↑

Properties

Label 2-2e10-32.29-c1-0-5
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  + (−2.90 − 1.20i)3-s + (1.21 + 2.92i)5-s + (0.933 − 0.933i)7-s + (4.84 + 4.84i)9-s + (−1.05 + 0.435i)11-s + (1.09 − 2.65i)13-s − 9.94i·15-s − 1.61i·17-s + (−2.34 + 5.66i)19-s + (−3.82 + 1.58i)21-s + (1.67 + 1.67i)23-s + (−3.56 + 3.56i)25-s + (−4.62 − 11.1i)27-s + (−7.06 − 2.92i)29-s + 1.53·31-s + ⋯
L(s)  = 1  + (−1.67 − 0.693i)3-s + (0.542 + 1.30i)5-s + (0.352 − 0.352i)7-s + (1.61 + 1.61i)9-s + (−0.317 + 0.131i)11-s + (0.304 − 0.735i)13-s − 2.56i·15-s − 0.390i·17-s + (−0.538 + 1.29i)19-s + (−0.835 + 0.346i)21-s + (0.350 + 0.350i)23-s + (−0.712 + 0.712i)25-s + (−0.890 − 2.15i)27-s + (−1.31 − 0.543i)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: $\chi_{1024} (641, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1024,\ (\ :1/2),\ 0.195 - 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7751478867\)
\(L(\frac12)\) \(\approx\) \(0.7751478867\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.90 + 1.20i)T + (2.12 + 2.12i)T^{2} \)
5 \( 1 + (-1.21 - 2.92i)T + (-3.53 + 3.53i)T^{2} \)
7 \( 1 + (-0.933 + 0.933i)T - 7iT^{2} \)
11 \( 1 + (1.05 - 0.435i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 + (-1.09 + 2.65i)T + (-9.19 - 9.19i)T^{2} \)
17 \( 1 + 1.61iT - 17T^{2} \)
19 \( 1 + (2.34 - 5.66i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (-1.67 - 1.67i)T + 23iT^{2} \)
29 \( 1 + (7.06 + 2.92i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 - 1.53T + 31T^{2} \)
37 \( 1 + (-2.08 - 5.04i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (1.03 + 1.03i)T + 41iT^{2} \)
43 \( 1 + (-3.98 + 1.64i)T + (30.4 - 30.4i)T^{2} \)
47 \( 1 - 2.97iT - 47T^{2} \)
53 \( 1 + (10.1 - 4.21i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (-3.34 - 8.08i)T + (-41.7 + 41.7i)T^{2} \)
61 \( 1 + (-12.6 - 5.23i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 + (-11.8 - 4.91i)T + (47.3 + 47.3i)T^{2} \)
71 \( 1 + (5.88 - 5.88i)T - 71iT^{2} \)
73 \( 1 + (-3.71 - 3.71i)T + 73iT^{2} \)
79 \( 1 - 10.4iT - 79T^{2} \)
83 \( 1 + (-1.13 + 2.73i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + (6.56 - 6.56i)T - 89iT^{2} \)
97 \( 1 - 1.01T + 97T^{2} \)
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

−10.35406111173858936311753385344, −9.802814232213784300987344896832, −8.050881644087416433935836190384, −7.36944963079297216632000460913, −6.65115842865687576538449704517, −5.88738009874583981599230159994, −5.36249047765461406214153480352, −4.04454378180228429801854128046, −2.53600268740320581879058014472, −1.27435942290469217947428056592, 0.48377395331635139745776454981, 1.83875741640425135004378891429, 3.94127636172455832756653961932, 4.89732269090960284834854614001, 5.20820361221094591226790960937, 6.10788507794049134933320442818, 6.88737555623235199936510421022, 8.352832083782493566470636627882, 9.195876279742039378654912843284, 9.677227437801216140992077232662

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