Properties

Label 2-273-91.54-c1-0-5
Degree $2$
Conductor $273$
Sign $-0.302 - 0.953i$
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  + (−1.30 + 1.30i)2-s + (−0.866 − 0.5i)3-s − 1.42i·4-s + (−0.0744 + 0.0199i)5-s + (1.78 − 0.478i)6-s + (2.55 + 0.700i)7-s + (−0.758 − 0.758i)8-s + (0.499 + 0.866i)9-s + (0.0713 − 0.123i)10-s + (0.672 − 0.180i)11-s + (−0.710 + 1.23i)12-s + (2.39 + 2.69i)13-s + (−4.25 + 2.42i)14-s + (0.0744 + 0.0199i)15-s + 4.82·16-s − 1.23·17-s + ⋯
L(s)  = 1  + (−0.924 + 0.924i)2-s + (−0.499 − 0.288i)3-s − 0.710i·4-s + (−0.0333 + 0.00892i)5-s + (0.729 − 0.195i)6-s + (0.964 + 0.264i)7-s + (−0.268 − 0.268i)8-s + (0.166 + 0.288i)9-s + (0.0225 − 0.0390i)10-s + (0.202 − 0.0543i)11-s + (−0.205 + 0.355i)12-s + (0.665 + 0.746i)13-s + (−1.13 + 0.647i)14-s + (0.0192 + 0.00515i)15-s + 1.20·16-s − 0.300·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.302 - 0.953i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.302 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.302 - 0.953i$
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: \((2,\ 273,\ (\ :1/2),\ -0.302 - 0.953i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.403030 + 0.550733i\)
\(L(\frac12)\) \(\approx\) \(0.403030 + 0.550733i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.55 - 0.700i)T \)
13 \( 1 + (-2.39 - 2.69i)T \)
good2 \( 1 + (1.30 - 1.30i)T - 2iT^{2} \)
5 \( 1 + (0.0744 - 0.0199i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (-0.672 + 0.180i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + 1.23T + 17T^{2} \)
19 \( 1 + (1.45 - 5.44i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 - 7.37iT - 23T^{2} \)
29 \( 1 + (-1.84 - 3.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.34 + 8.76i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (-3.25 - 3.25i)T + 37iT^{2} \)
41 \( 1 + (0.231 - 0.865i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (2.14 + 1.24i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.108 - 0.404i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-5.75 - 9.96i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-9.90 + 9.90i)T - 59iT^{2} \)
61 \( 1 + (-5.63 + 3.25i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.540 + 2.01i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-0.119 - 0.445i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (4.91 + 1.31i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-3.26 + 5.65i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (9.54 + 9.54i)T + 83iT^{2} \)
89 \( 1 + (-3.14 + 3.14i)T - 89iT^{2} \)
97 \( 1 + (15.6 - 4.19i)T + (84.0 - 48.5i)T^{2} \)
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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

−11.85527355438271431898781180341, −11.36289124100466282996840890542, −10.06983954903900782115715730584, −9.094091196971212007733418081731, −8.152335723313612594494776559667, −7.49007229949404126665850135068, −6.34330922695767295378182275526, −5.56107288355648685626845912197, −3.97049383723279130043709196218, −1.56681051468936849364596639084, 0.834839778805087352638468779279, 2.48148834632932955273715255929, 4.19533592393313402736507786483, 5.40271108605091530814244748115, 6.73199635967131791066708054897, 8.243427051154856729785093474371, 8.763359569806746042417471217101, 10.06765242635154188174636186373, 10.65514128905260602414141111436, 11.35620431929590586931229001512

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