Properties

Label 2-2e10-16.13-c1-0-0
Degree $2$
Conductor $1024$
Sign $-0.382 - 0.923i$
Analytic cond. $8.17668$
Root an. cond. $2.85948$
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  + (−0.414 − 0.414i)3-s − 2.65i·9-s + (−4.41 + 4.41i)11-s − 5.65·17-s + (5.24 + 5.24i)19-s + 5i·25-s + (−2.34 + 2.34i)27-s + 3.65·33-s − 6i·41-s + (−9.24 + 9.24i)43-s + 7·49-s + (2.34 + 2.34i)51-s − 4.34i·57-s + (−10.0 + 10.0i)59-s + (2.75 + 2.75i)67-s + ⋯
L(s)  = 1  + (−0.239 − 0.239i)3-s − 0.885i·9-s + (−1.33 + 1.33i)11-s − 1.37·17-s + (1.20 + 1.20i)19-s + i·25-s + (−0.450 + 0.450i)27-s + 0.636·33-s − 0.937i·41-s + (−1.40 + 1.40i)43-s + 49-s + (0.328 + 0.328i)51-s − 0.575i·57-s + (−1.31 + 1.31i)59-s + (0.336 + 0.336i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 - 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1024\)    =    \(2^{10}\)
Sign: $-0.382 - 0.923i$
Analytic conductor: \(8.17668\)
Root analytic conductor: \(2.85948\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1024} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1024,\ (\ :1/2),\ -0.382 - 0.923i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6502851402\)
\(L(\frac12)\) \(\approx\) \(0.6502851402\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
good3 \( 1 + (0.414 + 0.414i)T + 3iT^{2} \)
5 \( 1 - 5iT^{2} \)
7 \( 1 - 7T^{2} \)
11 \( 1 + (4.41 - 4.41i)T - 11iT^{2} \)
13 \( 1 + 13iT^{2} \)
17 \( 1 + 5.65T + 17T^{2} \)
19 \( 1 + (-5.24 - 5.24i)T + 19iT^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + 29iT^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 37iT^{2} \)
41 \( 1 + 6iT - 41T^{2} \)
43 \( 1 + (9.24 - 9.24i)T - 43iT^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 53iT^{2} \)
59 \( 1 + (10.0 - 10.0i)T - 59iT^{2} \)
61 \( 1 + 61iT^{2} \)
67 \( 1 + (-2.75 - 2.75i)T + 67iT^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 - 16.9iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + (7.58 + 7.58i)T + 83iT^{2} \)
89 \( 1 - 5.65iT - 89T^{2} \)
97 \( 1 + 16.9T + 97T^{2} \)
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

−10.06863023930523732802803062902, −9.539772300198722957139308700564, −8.543575747329207558032421293984, −7.49280995378075245156668909699, −7.01126384634165123003215594472, −5.89916428668121256222192473734, −5.08484612656141040723452359098, −4.04605528702453039880497134791, −2.84178696095739283497759850503, −1.57642784869622628734582406800, 0.29313741563596812817287784154, 2.28998756619477056284438854666, 3.19946971412468669710405256459, 4.65362040263685293880417108502, 5.22345022923748934383368639555, 6.17481309967527955008673703619, 7.22283249456590831896030918300, 8.128631745379989583821837749745, 8.733379719437086568597191625201, 9.788739478242926212696272276202

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