↑

Properties

Label 2-2e10-32.21-c1-0-19
Degree $2$
Conductor $1024$
Sign $0.195 + 0.980i$
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  + (−1.59 + 0.660i)3-s + (−0.212 + 0.512i)5-s + (1.69 + 1.69i)7-s + (−0.0177 + 0.0177i)9-s + (−3.44 − 1.42i)11-s + (−2.34 − 5.65i)13-s − 0.956i·15-s − 5.26i·17-s + (1.57 + 3.80i)19-s + (−3.82 − 1.58i)21-s + (−4.31 + 4.31i)23-s + (3.31 + 3.31i)25-s + (1.99 − 4.82i)27-s + (1.23 − 0.512i)29-s − 1.53·31-s + ⋯
L(s)  = 1  + (−0.920 + 0.381i)3-s + (−0.0949 + 0.229i)5-s + (0.642 + 0.642i)7-s + (−0.00590 + 0.00590i)9-s + (−1.03 − 0.429i)11-s + (−0.649 − 1.56i)13-s − 0.247i·15-s − 1.27i·17-s + (0.361 + 0.871i)19-s + (−0.835 − 0.346i)21-s + (−0.899 + 0.899i)23-s + (0.663 + 0.663i)25-s + (0.384 − 0.927i)27-s + (0.229 − 0.0951i)29-s − 0.274·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.195 + 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.195 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1024\)    =    \(2^{10}\)
Sign: $0.195 + 0.980i$
Analytic conductor: \(8.17668\)
Root analytic conductor: \(2.85948\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1024} (385, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1024,\ (\ :1/2),\ 0.195 + 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5963463508\)
\(L(\frac12)\) \(\approx\) \(0.5963463508\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.59 - 0.660i)T + (2.12 - 2.12i)T^{2} \)
5 \( 1 + (0.212 - 0.512i)T + (-3.53 - 3.53i)T^{2} \)
7 \( 1 + (-1.69 - 1.69i)T + 7iT^{2} \)
11 \( 1 + (3.44 + 1.42i)T + (7.77 + 7.77i)T^{2} \)
13 \( 1 + (2.34 + 5.65i)T + (-9.19 + 9.19i)T^{2} \)
17 \( 1 + 5.26iT - 17T^{2} \)
19 \( 1 + (-1.57 - 3.80i)T + (-13.4 + 13.4i)T^{2} \)
23 \( 1 + (4.31 - 4.31i)T - 23iT^{2} \)
29 \( 1 + (-1.23 + 0.512i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 + 1.53T + 31T^{2} \)
37 \( 1 + (-1.49 + 3.61i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (-8.69 + 8.69i)T - 41iT^{2} \)
43 \( 1 + (-0.511 - 0.211i)T + (30.4 + 30.4i)T^{2} \)
47 \( 1 + 9.73iT - 47T^{2} \)
53 \( 1 + (6.73 + 2.78i)T + (37.4 + 37.4i)T^{2} \)
59 \( 1 + (-1.14 + 2.76i)T + (-41.7 - 41.7i)T^{2} \)
61 \( 1 + (7.40 - 3.06i)T + (43.1 - 43.1i)T^{2} \)
67 \( 1 + (-5.33 + 2.20i)T + (47.3 - 47.3i)T^{2} \)
71 \( 1 + (9.45 + 9.45i)T + 71iT^{2} \)
73 \( 1 + (-1.69 + 1.69i)T - 73iT^{2} \)
79 \( 1 + 12.8iT - 79T^{2} \)
83 \( 1 + (0.360 + 0.870i)T + (-58.6 + 58.6i)T^{2} \)
89 \( 1 + (-1.14 - 1.14i)T + 89iT^{2} \)
97 \( 1 + 1.83T + 97T^{2} \)
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

−10.06702018310218689965581815254, −8.977089186291229145933888179802, −7.84593720280132102739256423915, −7.54025148248032544853132903429, −5.93752116799537346043386671287, −5.37550790705034415574638183603, −4.95765485523655012712565553886, −3.35984876501527087907563469262, −2.35269146799655787619811597282, −0.32540050011981413906178336287, 1.25717925560309613086482086409, 2.58206930411908315145112141071, 4.41068719288114514210047085000, 4.70097335929552900937032306490, 5.97458558880843867323162084537, 6.68819647400356700333553167361, 7.54180398408535115732325979527, 8.330262099810082104750227407267, 9.353463517480823487748745720971, 10.31652965470163454511158975249

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