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Properties

Label 2-273-91.19-c1-0-0
Degree $2$
Conductor $273$
Sign $-0.800 - 0.599i$
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  + (−1.49 − 0.400i)2-s + i·3-s + (0.345 + 0.199i)4-s + (0.481 − 0.128i)5-s + (0.400 − 1.49i)6-s + (−2.58 − 0.560i)7-s + (1.75 + 1.75i)8-s − 9-s − 0.771·10-s + (1.32 + 1.32i)11-s + (−0.199 + 0.345i)12-s + (−3.48 + 0.924i)13-s + (3.64 + 1.87i)14-s + (0.128 + 0.481i)15-s + (−2.31 − 4.01i)16-s + (−1.95 + 3.39i)17-s + ⋯
L(s)  = 1  + (−1.05 − 0.283i)2-s + 0.577i·3-s + (0.172 + 0.0997i)4-s + (0.215 − 0.0576i)5-s + (0.163 − 0.610i)6-s + (−0.977 − 0.211i)7-s + (0.619 + 0.619i)8-s − 0.333·9-s − 0.244·10-s + (0.398 + 0.398i)11-s + (−0.0576 + 0.0997i)12-s + (−0.966 + 0.256i)13-s + (0.973 + 0.501i)14-s + (0.0333 + 0.124i)15-s + (−0.579 − 1.00i)16-s + (−0.475 + 0.822i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.800 - 0.599i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.800 - 0.599i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.800 - 0.599i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.800 - 0.599i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0721024 + 0.216411i\)
\(L(\frac12)\) \(\approx\) \(0.0721024 + 0.216411i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - iT \)
7 \( 1 + (2.58 + 0.560i)T \)
13 \( 1 + (3.48 - 0.924i)T \)
good2 \( 1 + (1.49 + 0.400i)T + (1.73 + i)T^{2} \)
5 \( 1 + (-0.481 + 0.128i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (-1.32 - 1.32i)T + 11iT^{2} \)
17 \( 1 + (1.95 - 3.39i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.70 + 2.70i)T + 19iT^{2} \)
23 \( 1 + (4.25 - 2.45i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.28 - 3.96i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.12 - 4.21i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (-0.0389 + 0.145i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (-4.92 + 1.31i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (5.06 - 2.92i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.39 + 8.92i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-6.56 - 11.3i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.986 + 3.68i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + 11.7iT - 61T^{2} \)
67 \( 1 + (-6.99 + 6.99i)T - 67iT^{2} \)
71 \( 1 + (1.82 + 0.487i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (-6.55 - 1.75i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-2.22 + 3.85i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (5.85 + 5.85i)T + 83iT^{2} \)
89 \( 1 + (-11.9 - 3.20i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (2.09 - 7.81i)T + (-84.0 - 48.5i)T^{2} \)
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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

−12.06929168260183321357166250727, −10.93596074039960264731621366177, −10.15415050741617991988678885911, −9.485780338211687607896611744357, −8.886872306803321782469533369552, −7.62415438542603476472086591404, −6.49002760697873324252898542387, −5.08087185950562934676613081643, −3.83801901294048314887935300989, −2.07459250169217683611950014448, 0.23281341110576393899477193916, 2.34823330574284557651812719561, 4.07140453541014326592006500000, 5.88973830059059610396644413423, 6.77263431890529767440160088571, 7.72084075184551184637338243825, 8.624286394019547899811741704195, 9.627608727665333720299599651200, 10.11428980765982644724974839776, 11.51253195243645958216870548648

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