↑

Properties

Label 2-2e9-32.5-c1-0-7
Degree $2$
Conductor $512$
Sign $0.0775 + 0.996i$
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  + (−0.943 − 2.27i)3-s + (1.70 + 0.707i)5-s + (0.665 + 0.665i)7-s + (−2.18 + 2.18i)9-s + (1.52 − 3.69i)11-s + (4.26 − 1.76i)13-s − 4.55i·15-s + 3.61i·17-s + (−0.470 + 0.194i)19-s + (0.887 − 2.14i)21-s + (1.33 − 1.33i)23-s + (−1.12 − 1.12i)25-s + (0.191 + 0.0793i)27-s + (−2.37 − 5.73i)29-s + 1.17·31-s + ⋯
L(s)  = 1  + (−0.544 − 1.31i)3-s + (0.763 + 0.316i)5-s + (0.251 + 0.251i)7-s + (−0.726 + 0.726i)9-s + (0.461 − 1.11i)11-s + (1.18 − 0.489i)13-s − 1.17i·15-s + 0.877i·17-s + (−0.107 + 0.0446i)19-s + (0.193 − 0.467i)21-s + (0.278 − 0.278i)23-s + (−0.224 − 0.224i)25-s + (0.0368 + 0.0152i)27-s + (−0.441 − 1.06i)29-s + 0.210·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0775 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0775 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(512\)    =    \(2^{9}\)
Sign: $0.0775 + 0.996i$
Analytic conductor: \(4.08834\)
Root analytic conductor: \(2.02196\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{512} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 512,\ (\ :1/2),\ 0.0775 + 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.05389 - 0.975063i\)
\(L(\frac12)\) \(\approx\) \(1.05389 - 0.975063i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.943 + 2.27i)T + (-2.12 + 2.12i)T^{2} \)
5 \( 1 + (-1.70 - 0.707i)T + (3.53 + 3.53i)T^{2} \)
7 \( 1 + (-0.665 - 0.665i)T + 7iT^{2} \)
11 \( 1 + (-1.52 + 3.69i)T + (-7.77 - 7.77i)T^{2} \)
13 \( 1 + (-4.26 + 1.76i)T + (9.19 - 9.19i)T^{2} \)
17 \( 1 - 3.61iT - 17T^{2} \)
19 \( 1 + (0.470 - 0.194i)T + (13.4 - 13.4i)T^{2} \)
23 \( 1 + (-1.33 + 1.33i)T - 23iT^{2} \)
29 \( 1 + (2.37 + 5.73i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 - 1.17T + 31T^{2} \)
37 \( 1 + (1.23 + 0.510i)T + (26.1 + 26.1i)T^{2} \)
41 \( 1 + (1.66 - 1.66i)T - 41iT^{2} \)
43 \( 1 + (1.05 - 2.54i)T + (-30.4 - 30.4i)T^{2} \)
47 \( 1 + 1.49iT - 47T^{2} \)
53 \( 1 + (-1.90 + 4.59i)T + (-37.4 - 37.4i)T^{2} \)
59 \( 1 + (4.94 + 2.04i)T + (41.7 + 41.7i)T^{2} \)
61 \( 1 + (-5.67 - 13.7i)T + (-43.1 + 43.1i)T^{2} \)
67 \( 1 + (1.41 + 3.40i)T + (-47.3 + 47.3i)T^{2} \)
71 \( 1 + (-9.66 - 9.66i)T + 71iT^{2} \)
73 \( 1 + (-7.55 + 7.55i)T - 73iT^{2} \)
79 \( 1 - 17.2iT - 79T^{2} \)
83 \( 1 + (11.6 - 4.82i)T + (58.6 - 58.6i)T^{2} \)
89 \( 1 + (-5.43 - 5.43i)T + 89iT^{2} \)
97 \( 1 - 6.15T + 97T^{2} \)
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

−10.95665459643043041469132607214, −9.919149498122997967804701739737, −8.595266457658253713503880252560, −8.060223039418417349697881362546, −6.73358026660380947133415714340, −6.11233040773225674034912396914, −5.58716455340681456217679871361, −3.72116423077193514677147882553, −2.21359562302623711392501068250, −1.03675258619460326302507771177, 1.66621819722535568361411526299, 3.57488289095398309448282903540, 4.57743073755929591897358371402, 5.26190198203969615374853787746, 6.30094846974106549434259439676, 7.39319129796401363981072735407, 8.936444222639313181504817785877, 9.395343462457275895266784304452, 10.18464390209025849026817072820, 10.99317088608201580411167912662

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