Properties

Label 2-24e2-576.517-c1-0-68
Degree $2$
Conductor $576$
Sign $0.932 + 0.360i$
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  + (1.28 − 0.588i)2-s + (−0.509 + 1.65i)3-s + (1.30 − 1.51i)4-s + (0.263 − 0.300i)5-s + (0.318 + 2.42i)6-s + (2.25 − 0.297i)7-s + (0.791 − 2.71i)8-s + (−2.48 − 1.68i)9-s + (0.162 − 0.542i)10-s + (2.51 − 5.09i)11-s + (1.83 + 2.93i)12-s + (−0.728 + 2.14i)13-s + (2.72 − 1.71i)14-s + (0.363 + 0.590i)15-s + (−0.578 − 3.95i)16-s + (0.935 − 0.935i)17-s + ⋯
L(s)  = 1  + (0.909 − 0.415i)2-s + (−0.294 + 0.955i)3-s + (0.653 − 0.756i)4-s + (0.117 − 0.134i)5-s + (0.129 + 0.991i)6-s + (0.853 − 0.112i)7-s + (0.279 − 0.960i)8-s + (−0.826 − 0.562i)9-s + (0.0513 − 0.171i)10-s + (0.758 − 1.53i)11-s + (0.530 + 0.847i)12-s + (−0.202 + 0.595i)13-s + (0.729 − 0.457i)14-s + (0.0938 + 0.152i)15-s + (−0.144 − 0.989i)16-s + (0.227 − 0.227i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.932 + 0.360i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.932 + 0.360i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $0.932 + 0.360i$
Analytic conductor: \(4.59938\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (517, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ 0.932 + 0.360i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.48841 - 0.463988i\)
\(L(\frac12)\) \(\approx\) \(2.48841 - 0.463988i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.28 + 0.588i)T \)
3 \( 1 + (0.509 - 1.65i)T \)
good5 \( 1 + (-0.263 + 0.300i)T + (-0.652 - 4.95i)T^{2} \)
7 \( 1 + (-2.25 + 0.297i)T + (6.76 - 1.81i)T^{2} \)
11 \( 1 + (-2.51 + 5.09i)T + (-6.69 - 8.72i)T^{2} \)
13 \( 1 + (0.728 - 2.14i)T + (-10.3 - 7.91i)T^{2} \)
17 \( 1 + (-0.935 + 0.935i)T - 17iT^{2} \)
19 \( 1 + (-0.719 - 3.61i)T + (-17.5 + 7.27i)T^{2} \)
23 \( 1 + (0.735 - 5.59i)T + (-22.2 - 5.95i)T^{2} \)
29 \( 1 + (-8.78 - 0.575i)T + (28.7 + 3.78i)T^{2} \)
31 \( 1 + (3.85 + 2.22i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.930 + 4.67i)T + (-34.1 - 14.1i)T^{2} \)
41 \( 1 + (0.843 - 6.40i)T + (-39.6 - 10.6i)T^{2} \)
43 \( 1 + (10.5 + 5.19i)T + (26.1 + 34.1i)T^{2} \)
47 \( 1 + (6.67 + 1.78i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-0.873 - 1.30i)T + (-20.2 + 48.9i)T^{2} \)
59 \( 1 + (-2.57 + 2.93i)T + (-7.70 - 58.4i)T^{2} \)
61 \( 1 + (0.219 - 3.34i)T + (-60.4 - 7.96i)T^{2} \)
67 \( 1 + (5.54 - 2.73i)T + (40.7 - 53.1i)T^{2} \)
71 \( 1 + (-1.44 + 3.48i)T + (-50.2 - 50.2i)T^{2} \)
73 \( 1 + (-5.52 - 13.3i)T + (-51.6 + 51.6i)T^{2} \)
79 \( 1 + (-3.11 + 11.6i)T + (-68.4 - 39.5i)T^{2} \)
83 \( 1 + (5.97 - 5.24i)T + (10.8 - 82.2i)T^{2} \)
89 \( 1 + (2.28 + 0.947i)T + (62.9 + 62.9i)T^{2} \)
97 \( 1 + (8.69 - 5.01i)T + (48.5 - 84.0i)T^{2} \)
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

−10.95476098967933951623284198621, −9.984112923266399501563357251019, −9.180555836170651186116570744954, −8.135337133202792732311173031993, −6.68470093288042384172032317047, −5.70417079773032495014243472870, −5.06156977229546851563827609597, −3.96838979236499488634265284450, −3.22097639920429479936288793668, −1.39292704588641196894712131942, 1.73326868122605458776345584200, 2.80172349153289067646699152679, 4.54717825670509167739726268585, 5.10809627069998938919880300695, 6.46565660847223629638994816780, 6.87138086519621187877264744188, 7.919904971567657035473386967349, 8.541062078155014146560442187855, 10.12428899962205173161119551724, 11.09696202618648410176929501577

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