↑

Properties

Label 2-105e2-1.1-c1-0-5
Degree $2$
Conductor $11025$
Sign $1$
Analytic cond. $88.0350$
Root an. cond. $9.38270$
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  − 2·4-s + 2·13-s + 4·16-s − 8·19-s + 7·31-s − 11·37-s − 5·43-s − 4·52-s + 61-s − 8·64-s + 16·67-s + 17·73-s + 16·76-s + 17·79-s − 19·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  − 4-s + 0.554·13-s + 16-s − 1.83·19-s + 1.25·31-s − 1.80·37-s − 0.762·43-s − 0.554·52-s + 0.128·61-s − 64-s + 1.95·67-s + 1.98·73-s + 1.83·76-s + 1.91·79-s − 1.92·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 11025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(11025\)    =    \(3^{2} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(88.0350\)
Root analytic conductor: \(9.38270\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 11025,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.157214051\)
\(L(\frac12)\) \(\approx\) \(1.157214051\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 + 11 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 5 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - T + p T^{2} \)
67 \( 1 - 16 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 17 T + p T^{2} \)
79 \( 1 - 17 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 19 T + p T^{2} \)
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

−16.71714877920568, −15.85461877252992, −15.24792721216111, −14.86654960411351, −13.99485931529907, −13.76210028317134, −13.11108108909342, −12.48348278681751, −12.13708425747135, −11.14333733138538, −10.66354867121160, −10.04109888947759, −9.473812186944616, −8.637836266285948, −8.429203698586159, −7.798286048753875, −6.640940709654943, −6.417828232224431, −5.346933873496737, −4.923559575528694, −4.017938602216696, −3.656459207176328, −2.568476489958289, −1.605716865626998, −0.5121595037667615, 0.5121595037667615, 1.605716865626998, 2.568476489958289, 3.656459207176328, 4.017938602216696, 4.923559575528694, 5.346933873496737, 6.417828232224431, 6.640940709654943, 7.798286048753875, 8.429203698586159, 8.637836266285948, 9.473812186944616, 10.04109888947759, 10.66354867121160, 11.14333733138538, 12.13708425747135, 12.48348278681751, 13.11108108909342, 13.76210028317134, 13.99485931529907, 14.86654960411351, 15.24792721216111, 15.85461877252992, 16.71714877920568

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