↑

Properties

Label 2-2e9-32.29-c1-0-6
Degree $2$
Conductor $512$
Sign $0.788 + 0.615i$
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  + (−0.191 − 0.0794i)3-s + (0.292 + 0.707i)5-s + (2.27 − 2.27i)7-s + (−2.09 − 2.09i)9-s + (3.60 − 1.49i)11-s + (−1.86 + 4.50i)13-s − 0.158i·15-s − 3.05i·17-s + (1.60 − 3.87i)19-s + (−0.616 + 0.255i)21-s + (−0.271 − 0.271i)23-s + (3.12 − 3.12i)25-s + (0.473 + 1.14i)27-s + (2.24 + 0.931i)29-s + 6.82·31-s + ⋯
L(s)  = 1  + (−0.110 − 0.0458i)3-s + (0.130 + 0.316i)5-s + (0.858 − 0.858i)7-s + (−0.696 − 0.696i)9-s + (1.08 − 0.450i)11-s + (−0.517 + 1.24i)13-s − 0.0410i·15-s − 0.740i·17-s + (0.368 − 0.889i)19-s + (−0.134 + 0.0557i)21-s + (−0.0565 − 0.0565i)23-s + (0.624 − 0.624i)25-s + (0.0911 + 0.219i)27-s + (0.417 + 0.173i)29-s + 1.22·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.788 + 0.615i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.788 + 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(512\)    =    \(2^{9}\)
Sign: $0.788 + 0.615i$
Analytic conductor: \(4.08834\)
Root analytic conductor: \(2.02196\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{512} (321, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 512,\ (\ :1/2),\ 0.788 + 0.615i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.42743 - 0.490956i\)
\(L(\frac12)\) \(\approx\) \(1.42743 - 0.490956i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.191 + 0.0794i)T + (2.12 + 2.12i)T^{2} \)
5 \( 1 + (-0.292 - 0.707i)T + (-3.53 + 3.53i)T^{2} \)
7 \( 1 + (-2.27 + 2.27i)T - 7iT^{2} \)
11 \( 1 + (-3.60 + 1.49i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 + (1.86 - 4.50i)T + (-9.19 - 9.19i)T^{2} \)
17 \( 1 + 3.05iT - 17T^{2} \)
19 \( 1 + (-1.60 + 3.87i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (0.271 + 0.271i)T + 23iT^{2} \)
29 \( 1 + (-2.24 - 0.931i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 - 6.82T + 31T^{2} \)
37 \( 1 + (-1.50 - 3.63i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (-1.54 - 1.54i)T + 41iT^{2} \)
43 \( 1 + (1.80 - 0.748i)T + (30.4 - 30.4i)T^{2} \)
47 \( 1 + 7.37iT - 47T^{2} \)
53 \( 1 + (4.04 - 1.67i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (4.19 + 10.1i)T + (-41.7 + 41.7i)T^{2} \)
61 \( 1 + (3.28 + 1.35i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 + (-4.81 - 1.99i)T + (47.3 + 47.3i)T^{2} \)
71 \( 1 + (6.47 - 6.47i)T - 71iT^{2} \)
73 \( 1 + (-2.84 - 2.84i)T + 73iT^{2} \)
79 \( 1 - 9.74iT - 79T^{2} \)
83 \( 1 + (3.74 - 9.04i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + (7.58 - 7.58i)T - 89iT^{2} \)
97 \( 1 - 3.71T + 97T^{2} \)
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

−11.07793694361045542000011277861, −9.838031352589038449485980510795, −9.075469827901232834068763031916, −8.170961340732710281683821886757, −6.86251531280307871476106230842, −6.53735871528308613475071568536, −4.99730435741539223481813096880, −4.12034024001927669804832613771, −2.78246256488193253827444013473, −1.05561598336297316492855715690, 1.58986431222424686484865740674, 2.91519779611334118898603250632, 4.49528751850476131014802070235, 5.39398549953077912970037642712, 6.11791835041408663237280919512, 7.63292807988147334343373984679, 8.297376968319935240858362667294, 9.107139001788601773808375181709, 10.16175384611672119433583335530, 11.01342831861041373151293440990

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