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Properties

Label 2-273-91.59-c1-0-16
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: $\chi_{273} (241, \cdot )$
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

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

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