Properties

Label 2-2e9-512.125-c1-0-1
Degree $2$
Conductor $512$
Sign $-0.0111 + 0.999i$
Analytic cond. $4.08834$
Root an. cond. $2.02196$
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  + (0.746 + 1.20i)2-s + (−2.58 + 1.63i)3-s + (−0.884 + 1.79i)4-s + (0.800 + 0.0590i)5-s + (−3.89 − 1.87i)6-s + (−0.732 + 0.808i)7-s + (−2.81 + 0.278i)8-s + (2.70 − 5.72i)9-s + (0.527 + 1.00i)10-s + (−4.27 + 0.105i)11-s + (−0.650 − 6.07i)12-s + (−1.71 + 5.18i)13-s + (−1.51 − 0.276i)14-s + (−2.16 + 1.15i)15-s + (−2.43 − 3.17i)16-s + (3.01 − 5.63i)17-s + ⋯
L(s)  = 1  + (0.528 + 0.849i)2-s + (−1.49 + 0.943i)3-s + (−0.442 + 0.896i)4-s + (0.358 + 0.0264i)5-s + (−1.58 − 0.766i)6-s + (−0.276 + 0.305i)7-s + (−0.995 + 0.0983i)8-s + (0.902 − 1.90i)9-s + (0.166 + 0.318i)10-s + (−1.28 + 0.0316i)11-s + (−0.187 − 1.75i)12-s + (−0.475 + 1.43i)13-s + (−0.405 − 0.0737i)14-s + (−0.558 + 0.298i)15-s + (−0.609 − 0.793i)16-s + (0.730 − 1.36i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0111 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0111 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(512\)    =    \(2^{9}\)
Sign: $-0.0111 + 0.999i$
Analytic conductor: \(4.08834\)
Root analytic conductor: \(2.02196\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{512} (125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 512,\ (\ :1/2),\ -0.0111 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.246315 - 0.249083i\)
\(L(\frac12)\) \(\approx\) \(0.246315 - 0.249083i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.746 - 1.20i)T \)
good3 \( 1 + (2.58 - 1.63i)T + (1.28 - 2.71i)T^{2} \)
5 \( 1 + (-0.800 - 0.0590i)T + (4.94 + 0.733i)T^{2} \)
7 \( 1 + (0.732 - 0.808i)T + (-0.686 - 6.96i)T^{2} \)
11 \( 1 + (4.27 - 0.105i)T + (10.9 - 0.539i)T^{2} \)
13 \( 1 + (1.71 - 5.18i)T + (-10.4 - 7.74i)T^{2} \)
17 \( 1 + (-3.01 + 5.63i)T + (-9.44 - 14.1i)T^{2} \)
19 \( 1 + (-0.394 + 3.19i)T + (-18.4 - 4.61i)T^{2} \)
23 \( 1 + (-3.23 - 5.39i)T + (-10.8 + 20.2i)T^{2} \)
29 \( 1 + (-4.43 + 1.70i)T + (21.4 - 19.4i)T^{2} \)
31 \( 1 + (2.40 + 1.60i)T + (11.8 + 28.6i)T^{2} \)
37 \( 1 + (1.72 - 6.21i)T + (-31.7 - 19.0i)T^{2} \)
41 \( 1 + (2.23 - 0.331i)T + (39.2 - 11.9i)T^{2} \)
43 \( 1 + (1.20 + 5.37i)T + (-38.8 + 18.3i)T^{2} \)
47 \( 1 + (-2.59 - 0.256i)T + (46.0 + 9.16i)T^{2} \)
53 \( 1 + (4.40 + 1.70i)T + (39.2 + 35.5i)T^{2} \)
59 \( 1 + (13.2 - 4.38i)T + (47.3 - 35.1i)T^{2} \)
61 \( 1 + (6.22 - 1.08i)T + (57.4 - 20.5i)T^{2} \)
67 \( 1 + (0.714 - 1.01i)T + (-22.5 - 63.0i)T^{2} \)
71 \( 1 + (8.12 + 2.90i)T + (54.8 + 45.0i)T^{2} \)
73 \( 1 + (10.4 + 11.5i)T + (-7.15 + 72.6i)T^{2} \)
79 \( 1 + (0.894 + 0.734i)T + (15.4 + 77.4i)T^{2} \)
83 \( 1 + (-3.34 - 12.0i)T + (-71.1 + 42.6i)T^{2} \)
89 \( 1 + (7.37 - 12.3i)T + (-41.9 - 78.4i)T^{2} \)
97 \( 1 + (-0.773 - 3.88i)T + (-89.6 + 37.1i)T^{2} \)
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

−11.87676610893718152508108405597, −10.73937147246325305168986374210, −9.612480416165297372804109069607, −9.263746309826237454542188444516, −7.58490749442119092480380773752, −6.73873889230872134414068798584, −5.78590868558298716197173052488, −5.10069778554613409587479481267, −4.49135603728626638221445380738, −2.99815557440256797276783220047, 0.20705709924827310987296775101, 1.61302045247201522565784168270, 3.05159205586540760910197920669, 4.73110995516875745308650565132, 5.66760090648372103640581254436, 6.00513878957744572016132363003, 7.32576247056621337163457890142, 8.280784332570718544324577911738, 10.17574467459623292105117539472, 10.34715076030964507970321614431

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