Properties

Label 2-2e9-512.117-c1-0-10
Degree $2$
Conductor $512$
Sign $-0.442 + 0.896i$
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  + (0.126 + 1.40i)2-s + (0.0798 + 3.25i)3-s + (−1.96 + 0.356i)4-s + (−0.147 − 0.533i)5-s + (−4.56 + 0.523i)6-s + (−1.92 + 4.06i)7-s + (−0.750 − 2.72i)8-s + (−7.57 + 0.371i)9-s + (0.733 − 0.275i)10-s + (4.02 + 2.83i)11-s + (−1.31 − 6.37i)12-s + (4.49 − 0.554i)13-s + (−5.97 − 2.19i)14-s + (1.72 − 0.523i)15-s + (3.74 − 1.40i)16-s + (1.08 − 3.56i)17-s + ⋯
L(s)  = 1  + (0.0894 + 0.995i)2-s + (0.0460 + 1.87i)3-s + (−0.984 + 0.178i)4-s + (−0.0660 − 0.238i)5-s + (−1.86 + 0.213i)6-s + (−0.727 + 1.53i)7-s + (−0.265 − 0.964i)8-s + (−2.52 + 0.123i)9-s + (0.231 − 0.0871i)10-s + (1.21 + 0.855i)11-s + (−0.379 − 1.83i)12-s + (1.24 − 0.153i)13-s + (−1.59 − 0.586i)14-s + (0.445 − 0.135i)15-s + (0.936 − 0.350i)16-s + (0.262 − 0.864i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.442 + 0.896i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.442 + 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(512\)    =    \(2^{9}\)
Sign: $-0.442 + 0.896i$
Analytic conductor: \(4.08834\)
Root analytic conductor: \(2.02196\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{512} (117, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 512,\ (\ :1/2),\ -0.442 + 0.896i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.607960 - 0.977837i\)
\(L(\frac12)\) \(\approx\) \(0.607960 - 0.977837i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.126 - 1.40i)T \)
good3 \( 1 + (-0.0798 - 3.25i)T + (-2.99 + 0.147i)T^{2} \)
5 \( 1 + (0.147 + 0.533i)T + (-4.28 + 2.57i)T^{2} \)
7 \( 1 + (1.92 - 4.06i)T + (-4.44 - 5.41i)T^{2} \)
11 \( 1 + (-4.02 - 2.83i)T + (3.70 + 10.3i)T^{2} \)
13 \( 1 + (-4.49 + 0.554i)T + (12.6 - 3.15i)T^{2} \)
17 \( 1 + (-1.08 + 3.56i)T + (-14.1 - 9.44i)T^{2} \)
19 \( 1 + (-0.0701 + 0.951i)T + (-18.7 - 2.78i)T^{2} \)
23 \( 1 + (3.39 - 2.52i)T + (6.67 - 22.0i)T^{2} \)
29 \( 1 + (4.19 - 0.941i)T + (26.2 - 12.3i)T^{2} \)
31 \( 1 + (-2.82 - 4.22i)T + (-11.8 + 28.6i)T^{2} \)
37 \( 1 + (-0.201 - 0.399i)T + (-22.0 + 29.7i)T^{2} \)
41 \( 1 + (-6.47 - 3.88i)T + (19.3 + 36.1i)T^{2} \)
43 \( 1 + (3.43 + 3.60i)T + (-2.10 + 42.9i)T^{2} \)
47 \( 1 + (5.79 + 4.75i)T + (9.16 + 46.0i)T^{2} \)
53 \( 1 + (3.38 + 0.759i)T + (47.9 + 22.6i)T^{2} \)
59 \( 1 + (-0.624 + 5.06i)T + (-57.2 - 14.3i)T^{2} \)
61 \( 1 + (-3.74 - 8.45i)T + (-40.9 + 45.1i)T^{2} \)
67 \( 1 + (4.48 - 11.6i)T + (-49.6 - 44.9i)T^{2} \)
71 \( 1 + (-1.09 - 1.20i)T + (-6.95 + 70.6i)T^{2} \)
73 \( 1 + (1.70 + 3.60i)T + (-46.3 + 56.4i)T^{2} \)
79 \( 1 + (0.523 - 5.31i)T + (-77.4 - 15.4i)T^{2} \)
83 \( 1 + (-3.94 + 7.82i)T + (-49.4 - 66.6i)T^{2} \)
89 \( 1 + (-10.5 - 7.83i)T + (25.8 + 85.1i)T^{2} \)
97 \( 1 + (-12.3 - 2.46i)T + (89.6 + 37.1i)T^{2} \)
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

−11.53803532965717948116944576183, −10.16351810861239826983460948922, −9.402821013127212351149909847987, −9.030122816194596537643141895702, −8.335927591329912169542928053075, −6.61305290362070919748300489063, −5.76808893939464971315782072897, −5.00804724360726694738382481136, −4.01708831894442955135700239768, −3.12303957070206544105802868183, 0.72908436404806121723244588362, 1.58474598107896865232045057145, 3.25004439613298066395030909343, 3.93419692567939161405242900981, 6.11677785283242767843449144223, 6.41702019599063855243014482180, 7.65476236295908197317819840559, 8.418634247686941517842965597449, 9.395388570443878957970844573662, 10.73936379594858065501875826624

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