Properties

Label 4-80e4-1.1-c1e2-0-25
Degree $4$
Conductor $40960000$
Sign $1$
Analytic cond. $2611.64$
Root an. cond. $7.14872$
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  − 4·9-s − 4·13-s − 4·17-s + 16·29-s − 20·37-s − 12·49-s − 4·53-s + 20·61-s + 12·73-s + 7·81-s − 20·89-s − 28·97-s + 8·101-s − 4·109-s − 12·113-s + 16·117-s − 14·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 16·153-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  − 4/3·9-s − 1.10·13-s − 0.970·17-s + 2.97·29-s − 3.28·37-s − 1.71·49-s − 0.549·53-s + 2.56·61-s + 1.40·73-s + 7/9·81-s − 2.11·89-s − 2.84·97-s + 0.796·101-s − 0.383·109-s − 1.12·113-s + 1.47·117-s − 1.27·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 1.29·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 40960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40960000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(40960000\)    =    \(2^{16} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(2611.64\)
Root analytic conductor: \(7.14872\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{6400} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 40960000,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
5 \( 1 \)
good3$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
7$C_2^2$ \( 1 + 12 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
19$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 44 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2^2$ \( 1 + 36 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 132 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
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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

−7.924856596453866057467642175425, −7.58837561324100514935355256397, −6.84474745635581136660648093265, −6.81395972480546299041132073171, −6.57806798219420119004523404524, −6.26965954469776808989303424703, −5.43657286543204451111089472945, −5.39600081196474816560175108297, −5.04343504711090177733629467448, −4.76923727563996389514491124801, −4.15127244888056488229257636211, −3.90021435992391830790206852967, −3.21411533513852415825541645811, −2.98849673636576321247481408087, −2.53919199839450031910110930317, −2.24467587547750002395099680213, −1.61066136719801239993793103110, −1.01529407632240482519019454332, 0, 0, 1.01529407632240482519019454332, 1.61066136719801239993793103110, 2.24467587547750002395099680213, 2.53919199839450031910110930317, 2.98849673636576321247481408087, 3.21411533513852415825541645811, 3.90021435992391830790206852967, 4.15127244888056488229257636211, 4.76923727563996389514491124801, 5.04343504711090177733629467448, 5.39600081196474816560175108297, 5.43657286543204451111089472945, 6.26965954469776808989303424703, 6.57806798219420119004523404524, 6.81395972480546299041132073171, 6.84474745635581136660648093265, 7.58837561324100514935355256397, 7.924856596453866057467642175425

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