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Properties

Label 2-273-273.107-c0-0-1
Degree $2$
Conductor $273$
Sign $0.190 - 0.981i$
Analytic cond. $0.136244$
Root an. cond. $0.369113$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·2-s + (0.866 + 0.5i)3-s + (−0.866 − 0.5i)5-s + (−0.5 + 0.866i)6-s + (−0.5 − 0.866i)7-s + i·8-s + (0.499 + 0.866i)9-s + (0.5 − 0.866i)10-s − 13-s + (0.866 − 0.5i)14-s + (−0.499 − 0.866i)15-s − 16-s − i·17-s + (−0.866 + 0.499i)18-s − 0.999i·21-s + ⋯
L(s)  = 1  + i·2-s + (0.866 + 0.5i)3-s + (−0.866 − 0.5i)5-s + (−0.5 + 0.866i)6-s + (−0.5 − 0.866i)7-s + i·8-s + (0.499 + 0.866i)9-s + (0.5 − 0.866i)10-s − 13-s + (0.866 − 0.5i)14-s + (−0.499 − 0.866i)15-s − 16-s − i·17-s + (−0.866 + 0.499i)18-s − 0.999i·21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.190 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.190 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.190 - 0.981i$
Analytic conductor: \(0.136244\)
Root analytic conductor: \(0.369113\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :0),\ 0.190 - 0.981i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8902485401\)
\(L(\frac12)\) \(\approx\) \(0.8902485401\)
\(L(1)\) 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 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + T \)
good2 \( 1 - iT - T^{2} \)
5 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 + iT - T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + iT - T^{2} \)
29 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 - T + T^{2} \)
41 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
59 \( 1 - iT - T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
73 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - iT - T^{2} \)
97 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
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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

−12.45839087364309845829082429501, −11.34893602192075396821191455166, −10.27090992328387082700100767647, −9.261984649232099377938399527062, −8.231170295604727435905978789256, −7.51828293431547257653370785905, −6.79630155492941171133576287977, −5.07790150804553031667076782138, −4.21133916261879778572438295852, −2.75991142853910062421931303486, 2.09368771117900151686760285036, 3.08829971249185133363349303392, 3.99770012676609839692866788023, 6.11384775433892806220920232724, 7.22864699467596249750084034109, 8.007607814910001464625463047681, 9.337069498874731396699529351394, 9.946932265950353662835940003386, 11.28242428730413389059665891949, 11.90363920371632228224238796993

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