Properties

Degree $2$
Conductor $4410$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + 5-s − 8-s − 10-s − 4·11-s + 2·13-s + 16-s + 2·17-s + 4·19-s + 20-s + 4·22-s + 8·23-s + 25-s − 2·26-s − 6·29-s + 8·31-s − 32-s − 2·34-s − 2·37-s − 4·38-s − 40-s + 2·41-s − 12·43-s − 4·44-s − 8·46-s − 8·47-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.353·8-s − 0.316·10-s − 1.20·11-s + 0.554·13-s + 1/4·16-s + 0.485·17-s + 0.917·19-s + 0.223·20-s + 0.852·22-s + 1.66·23-s + 1/5·25-s − 0.392·26-s − 1.11·29-s + 1.43·31-s − 0.176·32-s − 0.342·34-s − 0.328·37-s − 0.648·38-s − 0.158·40-s + 0.312·41-s − 1.82·43-s − 0.603·44-s − 1.17·46-s − 1.16·47-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: $1$
Motivic weight: \(1\)
Character: $\chi_{4410} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4410,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.464139424\)
\(L(\frac12)\) \(\approx\) \(1.464139424\)
\(L(\frac{3}{2})\) 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 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
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

−17.98949152931852, −17.43577127500632, −16.72473231395532, −16.29003039978142, −15.51277230820482, −15.13174583110994, −14.29595781527627, −13.49291956430748, −13.09577169791431, −12.38025125404442, −11.42006741779274, −11.08541994457325, −10.20807095517637, −9.813293599598864, −9.084779277022377, −8.304251894090549, −7.792964377576660, −6.982555289393821, −6.313770592975154, −5.336537205106795, −4.960059351312345, −3.457756646789911, −2.868361289441156, −1.798229572676347, −0.7696301184397769, 0.7696301184397769, 1.798229572676347, 2.868361289441156, 3.457756646789911, 4.960059351312345, 5.336537205106795, 6.313770592975154, 6.982555289393821, 7.792964377576660, 8.304251894090549, 9.084779277022377, 9.813293599598864, 10.20807095517637, 11.08541994457325, 11.42006741779274, 12.38025125404442, 13.09577169791431, 13.49291956430748, 14.29595781527627, 15.13174583110994, 15.51277230820482, 16.29003039978142, 16.72473231395532, 17.43577127500632, 17.98949152931852

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