↑

Properties

Label 2-24e2-72.11-c1-0-20
Degree $2$
Conductor $576$
Sign $-0.941 + 0.335i$
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  + (0.231 − 1.71i)3-s + (0.959 + 1.66i)5-s + (−2.63 − 1.51i)7-s + (−2.89 − 0.795i)9-s + (−3.87 − 2.23i)11-s + (−3.64 + 2.10i)13-s + (3.07 − 1.26i)15-s − 5.94i·17-s − 1.59·19-s + (−3.21 + 4.16i)21-s + (−1.13 − 1.97i)23-s + (0.658 − 1.14i)25-s + (−2.03 + 4.78i)27-s + (1.90 − 3.29i)29-s + (8.67 − 5.00i)31-s + ⋯
L(s)  = 1  + (0.133 − 0.990i)3-s + (0.429 + 0.743i)5-s + (−0.994 − 0.573i)7-s + (−0.964 − 0.265i)9-s + (−1.16 − 0.673i)11-s + (−1.01 + 0.584i)13-s + (0.794 − 0.325i)15-s − 1.44i·17-s − 0.365·19-s + (−0.701 + 0.908i)21-s + (−0.237 − 0.411i)23-s + (0.131 − 0.228i)25-s + (−0.392 + 0.919i)27-s + (0.353 − 0.611i)29-s + (1.55 − 0.899i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.941 + 0.335i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.941 + 0.335i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $-0.941 + 0.335i$
Analytic conductor: \(4.59938\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (479, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ -0.941 + 0.335i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.118562 - 0.685914i\)
\(L(\frac12)\) \(\approx\) \(0.118562 - 0.685914i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.231 + 1.71i)T \)
good5 \( 1 + (-0.959 - 1.66i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (2.63 + 1.51i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (3.87 + 2.23i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.64 - 2.10i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 5.94iT - 17T^{2} \)
19 \( 1 + 1.59T + 19T^{2} \)
23 \( 1 + (1.13 + 1.97i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.90 + 3.29i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-8.67 + 5.00i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 7.53iT - 37T^{2} \)
41 \( 1 + (4.08 - 2.35i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.83 + 3.17i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.77 - 4.79i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 5.72T + 53T^{2} \)
59 \( 1 + (1.08 - 0.626i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.295 + 0.170i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.66 - 6.35i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 15.9T + 71T^{2} \)
73 \( 1 - 16.3T + 73T^{2} \)
79 \( 1 + (0.0479 + 0.0277i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-8.37 - 4.83i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 14.4iT - 89T^{2} \)
97 \( 1 + (1.99 - 3.44i)T + (-48.5 - 84.0i)T^{2} \)
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

−10.17736325129761486665753472675, −9.683595231275507741101950886756, −8.378197191939394799039234333834, −7.49712370542535926035181197580, −6.69325453769870276653889118011, −6.13961678648675405962894512417, −4.77374106986850521586586331380, −2.98343151105061022469053380297, −2.51592206671732975025665470982, −0.35188368029605454231757906577, 2.31030688012956496410421257394, 3.37836015619731073739181368477, 4.77346820626901257684429931035, 5.36653703667199087534860165246, 6.34788753685997895847361579348, 7.81818669258913393277296953880, 8.666706160151335841187799339844, 9.494226859860741927407158435072, 10.15538656815102739063266942260, 10.69481715123180492203600016319

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