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Properties

Label 2-273-13.10-c1-0-10
Degree $2$
Conductor $273$
Sign $0.496 + 0.868i$
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  + (−0.420 + 0.242i)2-s + (−0.5 − 0.866i)3-s + (−0.881 + 1.52i)4-s − 1.06i·5-s + (0.420 + 0.242i)6-s + (−0.866 − 0.5i)7-s − 1.82i·8-s + (−0.499 + 0.866i)9-s + (0.258 + 0.448i)10-s + (4.98 − 2.87i)11-s + 1.76·12-s + (1.77 − 3.13i)13-s + 0.485·14-s + (−0.922 + 0.532i)15-s + (−1.31 − 2.28i)16-s + (1.17 − 2.03i)17-s + ⋯
L(s)  = 1  + (−0.297 + 0.171i)2-s + (−0.288 − 0.499i)3-s + (−0.440 + 0.763i)4-s − 0.476i·5-s + (0.171 + 0.0992i)6-s + (−0.327 − 0.188i)7-s − 0.646i·8-s + (−0.166 + 0.288i)9-s + (0.0818 + 0.141i)10-s + (1.50 − 0.867i)11-s + 0.509·12-s + (0.492 − 0.870i)13-s + 0.129·14-s + (−0.238 + 0.137i)15-s + (−0.329 − 0.571i)16-s + (0.284 − 0.492i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.496 + 0.868i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.496 + 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.496 + 0.868i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.496 + 0.868i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.731620 - 0.424474i\)
\(L(\frac12)\) \(\approx\) \(0.731620 - 0.424474i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (0.866 + 0.5i)T \)
13 \( 1 + (-1.77 + 3.13i)T \)
good2 \( 1 + (0.420 - 0.242i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + 1.06iT - 5T^{2} \)
11 \( 1 + (-4.98 + 2.87i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-1.17 + 2.03i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (5.36 + 3.10i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.75 - 4.77i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.80 + 4.85i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 5.46iT - 31T^{2} \)
37 \( 1 + (4.08 - 2.35i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-7.69 + 4.44i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.77 - 8.27i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 5.21iT - 47T^{2} \)
53 \( 1 - 1.52T + 53T^{2} \)
59 \( 1 + (9.06 + 5.23i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.82 - 6.62i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.83 + 1.63i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-7.50 - 4.33i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 7.63iT - 73T^{2} \)
79 \( 1 + 15.6T + 79T^{2} \)
83 \( 1 - 2.63iT - 83T^{2} \)
89 \( 1 + (-10.4 + 6.03i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-14.5 - 8.38i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.78372924204237122508400729640, −11.01769642780326089486872518387, −9.500488407548494989515645318427, −8.827644917289117111467720156255, −7.941970685376779724862513750967, −6.87671899820044450373432278981, −5.89470506447465496129611076173, −4.37878783013853445896842950629, −3.21395782472208381261761294549, −0.822248566157718432086431418861, 1.71586109997505953043835547531, 3.78192334346048467857341451957, 4.78528232152730200246899385030, 6.20552809669654698642155834818, 6.80012693842937849171586370432, 8.774053343355966441821739186201, 9.153455239562415877367166266212, 10.33094274280118942289697951758, 10.77919618797031964823433889258, 11.92288685817902769766862347639

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