Properties

Label 2-4400-1.1-c1-0-44
Degree $2$
Conductor $4400$
Sign $-1$
Analytic cond. $35.1341$
Root an. cond. $5.92740$
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  − 2·3-s − 4·7-s + 9-s + 11-s + 4·13-s + 4·19-s + 8·21-s − 6·23-s + 4·27-s − 6·29-s − 8·31-s − 2·33-s − 2·37-s − 8·39-s + 6·41-s + 8·43-s + 6·47-s + 9·49-s + 6·53-s − 8·57-s + 12·59-s + 2·61-s − 4·63-s − 10·67-s + 12·69-s + 12·71-s + 16·73-s + ⋯
L(s)  = 1  − 1.15·3-s − 1.51·7-s + 1/3·9-s + 0.301·11-s + 1.10·13-s + 0.917·19-s + 1.74·21-s − 1.25·23-s + 0.769·27-s − 1.11·29-s − 1.43·31-s − 0.348·33-s − 0.328·37-s − 1.28·39-s + 0.937·41-s + 1.21·43-s + 0.875·47-s + 9/7·49-s + 0.824·53-s − 1.05·57-s + 1.56·59-s + 0.256·61-s − 0.503·63-s − 1.22·67-s + 1.44·69-s + 1.42·71-s + 1.87·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4400\)    =    \(2^{4} \cdot 5^{2} \cdot 11\)
Sign: $-1$
Analytic conductor: \(35.1341\)
Root analytic conductor: \(5.92740\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{4400} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4400,\ (\ :1/2),\ -1)\)

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 \( 1 \)
11 \( 1 - T \)
good3 \( 1 + 2 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 16 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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

−7.86499496632861719392953609421, −7.01609870378475161743947583743, −6.44241690116309043090441361867, −5.65889433188227421033845679887, −5.52958383648087111826963240906, −3.98244178628333179220455849882, −3.65149113231465708944230333700, −2.46535642408448141031036689969, −1.05453299331481824083824038720, 0, 1.05453299331481824083824038720, 2.46535642408448141031036689969, 3.65149113231465708944230333700, 3.98244178628333179220455849882, 5.52958383648087111826963240906, 5.65889433188227421033845679887, 6.44241690116309043090441361867, 7.01609870378475161743947583743, 7.86499496632861719392953609421

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