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Properties

Label 2-273-91.59-c1-0-11
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: $\chi_{273} (241, \cdot )$
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

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

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