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Properties

Label 2-273-91.41-c1-0-13
Degree $2$
Conductor $273$
Sign $0.840 + 0.541i$
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  + (0.604 + 0.161i)2-s + (0.866 − 0.5i)3-s + (−1.39 − 0.804i)4-s + (0.965 + 0.965i)5-s + (0.604 − 0.161i)6-s + (2.12 − 1.57i)7-s + (−1.59 − 1.59i)8-s + (0.499 − 0.866i)9-s + (0.427 + 0.739i)10-s + (1.26 − 4.72i)11-s − 1.60·12-s + (1.35 + 3.34i)13-s + (1.53 − 0.610i)14-s + (1.31 + 0.353i)15-s + (0.902 + 1.56i)16-s + (−2.72 + 4.72i)17-s + ⋯
L(s)  = 1  + (0.427 + 0.114i)2-s + (0.499 − 0.288i)3-s + (−0.696 − 0.402i)4-s + (0.431 + 0.431i)5-s + (0.246 − 0.0660i)6-s + (0.802 − 0.596i)7-s + (−0.564 − 0.564i)8-s + (0.166 − 0.288i)9-s + (0.135 + 0.233i)10-s + (0.382 − 1.42i)11-s − 0.464·12-s + (0.374 + 0.927i)13-s + (0.411 − 0.163i)14-s + (0.340 + 0.0912i)15-s + (0.225 + 0.390i)16-s + (−0.660 + 1.14i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.840 + 0.541i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.840 + 0.541i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.840 + 0.541i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.840 + 0.541i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.68495 - 0.496201i\)
\(L(\frac12)\) \(\approx\) \(1.68495 - 0.496201i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.12 + 1.57i)T \)
13 \( 1 + (-1.35 - 3.34i)T \)
good2 \( 1 + (-0.604 - 0.161i)T + (1.73 + i)T^{2} \)
5 \( 1 + (-0.965 - 0.965i)T + 5iT^{2} \)
11 \( 1 + (-1.26 + 4.72i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (2.72 - 4.72i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.47 + 1.19i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (3.14 - 1.81i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.00 - 1.74i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (5.91 + 5.91i)T + 31iT^{2} \)
37 \( 1 + (2.84 - 10.6i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (1.08 - 4.04i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (0.669 + 0.386i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (5.65 - 5.65i)T - 47iT^{2} \)
53 \( 1 + 6.72T + 53T^{2} \)
59 \( 1 + (-3.87 - 14.4i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-0.210 - 0.121i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.55 - 1.48i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (0.711 + 2.65i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (-2.17 + 2.17i)T - 73iT^{2} \)
79 \( 1 - 8.38T + 79T^{2} \)
83 \( 1 + (11.2 + 11.2i)T + 83iT^{2} \)
89 \( 1 + (-3.25 - 0.872i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-9.46 + 2.53i)T + (84.0 - 48.5i)T^{2} \)
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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

−11.76825186688765960480953175429, −10.92076105280017394419852638167, −9.860633123422547471146727964249, −8.859779797406763731136351220039, −8.096128653312604745783237339579, −6.63951045531348863424618271201, −5.86536178552234938454552869078, −4.44162910451527805241707266535, −3.44884864314000424461953869296, −1.47070234154534199670612762161, 2.10011142374173662311938021373, 3.60162261721047113836452850363, 4.88923967330161111738710015283, 5.38031347964797713809901103504, 7.29396051241053192801258781980, 8.304272928627919643273782163786, 9.173672327318881962823399548816, 9.771138212364613061782568006363, 11.20829825950857040045847744955, 12.28720418193517488127185952152

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