↑

Properties

Label 2-273-91.62-c2-0-8
Degree $2$
Conductor $273$
Sign $-0.618 - 0.785i$
Analytic cond. $7.43871$
Root an. cond. $2.72740$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.181 − 0.104i)2-s + (−1.5 + 0.866i)3-s + (−1.97 + 3.42i)4-s + 6.37·5-s + (−0.181 + 0.313i)6-s + (−3.53 + 6.04i)7-s + 1.66i·8-s + (1.5 − 2.59i)9-s + (1.15 − 0.666i)10-s + (10.6 − 6.13i)11-s − 6.85i·12-s + (−4.60 + 12.1i)13-s + (−0.00865 + 1.46i)14-s + (−9.56 + 5.52i)15-s + (−7.73 − 13.4i)16-s + (−21.0 − 12.1i)17-s + ⋯
L(s)  = 1  + (0.0905 − 0.0523i)2-s + (−0.5 + 0.288i)3-s + (−0.494 + 0.856i)4-s + 1.27·5-s + (−0.0301 + 0.0523i)6-s + (−0.505 + 0.863i)7-s + 0.208i·8-s + (0.166 − 0.288i)9-s + (0.115 − 0.0666i)10-s + (0.966 − 0.557i)11-s − 0.571i·12-s + (−0.353 + 0.935i)13-s + (−0.000618 + 0.104i)14-s + (−0.637 + 0.368i)15-s + (−0.483 − 0.837i)16-s + (−1.23 − 0.713i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.618 - 0.785i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.618 - 0.785i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.618 - 0.785i$
Analytic conductor: \(7.43871\)
Root analytic conductor: \(2.72740\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (244, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1),\ -0.618 - 0.785i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.519549 + 1.06988i\)
\(L(\frac12)\) \(\approx\) \(0.519549 + 1.06988i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.5 - 0.866i)T \)
7 \( 1 + (3.53 - 6.04i)T \)
13 \( 1 + (4.60 - 12.1i)T \)
good2 \( 1 + (-0.181 + 0.104i)T + (2 - 3.46i)T^{2} \)
5 \( 1 - 6.37T + 25T^{2} \)
11 \( 1 + (-10.6 + 6.13i)T + (60.5 - 104. i)T^{2} \)
17 \( 1 + (21.0 + 12.1i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (17.2 - 29.7i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (-16.9 - 29.3i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (10.3 + 18.0i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 - 21.1T + 961T^{2} \)
37 \( 1 + (43.2 - 24.9i)T + (684.5 - 1.18e3i)T^{2} \)
41 \( 1 + (-25.3 - 43.8i)T + (-840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-10.1 + 17.5i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + 42.2T + 2.20e3T^{2} \)
53 \( 1 + 6.63T + 2.80e3T^{2} \)
59 \( 1 + (-11.2 + 19.5i)T + (-1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-56.9 - 32.8i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-100. + 58.1i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + (-41.0 - 23.7i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + 18.1T + 5.32e3T^{2} \)
79 \( 1 - 85.7T + 6.24e3T^{2} \)
83 \( 1 - 61.5T + 6.88e3T^{2} \)
89 \( 1 + (37.7 + 65.3i)T + (-3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (29.0 - 50.2i)T + (-4.70e3 - 8.14e3i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.96229008340755423100261787031, −11.34180671877213244764432374376, −9.764960904400772395194996164763, −9.334136755336464480220613846282, −8.490400859307479015796889745119, −6.76699230215414224185623769341, −6.02930780762347671366127843247, −4.88608476649826411279306540442, −3.60881110933908509196601586593, −2.09511303035615061690958177403, 0.61582628054111245995989727915, 2.11785169259971406974231312764, 4.30316827508939663159595356917, 5.25543436528826760377391368636, 6.55859535393481528764986482014, 6.73088364630489901952909834316, 8.778730952303895941360207687947, 9.551902869602255841750389476548, 10.50018636301178335715827231951, 10.90712265908345138552104357561

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