# Properties

 Label 2-24e2-576.205-c1-0-80 Degree $2$ Conductor $576$ Sign $-0.999 + 0.00130i$ Analytic cond. $4.59938$ Root an. cond. $2.14461$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Learn more

## Dirichlet series

 L(s)  = 1 + (−1.39 + 0.208i)2-s + (0.0768 − 1.73i)3-s + (1.91 − 0.582i)4-s + (−2.08 − 2.38i)5-s + (0.252 + 2.43i)6-s + (2.53 + 0.333i)7-s + (−2.55 + 1.21i)8-s + (−2.98 − 0.266i)9-s + (3.41 + 2.89i)10-s + (−0.249 − 0.506i)11-s + (−0.860 − 3.35i)12-s + (−1.58 − 4.67i)13-s + (−3.61 + 0.0608i)14-s + (−4.27 + 3.42i)15-s + (3.32 − 2.22i)16-s + (1.15 + 1.15i)17-s + ⋯
 L(s)  = 1 + (−0.989 + 0.147i)2-s + (0.0443 − 0.999i)3-s + (0.956 − 0.291i)4-s + (−0.933 − 1.06i)5-s + (0.103 + 0.994i)6-s + (0.958 + 0.126i)7-s + (−0.903 + 0.428i)8-s + (−0.996 − 0.0886i)9-s + (1.08 + 0.915i)10-s + (−0.0752 − 0.152i)11-s + (−0.248 − 0.968i)12-s + (−0.439 − 1.29i)13-s + (−0.966 + 0.0162i)14-s + (−1.10 + 0.885i)15-s + (0.830 − 0.557i)16-s + (0.280 + 0.280i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.00130i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.00130i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$576$$    =    $$2^{6} \cdot 3^{2}$$ Sign: $-0.999 + 0.00130i$ Analytic conductor: $$4.59938$$ Root analytic conductor: $$2.14461$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{576} (205, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 576,\ (\ :1/2),\ -0.999 + 0.00130i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.000361988 - 0.555352i$$ $$L(\frac12)$$ $$\approx$$ $$0.000361988 - 0.555352i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (1.39 - 0.208i)T$$
3 $$1 + (-0.0768 + 1.73i)T$$
good5 $$1 + (2.08 + 2.38i)T + (-0.652 + 4.95i)T^{2}$$
7 $$1 + (-2.53 - 0.333i)T + (6.76 + 1.81i)T^{2}$$
11 $$1 + (0.249 + 0.506i)T + (-6.69 + 8.72i)T^{2}$$
13 $$1 + (1.58 + 4.67i)T + (-10.3 + 7.91i)T^{2}$$
17 $$1 + (-1.15 - 1.15i)T + 17iT^{2}$$
19 $$1 + (-0.747 + 3.75i)T + (-17.5 - 7.27i)T^{2}$$
23 $$1 + (-0.00142 - 0.0108i)T + (-22.2 + 5.95i)T^{2}$$
29 $$1 + (4.49 - 0.294i)T + (28.7 - 3.78i)T^{2}$$
31 $$1 + (3.52 - 2.03i)T + (15.5 - 26.8i)T^{2}$$
37 $$1 + (-0.974 - 4.90i)T + (-34.1 + 14.1i)T^{2}$$
41 $$1 + (-0.378 - 2.87i)T + (-39.6 + 10.6i)T^{2}$$
43 $$1 + (6.76 - 3.33i)T + (26.1 - 34.1i)T^{2}$$
47 $$1 + (-3.55 + 0.952i)T + (40.7 - 23.5i)T^{2}$$
53 $$1 + (-7.04 + 10.5i)T + (-20.2 - 48.9i)T^{2}$$
59 $$1 + (-0.853 - 0.973i)T + (-7.70 + 58.4i)T^{2}$$
61 $$1 + (-0.584 - 8.91i)T + (-60.4 + 7.96i)T^{2}$$
67 $$1 + (7.01 + 3.45i)T + (40.7 + 53.1i)T^{2}$$
71 $$1 + (4.98 + 12.0i)T + (-50.2 + 50.2i)T^{2}$$
73 $$1 + (6.20 - 14.9i)T + (-51.6 - 51.6i)T^{2}$$
79 $$1 + (4.48 + 16.7i)T + (-68.4 + 39.5i)T^{2}$$
83 $$1 + (4.91 + 4.30i)T + (10.8 + 82.2i)T^{2}$$
89 $$1 + (-3.90 + 1.61i)T + (62.9 - 62.9i)T^{2}$$
97 $$1 + (-12.9 - 7.50i)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

−10.29129858129538001228730157977, −8.962483156394720355255285862129, −8.371474390403219195799918529938, −7.80858821877946929564210485848, −7.16351357716178053734515592637, −5.75333290547199750682515108997, −4.94090908035187863431183839672, −3.08346885263933403435949532251, −1.59578944309933010249539563646, −0.42986154507983399988423184352, 2.13533742173739404829178110945, 3.44811833607096989960938479718, 4.30074763864824357260687066005, 5.74738765279955318943929587674, 7.12015819851608117232372384569, 7.64651590657266574927925957086, 8.613604544962218415424213235603, 9.495357623243124323771705427442, 10.33481457165507448245192403387, 11.09421316169065968935331606388