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Properties

Label 2-273-273.86-c1-0-4
Degree $2$
Conductor $273$
Sign $-0.563 - 0.825i$
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  + (0.258 + 0.965i)2-s + (−1.33 + 1.10i)3-s + (0.866 − 0.5i)4-s + (−0.151 − 0.565i)5-s + (−1.41 − i)6-s + (−1.15 + 2.38i)7-s + (2.12 + 2.12i)8-s + (0.548 − 2.94i)9-s + (0.507 − 0.292i)10-s + (−1.50 + 5.62i)11-s + (−0.599 + 1.62i)12-s + (2 + 3i)13-s + (−2.59 − 0.500i)14-s + (0.828 + 0.585i)15-s + (−0.500 + 0.866i)16-s + (−0.914 − 1.58i)17-s + ⋯
L(s)  = 1  + (0.183 + 0.683i)2-s + (−0.769 + 0.639i)3-s + (0.433 − 0.250i)4-s + (−0.0678 − 0.253i)5-s + (−0.577 − 0.408i)6-s + (−0.436 + 0.899i)7-s + (0.749 + 0.749i)8-s + (0.182 − 0.983i)9-s + (0.160 − 0.0926i)10-s + (−0.454 + 1.69i)11-s + (−0.173 + 0.469i)12-s + (0.554 + 0.832i)13-s + (−0.694 − 0.133i)14-s + (0.213 + 0.151i)15-s + (−0.125 + 0.216i)16-s + (−0.221 − 0.384i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.563 - 0.825i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.563 - 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.563 - 0.825i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (86, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.563 - 0.825i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.534455 + 1.01193i\)
\(L(\frac12)\) \(\approx\) \(0.534455 + 1.01193i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.33 - 1.10i)T \)
7 \( 1 + (1.15 - 2.38i)T \)
13 \( 1 + (-2 - 3i)T \)
good2 \( 1 + (-0.258 - 0.965i)T + (-1.73 + i)T^{2} \)
5 \( 1 + (0.151 + 0.565i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (1.50 - 5.62i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (0.914 + 1.58i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (6.43 - 1.72i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-3.29 + 5.70i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - iT - 29T^{2} \)
31 \( 1 + (-0.214 + 0.800i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (-1.03 - 3.86i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (-4 + 4i)T - 41iT^{2} \)
43 \( 1 - 6iT - 43T^{2} \)
47 \( 1 + (0.634 - 0.170i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-7.49 + 4.32i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.45 + 9.16i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.49 + 13.0i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (-0.464 + 0.464i)T - 71iT^{2} \)
73 \( 1 + (-12.3 - 3.32i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (0.707 - 1.22i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.07 + 3.07i)T - 83iT^{2} \)
89 \( 1 + (12.0 - 3.23i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (7 + 7i)T + 97iT^{2} \)
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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

−12.29643276479824553803319928162, −11.21798334540835532558587713997, −10.40706106315230865787418930004, −9.447302591863960627779129451989, −8.395759146556402609500030165148, −6.79698709309280513467033679746, −6.40143008306052271646721555633, −5.11698616955691382173734457301, −4.42793277944690722591180650607, −2.24932565920703328184973406783, 0.936426856354639800014310621549, 2.81039936739004820117198318883, 3.92769145520331031766332831476, 5.65094816313294648914920402245, 6.61585066875022942663266684340, 7.47754360924964133680409219839, 8.517746325553221598231837155389, 10.35231769365556893811868877704, 10.95225036216407018718480180524, 11.23766702484950556789804797229

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