Properties

Label 4-3528e2-1.1-c1e2-0-26
Degree $4$
Conductor $12446784$
Sign $1$
Analytic cond. $793.617$
Root an. cond. $5.30765$
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  − 8·13-s − 4·17-s − 4·19-s + 4·23-s + 5·25-s − 4·29-s + 8·31-s + 6·37-s + 24·41-s + 8·43-s + 8·47-s + 6·53-s − 12·59-s + 4·61-s + 4·67-s + 24·71-s + 8·73-s + 16·79-s − 8·83-s − 4·89-s − 32·97-s + 8·101-s − 8·103-s + 8·107-s + 14·109-s − 4·113-s + 11·121-s + ⋯
L(s)  = 1  − 2.21·13-s − 0.970·17-s − 0.917·19-s + 0.834·23-s + 25-s − 0.742·29-s + 1.43·31-s + 0.986·37-s + 3.74·41-s + 1.21·43-s + 1.16·47-s + 0.824·53-s − 1.56·59-s + 0.512·61-s + 0.488·67-s + 2.84·71-s + 0.936·73-s + 1.80·79-s − 0.878·83-s − 0.423·89-s − 3.24·97-s + 0.796·101-s − 0.788·103-s + 0.773·107-s + 1.34·109-s − 0.376·113-s + 121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12446784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12446784 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(12446784\)    =    \(2^{6} \cdot 3^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(793.617\)
Root analytic conductor: \(5.30765\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 12446784,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.628278793\)
\(L(\frac12)\) \(\approx\) \(2.628278793\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 + 4 T - T^{2} + 4 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 4 T - 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - 8 T + 33 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 6 T - T^{2} - 6 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 8 T + 17 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 4 T - 45 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 8 T - 9 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 16 T + 177 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 4 T - 73 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
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

−8.621769387669587346649037791083, −8.550181441505251941665826806283, −7.83383971831581124561357301231, −7.70901341739233562276063593342, −7.14308187644786823838709519227, −7.13287708950033197317733889306, −6.41217525695569191451383421613, −6.34879694395452939262081330808, −5.64391629436001272828147980636, −5.43730077629159443181338750981, −4.79042360778743511095277848122, −4.59790475035811651027839051400, −4.16185038324393241363626365194, −3.96482716833828362629177188076, −2.86820306804417651148157541783, −2.83334641012544794697011515094, −2.21350574336486342399471944220, −2.13314240196068769166453435986, −0.76559703669037767305700554422, −0.70961668738248180086026084938, 0.70961668738248180086026084938, 0.76559703669037767305700554422, 2.13314240196068769166453435986, 2.21350574336486342399471944220, 2.83334641012544794697011515094, 2.86820306804417651148157541783, 3.96482716833828362629177188076, 4.16185038324393241363626365194, 4.59790475035811651027839051400, 4.79042360778743511095277848122, 5.43730077629159443181338750981, 5.64391629436001272828147980636, 6.34879694395452939262081330808, 6.41217525695569191451383421613, 7.13287708950033197317733889306, 7.14308187644786823838709519227, 7.70901341739233562276063593342, 7.83383971831581124561357301231, 8.550181441505251941665826806283, 8.621769387669587346649037791083

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