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Properties

Label 2-273-21.20-c1-0-2
Degree $2$
Conductor $273$
Sign $-0.449 + 0.893i$
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  + 1.95i·2-s + (−1.02 + 1.39i)3-s − 1.84·4-s − 2.34·5-s + (−2.73 − 2.01i)6-s + (0.446 + 2.60i)7-s + 0.312i·8-s + (−0.883 − 2.86i)9-s − 4.59i·10-s − 2.13i·11-s + (1.89 − 2.56i)12-s + i·13-s + (−5.11 + 0.874i)14-s + (2.40 − 3.26i)15-s − 4.29·16-s + 0.386·17-s + ⋯
L(s)  = 1  + 1.38i·2-s + (−0.593 + 0.804i)3-s − 0.920·4-s − 1.04·5-s + (−1.11 − 0.823i)6-s + (0.168 + 0.985i)7-s + 0.110i·8-s + (−0.294 − 0.955i)9-s − 1.45i·10-s − 0.645i·11-s + (0.546 − 0.740i)12-s + 0.277i·13-s + (−1.36 + 0.233i)14-s + (0.622 − 0.842i)15-s − 1.07·16-s + 0.0936·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.449 + 0.893i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.449 + 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.449 + 0.893i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.449 + 0.893i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.315294 - 0.511843i\)
\(L(\frac12)\) \(\approx\) \(0.315294 - 0.511843i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.02 - 1.39i)T \)
7 \( 1 + (-0.446 - 2.60i)T \)
13 \( 1 - iT \)
good2 \( 1 - 1.95iT - 2T^{2} \)
5 \( 1 + 2.34T + 5T^{2} \)
11 \( 1 + 2.13iT - 11T^{2} \)
17 \( 1 - 0.386T + 17T^{2} \)
19 \( 1 + 7.19iT - 19T^{2} \)
23 \( 1 - 7.49iT - 23T^{2} \)
29 \( 1 - 8.05iT - 29T^{2} \)
31 \( 1 - 5.54iT - 31T^{2} \)
37 \( 1 + 9.24T + 37T^{2} \)
41 \( 1 - 3.60T + 41T^{2} \)
43 \( 1 - 6.60T + 43T^{2} \)
47 \( 1 + 7.17T + 47T^{2} \)
53 \( 1 - 4.94iT - 53T^{2} \)
59 \( 1 + 4.12T + 59T^{2} \)
61 \( 1 - 5.98iT - 61T^{2} \)
67 \( 1 - 1.15T + 67T^{2} \)
71 \( 1 + 8.44iT - 71T^{2} \)
73 \( 1 - 7.31iT - 73T^{2} \)
79 \( 1 - 13.6T + 79T^{2} \)
83 \( 1 - 6.26T + 83T^{2} \)
89 \( 1 - 8.04T + 89T^{2} \)
97 \( 1 - 6.02iT - 97T^{2} \)
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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

−12.31257398855570965735291734620, −11.48176443481629022821946351225, −10.91800646472572176808365829068, −9.159958876632362807049107040302, −8.753925902756177588192835288057, −7.54798888907194359131613709710, −6.57720594295932308983055412074, −5.47016099654770879887051764857, −4.81296250435701142504578043633, −3.33755043459791764355748634234, 0.49141550918721647252012960488, 2.02424935155840374998250096191, 3.70435174021127075077586087315, 4.58610749896656970483758787692, 6.35519786107749927222267560502, 7.50199312939084503363113713939, 8.156656184559083250261820269404, 9.914805648796605239788294966238, 10.60269988742062901233940291229, 11.37852097427091205927829656047

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