Properties

Label 4-1280e2-1.1-c1e2-0-25
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 + 4·5-s − 2·7-s + 2·9-s − 12·11-s + 2·13-s + 8·15-s + 2·17-s − 4·21-s + 10·23-s + 11·25-s + 6·27-s + 16·29-s − 24·33-s − 8·35-s + 10·37-s + 4·39-s − 12·41-s + 6·43-s + 8·45-s − 14·47-s + 2·49-s + 4·51-s − 2·53-s − 48·55-s − 4·63-s + 8·65-s + ⋯
L(s)  = 1  + 1.15·3-s + 1.78·5-s − 0.755·7-s + 2/3·9-s − 3.61·11-s + 0.554·13-s + 2.06·15-s + 0.485·17-s − 0.872·21-s + 2.08·23-s + 11/5·25-s + 1.15·27-s + 2.97·29-s − 4.17·33-s − 1.35·35-s + 1.64·37-s + 0.640·39-s − 1.87·41-s + 0.914·43-s + 1.19·45-s − 2.04·47-s + 2/7·49-s + 0.560·51-s − 0.274·53-s − 6.47·55-s − 0.503·63-s + 0.992·65-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

\(L(1)\) \(\approx\) \(3.984515805\)
\(L(\frac12)\) \(\approx\) \(3.984515805\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - 4 T + p T^{2} \)
good3$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
7$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 106 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 + 18 T + 162 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - p T^{2} )^{2} \)
97$C_2^2$ \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
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

−9.800862196148917985150105075181, −9.667528884723615665307041419012, −8.949366435985731420837697770313, −8.704802103274012488755441816104, −8.286939527238561190360143663378, −8.010523303179209619967310862233, −7.51461260341053648796666301248, −7.03314760966549803278375586568, −6.47192500526470257220498082761, −6.23854823234849410020436460528, −5.66443490782206789889608796529, −5.10406496873914563676237840731, −4.90659092493827824369394162835, −4.60246418321684472603823063312, −3.19298104510118680229330806411, −3.09646210670196029693825747092, −2.62367923068303506980941316916, −2.59877290287277884750544417019, −1.61017715746764157038280964060, −0.78009563855881793049807180819, 0.78009563855881793049807180819, 1.61017715746764157038280964060, 2.59877290287277884750544417019, 2.62367923068303506980941316916, 3.09646210670196029693825747092, 3.19298104510118680229330806411, 4.60246418321684472603823063312, 4.90659092493827824369394162835, 5.10406496873914563676237840731, 5.66443490782206789889608796529, 6.23854823234849410020436460528, 6.47192500526470257220498082761, 7.03314760966549803278375586568, 7.51461260341053648796666301248, 8.010523303179209619967310862233, 8.286939527238561190360143663378, 8.704802103274012488755441816104, 8.949366435985731420837697770313, 9.667528884723615665307041419012, 9.800862196148917985150105075181

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