↑

Properties

Label 2-810-9.7-c3-0-29
Degree $2$
Conductor $810$
Sign $-0.173 + 0.984i$
Analytic cond. $47.7915$
Root an. cond. $6.91314$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1 − 1.73i)2-s + (−1.99 − 3.46i)4-s + (−2.5 − 4.33i)5-s + (−16 + 27.7i)7-s − 7.99·8-s − 10·10-s + (30 − 51.9i)11-s + (17 + 29.4i)13-s + (31.9 + 55.4i)14-s + (−8 + 13.8i)16-s + 42·17-s − 76·19-s + (−10 + 17.3i)20-s + (−60 − 103. i)22-s + (−12.5 + 21.6i)25-s + 68·26-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.223 − 0.387i)5-s + (−0.863 + 1.49i)7-s − 0.353·8-s − 0.316·10-s + (0.822 − 1.42i)11-s + (0.362 + 0.628i)13-s + (0.610 + 1.05i)14-s + (−0.125 + 0.216i)16-s + 0.599·17-s − 0.917·19-s + (−0.111 + 0.193i)20-s + (−0.581 − 1.00i)22-s + (−0.100 + 0.173i)25-s + 0.512·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 + 0.984i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(810\)    =    \(2 \cdot 3^{4} \cdot 5\)
Sign: $-0.173 + 0.984i$
Analytic conductor: \(47.7915\)
Root analytic conductor: \(6.91314\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{810} (541, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 810,\ (\ :3/2),\ -0.173 + 0.984i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.792483415\)
\(L(\frac12)\) \(\approx\) \(1.792483415\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 + 1.73i)T \)
3 \( 1 \)
5 \( 1 + (2.5 + 4.33i)T \)
good7 \( 1 + (16 - 27.7i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (-30 + 51.9i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-17 - 29.4i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 - 42T + 4.91e3T^{2} \)
19 \( 1 + 76T + 6.85e3T^{2} \)
23 \( 1 + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (3 - 5.19i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (-116 - 200. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 - 134T + 5.06e4T^{2} \)
41 \( 1 + (117 + 202. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-206 + 356. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-180 + 311. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 222T + 1.48e5T^{2} \)
59 \( 1 + (330 + 571. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-245 + 424. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (406 + 703. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 120T + 3.57e5T^{2} \)
73 \( 1 - 746T + 3.89e5T^{2} \)
79 \( 1 + (76 - 131. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-402 + 696. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + 678T + 7.04e5T^{2} \)
97 \( 1 + (97 - 168. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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

−9.479096944857846219667136653748, −8.848214881277306948057079313164, −8.395208822307702350321096523819, −6.64842736603105063638981962837, −6.01368238490080781420393991091, −5.21393995428221737239292592381, −3.87400883850698906736110299304, −3.14765971809509804338832196463, −1.95693317656118713247152338428, −0.52284306338248460731448297078, 0.988237826964075403564759738576, 2.83060702468102597385727521336, 4.04265076488216819634517519659, 4.36608782396346041358153334149, 5.97891090054215909030413828864, 6.68369849093669125425275941762, 7.37802125234485328490807208166, 8.029740175808067916509861505565, 9.405671121307096273759102441707, 10.01911028492347273981096233158

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