↑

Properties

Label 2-330e2-1.1-c1-0-4
Degree $2$
Conductor $108900$
Sign $1$
Analytic cond. $869.570$
Root an. cond. $29.4884$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3·7-s − 6·13-s − 2·17-s + 19-s + 6·23-s − 4·29-s − 3·31-s + 3·37-s + 2·41-s − 8·43-s + 4·47-s + 2·49-s + 2·53-s − 8·59-s + 5·61-s + 67-s + 2·71-s − 13·73-s − 79-s − 12·83-s + 6·89-s + 18·91-s + 9·97-s + 101-s + 103-s + 107-s + 109-s + ⋯
L(s)  = 1  − 1.13·7-s − 1.66·13-s − 0.485·17-s + 0.229·19-s + 1.25·23-s − 0.742·29-s − 0.538·31-s + 0.493·37-s + 0.312·41-s − 1.21·43-s + 0.583·47-s + 2/7·49-s + 0.274·53-s − 1.04·59-s + 0.640·61-s + 0.122·67-s + 0.237·71-s − 1.52·73-s − 0.112·79-s − 1.31·83-s + 0.635·89-s + 1.88·91-s + 0.913·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(108900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(869.570\)
Root analytic conductor: \(29.4884\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 108900,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4403068272\)
\(L(\frac12)\) \(\approx\) \(0.4403068272\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
11 \( 1 \)
good7 \( 1 + 3 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 + 3 T + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 5 T + p T^{2} \)
67 \( 1 - T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 + 13 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 9 T + p T^{2} \)
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

−13.37397167133001, −13.23967293971594, −12.75164187279439, −12.26255457901675, −11.79674187824799, −11.25810478443246, −10.67237627347441, −10.20366781831321, −9.596075053427256, −9.387347359556738, −8.894900589862905, −8.198625021603143, −7.522026003139020, −7.101104692560431, −6.795015699482525, −6.100816147368800, −5.528292293538177, −4.990091767128420, −4.469841193834111, −3.780308972078571, −3.119900070205371, −2.699957201647206, −2.083207148992709, −1.202091294484688, −0.2100519139830845, 0.2100519139830845, 1.202091294484688, 2.083207148992709, 2.699957201647206, 3.119900070205371, 3.780308972078571, 4.469841193834111, 4.990091767128420, 5.528292293538177, 6.100816147368800, 6.795015699482525, 7.101104692560431, 7.522026003139020, 8.198625021603143, 8.894900589862905, 9.387347359556738, 9.596075053427256, 10.20366781831321, 10.67237627347441, 11.25810478443246, 11.79674187824799, 12.26255457901675, 12.75164187279439, 13.23967293971594, 13.37397167133001

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