↑

Properties

Label 2-864-9.4-c1-0-1
Degree $2$
Conductor $864$
Sign $-0.569 - 0.821i$
Analytic cond. $6.89907$
Root an. cond. $2.62660$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)5-s + (1.72 + 2.98i)7-s + (−0.724 − 1.25i)11-s + (−2.94 + 5.10i)13-s − 4.89·17-s − 4·19-s + (2.72 − 4.71i)23-s + (2 + 3.46i)25-s + (0.0505 + 0.0874i)29-s + (1.27 − 2.20i)31-s − 3.44·35-s − 0.898·37-s + (−5.94 + 10.3i)41-s + (−1.17 − 2.03i)43-s + (3.17 + 5.49i)47-s + ⋯
L(s)  = 1  + (−0.223 + 0.387i)5-s + (0.651 + 1.12i)7-s + (−0.218 − 0.378i)11-s + (−0.818 + 1.41i)13-s − 1.18·17-s − 0.917·19-s + (0.568 − 0.984i)23-s + (0.400 + 0.692i)25-s + (0.00937 + 0.0162i)29-s + (0.229 − 0.396i)31-s − 0.583·35-s − 0.147·37-s + (−0.929 + 1.60i)41-s + (−0.179 − 0.310i)43-s + (0.463 + 0.801i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.569 - 0.821i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.569 - 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $-0.569 - 0.821i$
Analytic conductor: \(6.89907\)
Root analytic conductor: \(2.62660\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{864} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 864,\ (\ :1/2),\ -0.569 - 0.821i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.481128 + 0.919055i\)
\(L(\frac12)\) \(\approx\) \(0.481128 + 0.919055i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (0.5 - 0.866i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-1.72 - 2.98i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.724 + 1.25i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.94 - 5.10i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 4.89T + 17T^{2} \)
19 \( 1 + 4T + 19T^{2} \)
23 \( 1 + (-2.72 + 4.71i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.0505 - 0.0874i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.27 + 2.20i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 0.898T + 37T^{2} \)
41 \( 1 + (5.94 - 10.3i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.17 + 2.03i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.17 - 5.49i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 8.89T + 53T^{2} \)
59 \( 1 + (7.17 - 12.4i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.94 - 6.84i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.17 + 10.6i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 7.79T + 71T^{2} \)
73 \( 1 + 4.89T + 73T^{2} \)
79 \( 1 + (-6.72 - 11.6i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.275 - 0.476i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 12.8T + 89T^{2} \)
97 \( 1 + (-1.94 - 3.37i)T + (-48.5 + 84.0i)T^{2} \)
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

−10.66622192078830394660702573777, −9.383108482012590698613163161265, −8.830096638432220304445747616515, −8.051850738531668019753918729572, −6.89312993571374996351988991092, −6.28440302381861851082274766102, −5.00641231546535804720315721015, −4.35939611524518985481579969079, −2.79934748383665522844416444455, −1.96527873155203264897516919106, 0.47841581401103970925867896547, 2.06788885321750734481405473282, 3.50716269744742240297203477266, 4.63557552129580965839884709151, 5.13943101309279274382693488963, 6.58179318671794532247993802529, 7.42268808209194544749927564598, 8.092118767951826425347910767243, 8.923035303949207738643266999553, 10.10528121698901886901619168896

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