + 0.741i)7-s − 0.616·8-s + (−0.684 − 0.729i)9-s + (−0.296 − 0.171i)10-s + (0.571 − 0.989i)11-s + (−0.888 + 0.102i)12-s + (−0.0721 − 0.997i)13-s + (−0.318 + 0.0682i)14-s + (−0.965 − 0.418i)15-s + (−0.346 + 0.600i)16-s + (−0.0272 − 0.0471i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.804 - 0.593i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.804 - 0.593i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.804 - 0.593i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.804 - 0.593i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.359940 + 1.09367i\)
\(L(\frac12)\) \(\approx\) \(0.359940 + 1.09367i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.688 - 1.58i)T \)
7 \( 1 + (-1.77 - 1.96i)T \)
13 \( 1 + (0.259 + 3.59i)T \)
good2 \( 1 + (0.230 - 0.398i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 - 2.35iT - 5T^{2} \)
11 \( 1 + (-1.89 + 3.28i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (0.112 + 0.194i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.495 - 0.858i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-5.19 - 2.99i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.01 + 1.74i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 8.13T + 31T^{2} \)
37 \( 1 + (-5.67 - 3.27i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.81 - 1.04i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.647 + 1.12i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 7.64iT - 47T^{2} \)
53 \( 1 - 7.19iT - 53T^{2} \)
59 \( 1 + (-6.86 + 3.96i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.77 - 1.02i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-9.58 - 5.53i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (5.36 + 9.29i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 1.05T + 73T^{2} \)
79 \( 1 + 8.02T + 79T^{2} \)
83 \( 1 + 8.85iT - 83T^{2} \)
89 \( 1 + (-8.09 - 4.67i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.92 + 3.32i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.86111469427037250839318028151, −11.26010808992479790268513314087, −10.69661429658939079965449583202, −9.317785497114700372804494719369, −8.484533286908577327756199792922, −7.41209054208809381294527652018, −6.23384232794894432787145750563, −5.39169626926554345242647556468, −3.65562976362796718910975483389, −2.83719507595636564277770223956, 1.06596985846047870714510121305, 2.00118141364205934925884303968, 4.48364540345738627639708388701, 5.37809144007132794585430069328, 6.72867211678366871732562831763, 7.36669613342081816486195651671, 8.757730161480409228706025463480, 9.581168212866527340529306802854, 10.92946430906132555954992121674, 11.42888966478061221312991694750

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