↑

Properties

Label 2-3200-1.1-c1-0-62
Degree $2$
Conductor $3200$
Sign $-1$
Analytic cond. $25.5521$
Root an. cond. $5.05491$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 2·9-s + 11-s − 2·13-s − 3·17-s − 3·19-s + 6·23-s − 5·27-s + 2·29-s − 2·31-s + 33-s + 4·37-s − 2·39-s − 3·41-s − 4·43-s − 6·47-s − 7·49-s − 3·51-s − 10·53-s − 3·57-s − 12·59-s + 12·61-s − 67-s + 6·69-s − 10·71-s − 73-s − 16·79-s + ⋯
L(s)  = 1  + 0.577·3-s − 2/3·9-s + 0.301·11-s − 0.554·13-s − 0.727·17-s − 0.688·19-s + 1.25·23-s − 0.962·27-s + 0.371·29-s − 0.359·31-s + 0.174·33-s + 0.657·37-s − 0.320·39-s − 0.468·41-s − 0.609·43-s − 0.875·47-s − 49-s − 0.420·51-s − 1.37·53-s − 0.397·57-s − 1.56·59-s + 1.53·61-s − 0.122·67-s + 0.722·69-s − 1.18·71-s − 0.117·73-s − 1.80·79-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(3200\)    =    \(2^{7} \cdot 5^{2}\)
Sign: $-1$
Analytic conductor: \(25.5521\)
Root analytic conductor: \(5.05491\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 3200,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 - T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + 3 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 12 T + p T^{2} \)
67 \( 1 + T + p T^{2} \)
71 \( 1 + 10 T + p T^{2} \)
73 \( 1 + T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 11 T + p T^{2} \)
89 \( 1 + 13 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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

−8.407785198048768450854036309786, −7.64982687941822778678848037597, −6.78694711426376595261097004543, −6.15266454668700470112520288453, −5.10663587184968574808448530688, −4.42990778465674443685244663727, −3.33594760822568850946346627067, −2.67318893443752795486555540794, −1.64962092814722458894301942775, 0, 1.64962092814722458894301942775, 2.67318893443752795486555540794, 3.33594760822568850946346627067, 4.42990778465674443685244663727, 5.10663587184968574808448530688, 6.15266454668700470112520288453, 6.78694711426376595261097004543, 7.64982687941822778678848037597, 8.407785198048768450854036309786

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