Properties

Label 4-320e2-1.1-c1e2-0-48
Degree $4$
Conductor $102400$
Sign $1$
Analytic cond. $6.52911$
Root an. cond. $1.59850$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 2·5-s − 2·7-s + 2·9-s − 8·11-s − 6·13-s + 4·15-s − 6·17-s + 4·21-s − 6·23-s − 25-s − 6·27-s + 4·29-s + 16·33-s + 4·35-s − 6·37-s + 12·39-s + 12·41-s + 6·43-s − 4·45-s − 18·47-s + 2·49-s + 12·51-s + 10·53-s + 16·55-s − 4·63-s + 12·65-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.894·5-s − 0.755·7-s + 2/3·9-s − 2.41·11-s − 1.66·13-s + 1.03·15-s − 1.45·17-s + 0.872·21-s − 1.25·23-s − 1/5·25-s − 1.15·27-s + 0.742·29-s + 2.78·33-s + 0.676·35-s − 0.986·37-s + 1.92·39-s + 1.87·41-s + 0.914·43-s − 0.596·45-s − 2.62·47-s + 2/7·49-s + 1.68·51-s + 1.37·53-s + 2.15·55-s − 0.503·63-s + 1.48·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 102400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(102400\)    =    \(2^{12} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(6.52911\)
Root analytic conductor: \(1.59850\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{320} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 102400,\ (\ :1/2, 1/2),\ 1)\)

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$C_2$ \( 1 + 2 T + p T^{2} \)
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$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
19$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
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$C_2^2$ \( 1 + 18 T + 162 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
59$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
67$C_2^2$ \( 1 - 18 T + 162 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 106 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - p T^{2} )^{2} \)
97$C_2^2$ \( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
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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

−11.18185980305712437639796722003, −11.16334154463584819392357277984, −10.42895204673257971173588717182, −10.24952265206505591234630510982, −9.549039434478250297792296135029, −9.387384252825253987340778726341, −8.228006242855630454386827497975, −8.029120002825090631820258281723, −7.63947010131915700151856943361, −6.93738955347480394716968290260, −6.64235707921365981905084689582, −5.90140119677323670090460866986, −5.19357968208382330038713759751, −5.14948456479960198295978311814, −4.28202416829487687819221214242, −3.79198998007098028885591972392, −2.57119187761471274617756645711, −2.39267600089662764382802844966, 0, 0, 2.39267600089662764382802844966, 2.57119187761471274617756645711, 3.79198998007098028885591972392, 4.28202416829487687819221214242, 5.14948456479960198295978311814, 5.19357968208382330038713759751, 5.90140119677323670090460866986, 6.64235707921365981905084689582, 6.93738955347480394716968290260, 7.63947010131915700151856943361, 8.029120002825090631820258281723, 8.228006242855630454386827497975, 9.387384252825253987340778726341, 9.549039434478250297792296135029, 10.24952265206505591234630510982, 10.42895204673257971173588717182, 11.16334154463584819392357277984, 11.18185980305712437639796722003

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