↑

Properties

Label 2-24e2-9.7-c1-0-9
Degree $2$
Conductor $576$
Sign $0.800 - 0.598i$
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  + (−1.35 + 1.07i)3-s + (1.18 + 2.05i)5-s + (1.10 − 1.91i)7-s + (0.686 − 2.92i)9-s + (2.96 − 5.14i)11-s + (2.18 + 3.78i)13-s + (−3.82 − 1.51i)15-s + 3.37·17-s − 3.72·19-s + (0.558 + 3.78i)21-s + (1.10 + 1.91i)23-s + (−0.313 + 0.543i)25-s + (2.20 + 4.70i)27-s + (0.186 − 0.322i)29-s + (4.83 + 8.36i)31-s + ⋯
L(s)  = 1  + (−0.783 + 0.621i)3-s + (0.530 + 0.918i)5-s + (0.417 − 0.723i)7-s + (0.228 − 0.973i)9-s + (0.894 − 1.54i)11-s + (0.606 + 1.05i)13-s + (−0.986 − 0.390i)15-s + 0.817·17-s − 0.854·19-s + (0.121 + 0.826i)21-s + (0.230 + 0.399i)23-s + (−0.0627 + 0.108i)25-s + (0.425 + 0.905i)27-s + (0.0345 − 0.0598i)29-s + (0.867 + 1.50i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.800 - 0.598i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.800 - 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $0.800 - 0.598i$
Analytic conductor: \(4.59938\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (385, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ 0.800 - 0.598i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.32845 + 0.441642i\)
\(L(\frac12)\) \(\approx\) \(1.32845 + 0.441642i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.35 - 1.07i)T \)
good5 \( 1 + (-1.18 - 2.05i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-1.10 + 1.91i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.96 + 5.14i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.18 - 3.78i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 3.37T + 17T^{2} \)
19 \( 1 + 3.72T + 19T^{2} \)
23 \( 1 + (-1.10 - 1.91i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.186 + 0.322i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-4.83 - 8.36i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 4T + 37T^{2} \)
41 \( 1 + (0.5 + 0.866i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.96 - 5.14i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.10 + 1.91i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 4T + 53T^{2} \)
59 \( 1 + (5.17 + 8.96i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7.55 + 13.0i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.17 - 8.96i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 4.41T + 71T^{2} \)
73 \( 1 - 4.62T + 73T^{2} \)
79 \( 1 + (4.83 - 8.36i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-7.04 + 12.1i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 1.25T + 89T^{2} \)
97 \( 1 + (4.5 - 7.79i)T + (-48.5 - 84.0i)T^{2} \)
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

−10.85935599149177074343056583564, −10.20295903907495772507659030273, −9.205195924957082817362223173323, −8.301350411431073571722914030630, −6.66551460199922228263098830548, −6.50210420256042733620388400129, −5.35202907133776481856332486765, −4.09056225303278332438360891828, −3.28636046877757041248334689394, −1.25362105105366285452440251435, 1.18111752386244678391138878497, 2.22777933758893959853346528689, 4.30298562117508305496540261411, 5.24833632935598799116507425102, 5.91040269918663873035426325106, 6.93846580985080609306721987968, 8.007661777408575879644315606738, 8.807383288244454580715418280238, 9.822339542975714290426339544157, 10.62027549349251443178193175482

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