Properties

Label 4-1280e2-1.1-c1e2-0-44
Degree $4$
Conductor $1638400$
Sign $1$
Analytic cond. $104.465$
Root an. cond. $3.19700$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s + 2·5-s + 6·7-s + 4·15-s − 4·19-s + 12·21-s + 6·23-s + 3·25-s − 2·27-s + 12·31-s + 12·35-s − 12·37-s − 12·41-s − 10·43-s + 6·47-s + 16·49-s − 8·57-s − 12·59-s + 12·61-s + 10·67-s + 12·69-s + 12·71-s + 8·73-s + 6·75-s + 24·79-s − 81-s + 6·83-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.894·5-s + 2.26·7-s + 1.03·15-s − 0.917·19-s + 2.61·21-s + 1.25·23-s + 3/5·25-s − 0.384·27-s + 2.15·31-s + 2.02·35-s − 1.97·37-s − 1.87·41-s − 1.52·43-s + 0.875·47-s + 16/7·49-s − 1.05·57-s − 1.56·59-s + 1.53·61-s + 1.22·67-s + 1.44·69-s + 1.42·71-s + 0.936·73-s + 0.692·75-s + 2.70·79-s − 1/9·81-s + 0.658·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1638400\)    =    \(2^{16} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(104.465\)
Root analytic conductor: \(3.19700\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1280} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1638400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.644380963\)
\(L(\frac12)\) \(\approx\) \(5.644380963\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
5$C_1$ \( ( 1 - T )^{2} \)
good3$D_{4}$ \( 1 - 2 T + 4 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 - 6 T + 20 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
23$D_{4}$ \( 1 - 6 T + 28 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$D_{4}$ \( 1 - 12 T + 86 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
41$D_{4}$ \( 1 + 12 T + 106 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 10 T + 84 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 6 T + 76 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
61$D_{4}$ \( 1 - 12 T + 110 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 10 T + 132 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 12 T + 70 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
83$D_{4}$ \( 1 - 6 T + 172 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 12 T + 166 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.709365929211758930633779755935, −9.445026292212794170261431296328, −8.837846762796221502353727919303, −8.564479835042351229881647927291, −8.259165428997083822799594819274, −8.190125431894169279004739824631, −7.63069463683182740283531031273, −6.99883669582351890845569233218, −6.49645254919945655010070597892, −6.42542475179588501980946204133, −5.32115649095844684880089566713, −5.22687861877540423997021405002, −4.84973073848680922881631775701, −4.48527093005597921647991269512, −3.48293374064653797372379883625, −3.40438788261367541822551840578, −2.38192986440091497987991863021, −2.24634652414582701977966595200, −1.66452138663424647605199742595, −0.995301817899201935474387564944, 0.995301817899201935474387564944, 1.66452138663424647605199742595, 2.24634652414582701977966595200, 2.38192986440091497987991863021, 3.40438788261367541822551840578, 3.48293374064653797372379883625, 4.48527093005597921647991269512, 4.84973073848680922881631775701, 5.22687861877540423997021405002, 5.32115649095844684880089566713, 6.42542475179588501980946204133, 6.49645254919945655010070597892, 6.99883669582351890845569233218, 7.63069463683182740283531031273, 8.190125431894169279004739824631, 8.259165428997083822799594819274, 8.564479835042351229881647927291, 8.837846762796221502353727919303, 9.445026292212794170261431296328, 9.709365929211758930633779755935

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