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Properties

Label 2-273-91.54-c1-0-2
Degree $2$
Conductor $273$
Sign $-0.928 - 0.372i$
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  + (0.111 − 0.111i)2-s + (−0.866 − 0.5i)3-s + 1.97i·4-s + (−2.13 + 0.570i)5-s + (−0.151 + 0.0407i)6-s + (−0.399 − 2.61i)7-s + (0.442 + 0.442i)8-s + (0.499 + 0.866i)9-s + (−0.173 + 0.300i)10-s + (−5.42 + 1.45i)11-s + (0.987 − 1.71i)12-s + (−1.34 + 3.34i)13-s + (−0.335 − 0.246i)14-s + (2.13 + 0.570i)15-s − 3.85·16-s − 3.59·17-s + ⋯
L(s)  = 1  + (0.0786 − 0.0786i)2-s + (−0.499 − 0.288i)3-s + 0.987i·4-s + (−0.952 + 0.255i)5-s + (−0.0620 + 0.0166i)6-s + (−0.151 − 0.988i)7-s + (0.156 + 0.156i)8-s + (0.166 + 0.288i)9-s + (−0.0548 + 0.0950i)10-s + (−1.63 + 0.438i)11-s + (0.285 − 0.493i)12-s + (−0.371 + 0.928i)13-s + (−0.0896 − 0.0658i)14-s + (0.550 + 0.147i)15-s − 0.963·16-s − 0.870·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.928 - 0.372i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.928 - 0.372i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.928 - 0.372i$
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: \((2,\ 273,\ (\ :1/2),\ -0.928 - 0.372i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0498781 + 0.258190i\)
\(L(\frac12)\) \(\approx\) \(0.0498781 + 0.258190i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.399 + 2.61i)T \)
13 \( 1 + (1.34 - 3.34i)T \)
good2 \( 1 + (-0.111 + 0.111i)T - 2iT^{2} \)
5 \( 1 + (2.13 - 0.570i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (5.42 - 1.45i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + 3.59T + 17T^{2} \)
19 \( 1 + (-1.24 + 4.65i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 - 4.45iT - 23T^{2} \)
29 \( 1 + (-1.02 - 1.77i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.643 + 2.40i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (-7.22 - 7.22i)T + 37iT^{2} \)
41 \( 1 + (-1.34 + 5.01i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-4.12 - 2.38i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.22 + 4.56i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-0.201 - 0.348i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.93 - 1.93i)T - 59iT^{2} \)
61 \( 1 + (-1.61 + 0.930i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.78 - 6.65i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (3.63 + 13.5i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (4.45 + 1.19i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (2.45 - 4.25i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-10.3 - 10.3i)T + 83iT^{2} \)
89 \( 1 + (12.7 - 12.7i)T - 89iT^{2} \)
97 \( 1 + (3.52 - 0.944i)T + (84.0 - 48.5i)T^{2} \)
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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

−12.22460229374897454797265511431, −11.37575291895612751524202611462, −10.83568936454521610119012207618, −9.516029954540169508766525465066, −8.064201881255273553162109251251, −7.41340633532362898967102259872, −6.82291827743535080268318577859, −4.88552463002208818180464097187, −4.04298589790028102114148320522, −2.63737742968768811338102318287, 0.19536755865913418906481704608, 2.66031679440000645524697084376, 4.45871191257696238278617338243, 5.43016942835410922482301253558, 6.10754108334962240967926043174, 7.67222683685014920122116540114, 8.554412506152073298298306850900, 9.796800027284420612571466579602, 10.61374290329283495803390653577, 11.35044019626002052606395068136

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