↑

Properties

Label 2-24e2-144.83-c1-0-5
Degree $2$
Conductor $576$
Sign $-0.769 - 0.638i$
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  + (0.427 + 1.67i)3-s + (−0.0458 + 0.170i)5-s + (−1.17 + 2.03i)7-s + (−2.63 + 1.43i)9-s + (0.0913 + 0.340i)11-s + (0.399 − 1.49i)13-s + (−0.306 − 0.00382i)15-s + 3.58i·17-s + (−5.36 + 5.36i)19-s + (−3.91 − 1.10i)21-s + (−0.165 + 0.0953i)23-s + (4.30 + 2.48i)25-s + (−3.53 − 3.80i)27-s + (−2.43 − 9.10i)29-s + (−3.43 + 1.98i)31-s + ⋯
L(s)  = 1  + (0.246 + 0.969i)3-s + (−0.0204 + 0.0764i)5-s + (−0.443 + 0.768i)7-s + (−0.878 + 0.478i)9-s + (0.0275 + 0.102i)11-s + (0.110 − 0.413i)13-s + (−0.0791 − 0.000988i)15-s + 0.868i·17-s + (−1.23 + 1.23i)19-s + (−0.854 − 0.240i)21-s + (−0.0344 + 0.0198i)23-s + (0.860 + 0.496i)25-s + (−0.680 − 0.733i)27-s + (−0.452 − 1.69i)29-s + (−0.617 + 0.356i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.769 - 0.638i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.769 - 0.638i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $-0.769 - 0.638i$
Analytic conductor: \(4.59938\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ -0.769 - 0.638i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.388408 + 1.07593i\)
\(L(\frac12)\) \(\approx\) \(0.388408 + 1.07593i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.427 - 1.67i)T \)
good5 \( 1 + (0.0458 - 0.170i)T + (-4.33 - 2.5i)T^{2} \)
7 \( 1 + (1.17 - 2.03i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.0913 - 0.340i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (-0.399 + 1.49i)T + (-11.2 - 6.5i)T^{2} \)
17 \( 1 - 3.58iT - 17T^{2} \)
19 \( 1 + (5.36 - 5.36i)T - 19iT^{2} \)
23 \( 1 + (0.165 - 0.0953i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.43 + 9.10i)T + (-25.1 + 14.5i)T^{2} \)
31 \( 1 + (3.43 - 1.98i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.28 - 3.28i)T - 37iT^{2} \)
41 \( 1 + (-4.25 - 7.37i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.09 + 1.09i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (-4.93 + 8.53i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.83 - 4.83i)T + 53iT^{2} \)
59 \( 1 + (-2.68 - 0.720i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (7.97 - 2.13i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (11.4 + 3.06i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 1.13iT - 71T^{2} \)
73 \( 1 - 5.67iT - 73T^{2} \)
79 \( 1 + (-12.8 - 7.42i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-12.3 + 3.31i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + 3.05T + 89T^{2} \)
97 \( 1 + (0.996 - 1.72i)T + (-48.5 - 84.0i)T^{2} \)
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

−10.75546790147094823202502990573, −10.27772330157513393828592521034, −9.288776500991284865357782371781, −8.603692059519650053682575623553, −7.76816568851204026148756944978, −6.22415412467884086355588987076, −5.63240080703539929660874018376, −4.35227388523480307285147481407, −3.44098373339593491268142897822, −2.24108988634808591697441503999, 0.60838022626162135068420152669, 2.23556737363903290752247014418, 3.44166959473636057192440469997, 4.71269455356004737715356676442, 6.04583878548493128068806189599, 7.01736908919818669035952308573, 7.37844408053870823169626024554, 8.802891343242643251780061542309, 9.138598027866464519197536854826, 10.61197271101100922520693985046

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