Properties

Label 2-4410-1.1-c1-0-58
Degree $2$
Conductor $4410$
Sign $-1$
Analytic cond. $35.2140$
Root an. cond. $5.93414$
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  + 2-s + 4-s − 5-s + 8-s − 10-s − 2·13-s + 16-s − 2·19-s − 20-s + 25-s − 2·26-s − 6·29-s − 8·31-s + 32-s − 4·37-s − 2·38-s − 40-s + 6·41-s + 2·43-s − 6·47-s + 50-s − 2·52-s − 6·53-s − 6·58-s + 12·59-s − 8·61-s − 8·62-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.353·8-s − 0.316·10-s − 0.554·13-s + 1/4·16-s − 0.458·19-s − 0.223·20-s + 1/5·25-s − 0.392·26-s − 1.11·29-s − 1.43·31-s + 0.176·32-s − 0.657·37-s − 0.324·38-s − 0.158·40-s + 0.937·41-s + 0.304·43-s − 0.875·47-s + 0.141·50-s − 0.277·52-s − 0.824·53-s − 0.787·58-s + 1.56·59-s − 1.02·61-s − 1.01·62-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4410\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(35.2140\)
Root analytic conductor: \(5.93414\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4410,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
3 \( 1 \)
5 \( 1 + T \)
7 \( 1 \)
good11 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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

−7.72884256525441388224603340660, −7.31197475215030351263291600034, −6.49895271238158734940980188459, −5.67215196547072475539085458023, −5.00679853986208418603505167842, −4.16364492087773222905282744018, −3.52571722299951763856854309445, −2.57091189800820073190585026882, −1.61549144219507342431900476742, 0, 1.61549144219507342431900476742, 2.57091189800820073190585026882, 3.52571722299951763856854309445, 4.16364492087773222905282744018, 5.00679853986208418603505167842, 5.67215196547072475539085458023, 6.49895271238158734940980188459, 7.31197475215030351263291600034, 7.72884256525441388224603340660

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