↑

Properties

Label 2-24e2-72.59-c1-0-11
Degree $2$
Conductor $576$
Sign $0.691 - 0.722i$
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  + (−1.31 + 1.12i)3-s + (−1.21 + 2.09i)5-s + (3.96 − 2.28i)7-s + (0.456 − 2.96i)9-s + (3.73 − 2.15i)11-s + (−1.88 − 1.08i)13-s + (−0.774 − 4.12i)15-s + 3.90i·17-s + 5.93·19-s + (−2.62 + 7.47i)21-s + (−2.94 + 5.10i)23-s + (−0.437 − 0.757i)25-s + (2.74 + 4.41i)27-s + (−0.776 − 1.34i)29-s + (0.925 + 0.534i)31-s + ⋯
L(s)  = 1  + (−0.758 + 0.651i)3-s + (−0.541 + 0.938i)5-s + (1.49 − 0.864i)7-s + (0.152 − 0.988i)9-s + (1.12 − 0.650i)11-s + (−0.522 − 0.301i)13-s + (−0.199 − 1.06i)15-s + 0.947i·17-s + 1.36·19-s + (−0.573 + 1.63i)21-s + (−0.614 + 1.06i)23-s + (−0.0874 − 0.151i)25-s + (0.528 + 0.849i)27-s + (−0.144 − 0.249i)29-s + (0.166 + 0.0959i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.691 - 0.722i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.691 - 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $0.691 - 0.722i$
Analytic conductor: \(4.59938\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (95, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ 0.691 - 0.722i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.17423 + 0.501450i\)
\(L(\frac12)\) \(\approx\) \(1.17423 + 0.501450i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.31 - 1.12i)T \)
good5 \( 1 + (1.21 - 2.09i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-3.96 + 2.28i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-3.73 + 2.15i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.88 + 1.08i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 3.90iT - 17T^{2} \)
19 \( 1 - 5.93T + 19T^{2} \)
23 \( 1 + (2.94 - 5.10i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.776 + 1.34i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.925 - 0.534i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 2.02iT - 37T^{2} \)
41 \( 1 + (-10.4 - 6.04i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.995 + 1.72i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.70 + 6.41i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 3.97T + 53T^{2} \)
59 \( 1 + (0.294 + 0.169i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (8.33 - 4.80i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.71 + 2.97i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 2.32T + 71T^{2} \)
73 \( 1 + 2.37T + 73T^{2} \)
79 \( 1 + (-5.38 + 3.10i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.40 + 3.11i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 4.90iT - 89T^{2} \)
97 \( 1 + (1.32 + 2.28i)T + (-48.5 + 84.0i)T^{2} \)
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

−10.90678099311329340947095068699, −10.26395557395034814689451360225, −9.253233260882468608107397644716, −7.959948604034986335755666418706, −7.32435580177310930601697386603, −6.25503603016738864171941177686, −5.21390679322731788257558064394, −4.13419682363143177074693862537, −3.43907368788077648396535321889, −1.23734647006744256142850373584, 1.06097117813507870020647224634, 2.22047156592260495963151271332, 4.47840744854785293303626846590, 4.89807066770983551091809837076, 5.90108548077156834247804012836, 7.19181033036848999342055542366, 7.84606537837826379652030634901, 8.767146783866427229875028007216, 9.570943564904109440226721743662, 11.00336482325133260738197531072

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