↑

Properties

Label 2-3200-40.29-c1-0-18
Degree $2$
Conductor $3200$
Sign $-0.316 - 0.948i$
Analytic cond. $25.5521$
Root an. cond. $5.05491$
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  + 1.41·3-s + 4.24i·7-s − 0.999·9-s − 5.65i·11-s + 2·13-s + 6i·17-s + 2.82i·19-s + 6i·21-s + 7.07i·23-s − 5.65·27-s − 4i·29-s + 2.82·31-s − 8.00i·33-s − 2·37-s + 2.82·39-s + ⋯
L(s)  = 1  + 0.816·3-s + 1.60i·7-s − 0.333·9-s − 1.70i·11-s + 0.554·13-s + 1.45i·17-s + 0.648i·19-s + 1.30i·21-s + 1.47i·23-s − 1.08·27-s − 0.742i·29-s + 0.508·31-s − 1.39i·33-s − 0.328·37-s + 0.452·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.316 - 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.316 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3200\)    =    \(2^{7} \cdot 5^{2}\)
Sign: $-0.316 - 0.948i$
Analytic conductor: \(25.5521\)
Root analytic conductor: \(5.05491\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3200} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3200,\ (\ :1/2),\ -0.316 - 0.948i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.812931121\)
\(L(\frac12)\) \(\approx\) \(1.812931121\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 - 1.41T + 3T^{2} \)
7 \( 1 - 4.24iT - 7T^{2} \)
11 \( 1 + 5.65iT - 11T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 - 6iT - 17T^{2} \)
19 \( 1 - 2.82iT - 19T^{2} \)
23 \( 1 - 7.07iT - 23T^{2} \)
29 \( 1 + 4iT - 29T^{2} \)
31 \( 1 - 2.82T + 31T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 + 8T + 41T^{2} \)
43 \( 1 - 1.41T + 43T^{2} \)
47 \( 1 - 1.41iT - 47T^{2} \)
53 \( 1 + 2T + 53T^{2} \)
59 \( 1 - 2.82iT - 59T^{2} \)
61 \( 1 - 14iT - 61T^{2} \)
67 \( 1 - 4.24T + 67T^{2} \)
71 \( 1 + 2.82T + 71T^{2} \)
73 \( 1 - 6iT - 73T^{2} \)
79 \( 1 - 16.9T + 79T^{2} \)
83 \( 1 + 12.7T + 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 - 10iT - 97T^{2} \)
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

−8.685818051600479326745316972475, −8.368749169105749600195756707918, −7.81832677331179111094078151188, −6.34742912170191148587928972218, −5.82710150545774756675071259330, −5.42238522062274864430675585209, −3.83587323547253406301310411436, −3.31109912014791385844933773083, −2.49202707908488327159263171217, −1.49063120655329957533060908613, 0.47775194775641930041743997312, 1.82422069127922980614930601706, 2.81327081942034249720030358917, 3.65668975327959325600108967848, 4.56433654345837676627655618821, 5.03054640199900825999542865008, 6.60765061319520479878551008123, 6.97676807215957426159907811084, 7.67873879119368513397193398160, 8.377203527206289375087068087142

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