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Properties

Label 2-273-13.10-c1-0-1
Degree $2$
Conductor $273$
Sign $-0.382 - 0.923i$
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  + (0.0921 − 0.0531i)2-s + (−0.5 − 0.866i)3-s + (−0.994 + 1.72i)4-s + 1.41i·5-s + (−0.0921 − 0.0531i)6-s + (−0.866 − 0.5i)7-s + 0.424i·8-s + (−0.499 + 0.866i)9-s + (0.0751 + 0.130i)10-s + (−2.59 + 1.50i)11-s + 1.98·12-s + (−2.15 + 2.88i)13-s − 0.106·14-s + (1.22 − 0.706i)15-s + (−1.96 − 3.40i)16-s + (−3.18 + 5.51i)17-s + ⋯
L(s)  = 1  + (0.0651 − 0.0376i)2-s + (−0.288 − 0.499i)3-s + (−0.497 + 0.861i)4-s + 0.631i·5-s + (−0.0376 − 0.0217i)6-s + (−0.327 − 0.188i)7-s + 0.149i·8-s + (−0.166 + 0.288i)9-s + (0.0237 + 0.0411i)10-s + (−0.783 + 0.452i)11-s + 0.574·12-s + (−0.598 + 0.801i)13-s − 0.0284·14-s + (0.315 − 0.182i)15-s + (−0.491 − 0.851i)16-s + (−0.772 + 1.33i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 - 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.382 - 0.923i$
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: \((2,\ 273,\ (\ :1/2),\ -0.382 - 0.923i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.392237 + 0.586969i\)
\(L(\frac12)\) \(\approx\) \(0.392237 + 0.586969i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.15 - 2.88i)T \)
good2 \( 1 + (-0.0921 + 0.0531i)T + (1 - 1.73i)T^{2} \)
5 \( 1 - 1.41iT - 5T^{2} \)
11 \( 1 + (2.59 - 1.50i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (3.18 - 5.51i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.59 - 2.65i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.335 + 0.581i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.258 + 0.447i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 0.282iT - 31T^{2} \)
37 \( 1 + (-5.58 + 3.22i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.43 - 1.98i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.47 + 4.28i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 5.91iT - 47T^{2} \)
53 \( 1 + 12.2T + 53T^{2} \)
59 \( 1 + (1.99 + 1.15i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.01 - 3.49i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.83 + 1.05i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-12.7 - 7.33i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 6.44iT - 73T^{2} \)
79 \( 1 - 12.6T + 79T^{2} \)
83 \( 1 - 8.49iT - 83T^{2} \)
89 \( 1 + (-9.89 + 5.71i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.317 - 0.183i)T + (48.5 + 84.0i)T^{2} \)
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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

−12.37380416457797651881656364015, −11.38488405527607937596260580364, −10.38557745719851873513345901260, −9.342608363482825279202165510474, −8.130668017883848437317695633768, −7.33588532568165226155209168344, −6.44716165312857538543068186494, −4.98056925671970622669827027342, −3.73053669407676469007722550390, −2.36916759424727980800318820373, 0.54162034020040983132923793345, 2.95372017102436677146215278842, 4.83344314546502243424178501133, 5.14274930772516060626818085671, 6.34337942063352366257083358577, 7.79935596353010726788628672304, 9.132078130523595413708638391976, 9.567607483041878899122077677603, 10.60757303112628751593354901516, 11.44529016761128860832504209156

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