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Properties

Label 2-273-91.54-c1-0-3
Degree $2$
Conductor $273$
Sign $-0.968 + 0.248i$
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  + (−1.96 + 1.96i)2-s + (0.866 + 0.5i)3-s − 5.75i·4-s + (−1.75 + 0.470i)5-s + (−2.68 + 0.720i)6-s + (2.27 + 1.35i)7-s + (7.38 + 7.38i)8-s + (0.499 + 0.866i)9-s + (2.52 − 4.37i)10-s + (−4.28 + 1.14i)11-s + (2.87 − 4.97i)12-s + (−2.12 + 2.91i)13-s + (−7.13 + 1.81i)14-s + (−1.75 − 0.470i)15-s − 17.5·16-s + 0.0643·17-s + ⋯
L(s)  = 1  + (−1.39 + 1.39i)2-s + (0.499 + 0.288i)3-s − 2.87i·4-s + (−0.784 + 0.210i)5-s + (−1.09 + 0.294i)6-s + (0.859 + 0.510i)7-s + (2.61 + 2.61i)8-s + (0.166 + 0.288i)9-s + (0.799 − 1.38i)10-s + (−1.29 + 0.346i)11-s + (0.829 − 1.43i)12-s + (−0.590 + 0.807i)13-s + (−1.90 + 0.486i)14-s + (−0.452 − 0.121i)15-s − 4.39·16-s + 0.0156·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 + 0.248i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.968 + 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.968 + 0.248i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.968 + 0.248i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0597171 - 0.473504i\)
\(L(\frac12)\) \(\approx\) \(0.0597171 - 0.473504i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.27 - 1.35i)T \)
13 \( 1 + (2.12 - 2.91i)T \)
good2 \( 1 + (1.96 - 1.96i)T - 2iT^{2} \)
5 \( 1 + (1.75 - 0.470i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (4.28 - 1.14i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 - 0.0643T + 17T^{2} \)
19 \( 1 + (0.788 - 2.94i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + 3.21iT - 23T^{2} \)
29 \( 1 + (-2.41 - 4.18i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.34 - 5.02i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (3.08 + 3.08i)T + 37iT^{2} \)
41 \( 1 + (-2.12 + 7.93i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (3.22 + 1.86i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.08 - 4.06i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-3.30 - 5.71i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.28 - 3.28i)T - 59iT^{2} \)
61 \( 1 + (-1.90 + 1.09i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.62 - 6.07i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (1.98 + 7.40i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (-6.03 - 1.61i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-0.639 + 1.10i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-8.90 - 8.90i)T + 83iT^{2} \)
89 \( 1 + (1.02 - 1.02i)T - 89iT^{2} \)
97 \( 1 + (-8.79 + 2.35i)T + (84.0 - 48.5i)T^{2} \)
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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

−12.21353173688123522798035253358, −10.89641856849177121559537205244, −10.32925597878593438545957783966, −9.172243662889758555580996145047, −8.426032901514374125035961144185, −7.70633326392615680831880318424, −7.03497275255780784662612853759, −5.51965846340573924868614518650, −4.62145337671258667614350465312, −2.08107869112410323446615061925, 0.52866566809800929713429875302, 2.25123804186960528770718055204, 3.39138122661670869063899507841, 4.66040688422665351040479540283, 7.30955425625133758967495870764, 8.020185852594662797595801557406, 8.297042807366277888083907819243, 9.647186515036312402090366021646, 10.44902322686833570911543924933, 11.30476943630791725685253398193

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