↑

Properties

Label 2-3584-16.5-c1-0-91
Degree $2$
Conductor $3584$
Sign $-0.923 - 0.382i$
Analytic cond. $28.6183$
Root an. cond. $5.34961$
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  + (1.73 − 1.73i)3-s + (−2.73 − 2.73i)5-s − i·7-s − 2.99i·9-s + (−2 − 2i)11-s + (4.73 − 4.73i)13-s − 9.46·15-s − 3.46·17-s + (1.73 − 1.73i)19-s + (−1.73 − 1.73i)21-s − 9.46i·23-s + 9.92i·25-s + (2.46 − 2.46i)29-s + 7.46·31-s − 6.92·33-s + ⋯
L(s)  = 1  + (0.999 − 0.999i)3-s + (−1.22 − 1.22i)5-s − 0.377i·7-s − 0.999i·9-s + (−0.603 − 0.603i)11-s + (1.31 − 1.31i)13-s − 2.44·15-s − 0.840·17-s + (0.397 − 0.397i)19-s + (−0.377 − 0.377i)21-s − 1.97i·23-s + 1.98i·25-s + (0.457 − 0.457i)29-s + 1.34·31-s − 1.20·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.923 - 0.382i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3584\)    =    \(2^{9} \cdot 7\)
Sign: $-0.923 - 0.382i$
Analytic conductor: \(28.6183\)
Root analytic conductor: \(5.34961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3584} (897, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3584,\ (\ :1/2),\ -0.923 - 0.382i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.872778805\)
\(L(\frac12)\) \(\approx\) \(1.872778805\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + iT \)
good3 \( 1 + (-1.73 + 1.73i)T - 3iT^{2} \)
5 \( 1 + (2.73 + 2.73i)T + 5iT^{2} \)
11 \( 1 + (2 + 2i)T + 11iT^{2} \)
13 \( 1 + (-4.73 + 4.73i)T - 13iT^{2} \)
17 \( 1 + 3.46T + 17T^{2} \)
19 \( 1 + (-1.73 + 1.73i)T - 19iT^{2} \)
23 \( 1 + 9.46iT - 23T^{2} \)
29 \( 1 + (-2.46 + 2.46i)T - 29iT^{2} \)
31 \( 1 - 7.46T + 31T^{2} \)
37 \( 1 + (0.464 + 0.464i)T + 37iT^{2} \)
41 \( 1 + 3.46iT - 41T^{2} \)
43 \( 1 + (-7.46 - 7.46i)T + 43iT^{2} \)
47 \( 1 + 11.4T + 47T^{2} \)
53 \( 1 + (1 + i)T + 53iT^{2} \)
59 \( 1 + (-2.26 - 2.26i)T + 59iT^{2} \)
61 \( 1 + (5.26 - 5.26i)T - 61iT^{2} \)
67 \( 1 + (-2.92 + 2.92i)T - 67iT^{2} \)
71 \( 1 - 4iT - 71T^{2} \)
73 \( 1 - 6iT - 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 + (-2.26 + 2.26i)T - 83iT^{2} \)
89 \( 1 - 8.92iT - 89T^{2} \)
97 \( 1 + 2.39T + 97T^{2} \)
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

−8.306275952468505359068728646739, −7.78728506143374357466108029505, −6.84292268873419559427515328945, −6.04216730879437195477032727181, −4.90211847637134218441674821884, −4.23420740076543794963872337228, −3.26375940434993576084773034259, −2.58926221372796328475217424242, −1.08066482914384764617660312159, −0.56614960638270059291854645167, 1.87565712609996423837289842626, 2.95980943745217815252837346302, 3.48476918091774394769566076807, 4.13961673610059039520304127496, 4.82147955293034466555664807909, 6.15054142752406989062205384341, 6.87529956962821689666676794209, 7.64476521741458621219684026808, 8.280204355498539330588337521935, 8.925231138519294852465125165764

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