Properties

Label 4-1232e2-1.1-c1e2-0-69
Degree $4$
Conductor $1517824$
Sign $1$
Analytic cond. $96.7777$
Root an. cond. $3.13649$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 4·5-s − 2·7-s + 2·9-s + 2·11-s + 2·13-s + 8·15-s − 2·17-s − 4·19-s + 4·21-s + 4·23-s + 2·25-s − 6·27-s + 8·29-s + 10·31-s − 4·33-s + 8·35-s − 8·37-s − 4·39-s − 18·41-s − 16·43-s − 8·45-s − 10·47-s + 3·49-s + 4·51-s + 8·53-s − 8·55-s + ⋯
L(s)  = 1  − 1.15·3-s − 1.78·5-s − 0.755·7-s + 2/3·9-s + 0.603·11-s + 0.554·13-s + 2.06·15-s − 0.485·17-s − 0.917·19-s + 0.872·21-s + 0.834·23-s + 2/5·25-s − 1.15·27-s + 1.48·29-s + 1.79·31-s − 0.696·33-s + 1.35·35-s − 1.31·37-s − 0.640·39-s − 2.81·41-s − 2.43·43-s − 1.19·45-s − 1.45·47-s + 3/7·49-s + 0.560·51-s + 1.09·53-s − 1.07·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1517824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1517824 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1517824\)    =    \(2^{8} \cdot 7^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(96.7777\)
Root analytic conductor: \(3.13649\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 1517824,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_1$ \( ( 1 + T )^{2} \)
11$C_1$ \( ( 1 - T )^{2} \)
good3$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
5$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
13$D_{4}$ \( 1 - 2 T + 22 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 2 T + 30 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
29$C_4$ \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 10 T + 82 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 18 T + 158 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
43$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
47$D_{4}$ \( 1 + 10 T + 114 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 2 T + 114 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 10 T + 142 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 20 T + 214 T^{2} + 20 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 12 T + 158 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 6 T + 150 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 78 T^{2} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 4 T - 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
97$D_{4}$ \( 1 - 8 T + 30 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.658007683079271225961522872242, −8.980425650874201123957128811750, −8.481311601414679993434083218113, −8.393942591645093172894639275399, −8.005603817371518168713372867379, −7.29469539437670736231617250406, −6.83102944525725550088189005455, −6.72933873900394837537321773769, −6.23649246039197019320730642261, −5.92833186216556798005243553184, −5.05889013210013028522599446080, −4.81185089287575851545426692475, −4.37470572462229169455552016936, −3.85507384856613512217544554264, −3.25944177071667590732404141922, −3.19468840824857225664526545575, −1.95299001700720966029600333752, −1.23675716743807956303806382658, 0, 0, 1.23675716743807956303806382658, 1.95299001700720966029600333752, 3.19468840824857225664526545575, 3.25944177071667590732404141922, 3.85507384856613512217544554264, 4.37470572462229169455552016936, 4.81185089287575851545426692475, 5.05889013210013028522599446080, 5.92833186216556798005243553184, 6.23649246039197019320730642261, 6.72933873900394837537321773769, 6.83102944525725550088189005455, 7.29469539437670736231617250406, 8.005603817371518168713372867379, 8.393942591645093172894639275399, 8.481311601414679993434083218113, 8.980425650874201123957128811750, 9.658007683079271225961522872242

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