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Properties

Label 2-273-91.69-c2-0-12
Degree $2$
Conductor $273$
Sign $-0.299 + 0.954i$
Analytic cond. $7.43871$
Root an. cond. $2.72740$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.70 − 1.56i)2-s + (−1.5 − 0.866i)3-s + (2.88 + 4.99i)4-s − 2.70·5-s + (2.70 + 4.68i)6-s + (−3.66 + 5.96i)7-s − 5.54i·8-s + (1.5 + 2.59i)9-s + (7.31 + 4.22i)10-s + (−5.80 − 3.35i)11-s − 9.99i·12-s + (12.6 − 3.01i)13-s + (19.2 − 10.4i)14-s + (4.05 + 2.34i)15-s + (2.88 − 4.99i)16-s + (−11.9 + 6.88i)17-s + ⋯
L(s)  = 1  + (−1.35 − 0.781i)2-s + (−0.5 − 0.288i)3-s + (0.721 + 1.24i)4-s − 0.540·5-s + (0.451 + 0.781i)6-s + (−0.522 + 0.852i)7-s − 0.692i·8-s + (0.166 + 0.288i)9-s + (0.731 + 0.422i)10-s + (−0.527 − 0.304i)11-s − 0.833i·12-s + (0.972 − 0.231i)13-s + (1.37 − 0.745i)14-s + (0.270 + 0.156i)15-s + (0.180 − 0.312i)16-s + (−0.701 + 0.405i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.299 + 0.954i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.299 + 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.299 + 0.954i$
Analytic conductor: \(7.43871\)
Root analytic conductor: \(2.72740\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (160, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1),\ -0.299 + 0.954i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.238762 - 0.325064i\)
\(L(\frac12)\) \(\approx\) \(0.238762 - 0.325064i\)
\(L(2)\) 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 \( 1 + (3.66 - 5.96i)T \)
13 \( 1 + (-12.6 + 3.01i)T \)
good2 \( 1 + (2.70 + 1.56i)T + (2 + 3.46i)T^{2} \)
5 \( 1 + 2.70T + 25T^{2} \)
11 \( 1 + (5.80 + 3.35i)T + (60.5 + 104. i)T^{2} \)
17 \( 1 + (11.9 - 6.88i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-14.1 - 24.4i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (-19.5 + 33.8i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (-9.25 + 16.0i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 - 2.09T + 961T^{2} \)
37 \( 1 + (27.3 + 15.7i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 + (-22.7 + 39.4i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (17.1 + 29.6i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 - 6.43T + 2.20e3T^{2} \)
53 \( 1 + 74.3T + 2.80e3T^{2} \)
59 \( 1 + (-8.75 - 15.1i)T + (-1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-70.7 + 40.8i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-9.40 - 5.43i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + (-102. + 59.3i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 - 105.T + 5.32e3T^{2} \)
79 \( 1 - 147.T + 6.24e3T^{2} \)
83 \( 1 - 49.6T + 6.88e3T^{2} \)
89 \( 1 + (65.1 - 112. i)T + (-3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (46.2 + 80.1i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−11.14491579951644206286152913795, −10.59655522892242261107228322438, −9.548206085897652165515903849022, −8.506811098919447831349561459986, −7.981783538696937120533120404865, −6.58577068826130741466754375913, −5.45568410604339088093338851932, −3.52477499707407095333375046427, −2.14442458637591431446094510124, −0.46631627231935247303772900059, 0.932651043722079011735844134127, 3.55194119913617959408567157471, 5.00538221149264731997309118282, 6.47524868122898325732794679241, 7.13701484244918961710015987512, 8.001622621328513541176551119491, 9.207836887788299066474252595739, 9.785550733360711877439165278875, 10.92436115083277766744069031565, 11.38441804000999470360685792975

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