↑

Properties

Label 4-1776e2-1.1-c0e2-0-5
Degree $4$
Conductor $3154176$
Sign $1$
Analytic cond. $0.785597$
Root an. cond. $0.941456$
Motivic weight $0$
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 + 3·9-s + 4·27-s − 2·37-s − 2·49-s + 5·81-s − 4·111-s + 2·121-s + 127-s + 131-s + 137-s + 139-s − 4·147-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
L(s)  = 1  + 2·3-s + 3·9-s + 4·27-s − 2·37-s − 2·49-s + 5·81-s − 4·111-s + 2·121-s + 127-s + 131-s + 137-s + 139-s − 4·147-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3154176 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3154176 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3154176\)    =    \(2^{8} \cdot 3^{2} \cdot 37^{2}\)
Sign: $1$
Analytic conductor: \(0.785597\)
Root analytic conductor: \(0.941456\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 3154176,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.606777401\)
\(L(\frac12)\) \(\approx\) \(2.606777401\)
\(L(1)\) 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 \( 1 \)
3$C_1$ \( ( 1 - T )^{2} \)
37$C_1$ \( ( 1 + T )^{2} \)
good5$C_2^2$ \( 1 + T^{4} \)
7$C_2$ \( ( 1 + T^{2} )^{2} \)
11$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
13$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
17$C_2^2$ \( 1 + T^{4} \)
19$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
23$C_2^2$ \( 1 + T^{4} \)
29$C_2^2$ \( 1 + T^{4} \)
31$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
41$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
43$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
47$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
53$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
59$C_2^2$ \( 1 + T^{4} \)
61$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
67$C_2$ \( ( 1 + T^{2} )^{2} \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
73$C_2$ \( ( 1 + T^{2} )^{2} \)
79$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
83$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
89$C_2^2$ \( 1 + T^{4} \)
97$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
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

−9.426173882223015863852285668220, −9.325054677920461449105578845317, −8.844391903596422675846563968073, −8.523209355728626700386277236911, −8.060460267554353381103231348907, −7.965544001346385165724090064048, −7.35181616075604460950650101971, −7.07253664058948919393827658611, −6.61100799957585136596190739615, −6.35329881595573979449596344814, −5.48856217915871096384010677831, −5.16481271529654733240152396222, −4.45666397988292942789989544860, −4.35580750414592608714486247682, −3.49056794757306326801487732408, −3.43080518028761430301594679409, −2.90887132766063790590384141885, −2.24946433102945990229629677595, −1.82859637069686783306816551439, −1.25810513308472643295802697626, 1.25810513308472643295802697626, 1.82859637069686783306816551439, 2.24946433102945990229629677595, 2.90887132766063790590384141885, 3.43080518028761430301594679409, 3.49056794757306326801487732408, 4.35580750414592608714486247682, 4.45666397988292942789989544860, 5.16481271529654733240152396222, 5.48856217915871096384010677831, 6.35329881595573979449596344814, 6.61100799957585136596190739615, 7.07253664058948919393827658611, 7.35181616075604460950650101971, 7.965544001346385165724090064048, 8.060460267554353381103231348907, 8.523209355728626700386277236911, 8.844391903596422675846563968073, 9.325054677920461449105578845317, 9.426173882223015863852285668220

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