Properties

Label 2-648-9.5-c2-0-7
Degree $2$
Conductor $648$
Sign $-0.173 - 0.984i$
Analytic cond. $17.6567$
Root an. cond. $4.20199$
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  + (4.89 + 2.82i)5-s + (3 + 5.19i)7-s + (−4.89 + 2.82i)11-s + (−5 + 8.66i)13-s + 22.6i·17-s + 2·19-s + (−9.79 − 5.65i)23-s + (3.49 + 6.06i)25-s + (−14.6 + 8.48i)29-s + (11 − 19.0i)31-s + 33.9i·35-s − 6·37-s + (29.3 + 16.9i)41-s + (−41 − 71.0i)43-s + (−58.7 + 33.9i)47-s + ⋯
L(s)  = 1  + (0.979 + 0.565i)5-s + (0.428 + 0.742i)7-s + (−0.445 + 0.257i)11-s + (−0.384 + 0.666i)13-s + 1.33i·17-s + 0.105·19-s + (−0.425 − 0.245i)23-s + (0.139 + 0.242i)25-s + (−0.506 + 0.292i)29-s + (0.354 − 0.614i)31-s + 0.969i·35-s − 0.162·37-s + (0.716 + 0.413i)41-s + (−0.953 − 1.65i)43-s + (−1.25 + 0.722i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(648\)    =    \(2^{3} \cdot 3^{4}\)
Sign: $-0.173 - 0.984i$
Analytic conductor: \(17.6567\)
Root analytic conductor: \(4.20199\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{648} (377, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 648,\ (\ :1),\ -0.173 - 0.984i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.873292937\)
\(L(\frac12)\) \(\approx\) \(1.873292937\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (-4.89 - 2.82i)T + (12.5 + 21.6i)T^{2} \)
7 \( 1 + (-3 - 5.19i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (4.89 - 2.82i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (5 - 8.66i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 - 22.6iT - 289T^{2} \)
19 \( 1 - 2T + 361T^{2} \)
23 \( 1 + (9.79 + 5.65i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (14.6 - 8.48i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (-11 + 19.0i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + 6T + 1.36e3T^{2} \)
41 \( 1 + (-29.3 - 16.9i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (41 + 71.0i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (58.7 - 33.9i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 62.2iT - 2.80e3T^{2} \)
59 \( 1 + (-63.6 - 36.7i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-43 - 74.4i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (1 - 1.73i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 124. iT - 5.04e3T^{2} \)
73 \( 1 - 82T + 5.32e3T^{2} \)
79 \( 1 + (5 + 8.66i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-63.6 + 36.7i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 33.9iT - 7.92e3T^{2} \)
97 \( 1 + (-47 - 81.4i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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

−10.42443897680818795190462191824, −9.865002050798680009969938024307, −8.894960391718214939352395489649, −8.056201222203477820314958926091, −6.92673510746324879163474948228, −6.03922595575619394758102089440, −5.31401405845741258720619847810, −4.06737185896976583603817403615, −2.54245518389451309474752685681, −1.81495839391927337244744552181, 0.65917763751902512153555662181, 2.01251257145368673643389970499, 3.33960698552494927329551399692, 4.84911888966896128966657426662, 5.32754310119831671937951955415, 6.50152581484569674922836621208, 7.57143116983371955603583918920, 8.297058207061666313609714454381, 9.503921394475568841713098857582, 9.920278968002095867263127988556

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