↑

Properties

Label 2-24e2-144.11-c1-0-21
Degree $2$
Conductor $576$
Sign $-0.878 - 0.478i$
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  + (−0.117 − 1.72i)3-s + (−3.58 − 0.961i)5-s + (1.29 − 2.23i)7-s + (−2.97 + 0.406i)9-s + (2.02 − 0.541i)11-s + (−0.998 − 0.267i)13-s + (−1.23 + 6.31i)15-s + 4.13i·17-s + (−2.49 − 2.49i)19-s + (−4.01 − 1.96i)21-s + (−7.01 + 4.05i)23-s + (7.62 + 4.40i)25-s + (1.05 + 5.08i)27-s + (−7.02 + 1.88i)29-s + (2.03 − 1.17i)31-s + ⋯
L(s)  = 1  + (−0.0679 − 0.997i)3-s + (−1.60 − 0.430i)5-s + (0.488 − 0.845i)7-s + (−0.990 + 0.135i)9-s + (0.609 − 0.163i)11-s + (−0.276 − 0.0741i)13-s + (−0.319 + 1.63i)15-s + 1.00i·17-s + (−0.571 − 0.571i)19-s + (−0.877 − 0.429i)21-s + (−1.46 + 0.845i)23-s + (1.52 + 0.880i)25-s + (0.202 + 0.979i)27-s + (−1.30 + 0.349i)29-s + (0.365 − 0.210i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.878 - 0.478i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.878 - 0.478i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $-0.878 - 0.478i$
Analytic conductor: \(4.59938\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (335, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ -0.878 - 0.478i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.101177 + 0.397217i\)
\(L(\frac12)\) \(\approx\) \(0.101177 + 0.397217i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.117 + 1.72i)T \)
good5 \( 1 + (3.58 + 0.961i)T + (4.33 + 2.5i)T^{2} \)
7 \( 1 + (-1.29 + 2.23i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.02 + 0.541i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (0.998 + 0.267i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 - 4.13iT - 17T^{2} \)
19 \( 1 + (2.49 + 2.49i)T + 19iT^{2} \)
23 \( 1 + (7.01 - 4.05i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (7.02 - 1.88i)T + (25.1 - 14.5i)T^{2} \)
31 \( 1 + (-2.03 + 1.17i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.75 + 4.75i)T + 37iT^{2} \)
41 \( 1 + (-0.636 - 1.10i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.389 + 1.45i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (-3.60 + 6.24i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.546 + 0.546i)T - 53iT^{2} \)
59 \( 1 + (-2.14 + 8.00i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-1.81 - 6.77i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (0.751 - 2.80i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 6.59iT - 71T^{2} \)
73 \( 1 + 8.78iT - 73T^{2} \)
79 \( 1 + (5.64 + 3.26i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.06 + 7.71i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 + 14.1T + 89T^{2} \)
97 \( 1 + (-6.54 + 11.3i)T + (-48.5 - 84.0i)T^{2} \)
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

−10.52902248493732785071550550557, −8.985897164325686977227036432420, −8.186477215329349970814375211497, −7.59447285588256397702883843741, −6.91463937935464685611014259228, −5.66564355503608477214314950992, −4.29672411130096891100596623407, −3.61244471507880729258901407029, −1.69719156659346661498582571534, −0.23273787827424192568697650263, 2.57628320836587194158559507997, 3.82516062136685680341241321866, 4.43700863131427647064668229248, 5.56605148123534512649151555231, 6.77156503918865180512258443487, 7.932607250083427387255715025189, 8.523211420378442068772829641897, 9.481180290834925707009086305508, 10.45242267326785492722665976291, 11.36613840602248143325668776314

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