↑

Properties

Label 2-420e2-1.1-c1-0-14
Degree $2$
Conductor $176400$
Sign $1$
Analytic cond. $1408.56$
Root an. cond. $37.5308$
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  − 11-s − 2·13-s − 8·17-s − 2·19-s − 23-s − 29-s + 6·31-s − 9·37-s − 43-s + 6·47-s + 2·53-s + 6·59-s − 8·61-s − 3·67-s + 7·71-s + 16·73-s − 79-s − 6·83-s − 14·89-s + 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  − 0.301·11-s − 0.554·13-s − 1.94·17-s − 0.458·19-s − 0.208·23-s − 0.185·29-s + 1.07·31-s − 1.47·37-s − 0.152·43-s + 0.875·47-s + 0.274·53-s + 0.781·59-s − 1.02·61-s − 0.366·67-s + 0.830·71-s + 1.87·73-s − 0.112·79-s − 0.658·83-s − 1.48·89-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(176400\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(1408.56\)
Root analytic conductor: \(37.5308\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 176400,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5232891620\)
\(L(\frac12)\) \(\approx\) \(0.5232891620\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 \)
good11 \( 1 + T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 8 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 + T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 + 9 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + 3 T + p T^{2} \)
71 \( 1 - 7 T + p T^{2} \)
73 \( 1 - 16 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
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

−13.29179323846153, −12.58476277185961, −12.34417398771892, −11.80084645762055, −11.24217392890976, −10.74631411478308, −10.51625712481043, −9.781315622471684, −9.445739425450210, −8.787656702729895, −8.448229672853664, −8.014252036025249, −7.209033514080637, −6.962318568493929, −6.392091267981933, −5.948597695058016, −5.031800454769882, −4.997762091130204, −4.081370777978929, −3.915820545704334, −2.918482131578365, −2.447827257590896, −2.003839921556154, −1.202525434463795, −0.2075843646086525, 0.2075843646086525, 1.202525434463795, 2.003839921556154, 2.447827257590896, 2.918482131578365, 3.915820545704334, 4.081370777978929, 4.997762091130204, 5.031800454769882, 5.948597695058016, 6.392091267981933, 6.962318568493929, 7.209033514080637, 8.014252036025249, 8.448229672853664, 8.787656702729895, 9.445739425450210, 9.781315622471684, 10.51625712481043, 10.74631411478308, 11.24217392890976, 11.80084645762055, 12.34417398771892, 12.58476277185961, 13.29179323846153

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