↑

Properties

Label 2-24e2-576.205-c1-0-52
Degree $2$
Conductor $576$
Sign $0.587 + 0.809i$
Analytic cond. $4.59938$
Root an. cond. $2.14461$
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  + (−1.15 + 0.809i)2-s + (−0.561 + 1.63i)3-s + (0.690 − 1.87i)4-s + (−0.301 − 0.343i)5-s + (−0.675 − 2.35i)6-s + (0.560 + 0.0737i)7-s + (0.717 + 2.73i)8-s + (−2.37 − 1.83i)9-s + (0.628 + 0.154i)10-s + (−1.45 − 2.94i)11-s + (2.68 + 2.18i)12-s + (0.878 + 2.58i)13-s + (−0.709 + 0.367i)14-s + (0.732 − 0.301i)15-s + (−3.04 − 2.59i)16-s + (−5.57 − 5.57i)17-s + ⋯
L(s)  = 1  + (−0.820 + 0.572i)2-s + (−0.323 + 0.946i)3-s + (0.345 − 0.938i)4-s + (−0.134 − 0.153i)5-s + (−0.275 − 0.961i)6-s + (0.211 + 0.0278i)7-s + (0.253 + 0.967i)8-s + (−0.790 − 0.612i)9-s + (0.198 + 0.0489i)10-s + (−0.437 − 0.887i)11-s + (0.776 + 0.630i)12-s + (0.243 + 0.718i)13-s + (−0.189 + 0.0983i)14-s + (0.189 − 0.0777i)15-s + (−0.761 − 0.648i)16-s + (−1.35 − 1.35i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.587 + 0.809i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.587 + 0.809i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $0.587 + 0.809i$
Analytic conductor: \(4.59938\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (205, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ 0.587 + 0.809i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.382095 - 0.194770i\)
\(L(\frac12)\) \(\approx\) \(0.382095 - 0.194770i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.15 - 0.809i)T \)
3 \( 1 + (0.561 - 1.63i)T \)
good5 \( 1 + (0.301 + 0.343i)T + (-0.652 + 4.95i)T^{2} \)
7 \( 1 + (-0.560 - 0.0737i)T + (6.76 + 1.81i)T^{2} \)
11 \( 1 + (1.45 + 2.94i)T + (-6.69 + 8.72i)T^{2} \)
13 \( 1 + (-0.878 - 2.58i)T + (-10.3 + 7.91i)T^{2} \)
17 \( 1 + (5.57 + 5.57i)T + 17iT^{2} \)
19 \( 1 + (0.176 - 0.884i)T + (-17.5 - 7.27i)T^{2} \)
23 \( 1 + (0.338 + 2.57i)T + (-22.2 + 5.95i)T^{2} \)
29 \( 1 + (8.99 - 0.589i)T + (28.7 - 3.78i)T^{2} \)
31 \( 1 + (-4.45 + 2.57i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.00 + 5.07i)T + (-34.1 + 14.1i)T^{2} \)
41 \( 1 + (-1.11 - 8.49i)T + (-39.6 + 10.6i)T^{2} \)
43 \( 1 + (-0.681 + 0.336i)T + (26.1 - 34.1i)T^{2} \)
47 \( 1 + (-5.21 + 1.39i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-7.17 + 10.7i)T + (-20.2 - 48.9i)T^{2} \)
59 \( 1 + (-7.91 - 9.02i)T + (-7.70 + 58.4i)T^{2} \)
61 \( 1 + (0.128 + 1.96i)T + (-60.4 + 7.96i)T^{2} \)
67 \( 1 + (-8.15 - 4.01i)T + (40.7 + 53.1i)T^{2} \)
71 \( 1 + (4.61 + 11.1i)T + (-50.2 + 50.2i)T^{2} \)
73 \( 1 + (0.512 - 1.23i)T + (-51.6 - 51.6i)T^{2} \)
79 \( 1 + (3.36 + 12.5i)T + (-68.4 + 39.5i)T^{2} \)
83 \( 1 + (1.42 + 1.25i)T + (10.8 + 82.2i)T^{2} \)
89 \( 1 + (12.8 - 5.30i)T + (62.9 - 62.9i)T^{2} \)
97 \( 1 + (0.166 + 0.0959i)T + (48.5 + 84.0i)T^{2} \)
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

−10.51027790499758727140606036217, −9.570224840007513726838531485477, −8.880541862330910943207880397134, −8.218778835208681798081152683558, −6.95875402256199960373553748043, −6.06254237471709820511334564622, −5.12729499155457510609229429124, −4.19160041143593314628410827136, −2.49697175322333827302137076095, −0.32712313425996608111693985768, 1.52216763144498054526279054825, 2.53875861867737832984890839179, 3.97569352602417089637457112769, 5.46938563784168282562210494007, 6.69495389838287214534542132445, 7.41917860693289355569323818232, 8.181247694516198819402397936073, 8.984253591863792696422847456694, 10.17604785324226973375957811881, 10.96336337674223453547773549454

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