Properties

Label 4-1280e2-1.1-c1e2-0-9
Degree $4$
Conductor $1638400$
Sign $1$
Analytic cond. $104.465$
Root an. cond. $3.19700$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·3-s − 2·5-s − 4·7-s + 8·9-s + 2·13-s − 8·15-s − 10·17-s − 16·21-s − 4·23-s − 25-s + 12·27-s − 8·29-s + 8·35-s − 2·37-s + 8·39-s + 12·43-s − 16·45-s − 4·47-s + 8·49-s − 40·51-s − 14·53-s − 32·63-s − 4·65-s + 20·67-s − 16·69-s + 6·73-s − 4·75-s + ⋯
L(s)  = 1  + 2.30·3-s − 0.894·5-s − 1.51·7-s + 8/3·9-s + 0.554·13-s − 2.06·15-s − 2.42·17-s − 3.49·21-s − 0.834·23-s − 1/5·25-s + 2.30·27-s − 1.48·29-s + 1.35·35-s − 0.328·37-s + 1.28·39-s + 1.82·43-s − 2.38·45-s − 0.583·47-s + 8/7·49-s − 5.60·51-s − 1.92·53-s − 4.03·63-s − 0.496·65-s + 2.44·67-s − 1.92·69-s + 0.702·73-s − 0.461·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1638400\)    =    \(2^{16} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(104.465\)
Root analytic conductor: \(3.19700\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1280} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1638400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.079274822\)
\(L(\frac12)\) \(\approx\) \(2.079274822\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
7$C_2^2$ \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
19$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 106 T^{2} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 20 T + 200 T^{2} - 20 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - p T^{2} )^{2} \)
97$C_2^2$ \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
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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

−9.973679911939139526748356336571, −9.413182050107134106519429533585, −9.126736827397984662897507742928, −8.584297267753593967619216433470, −8.385563852228643003862097865187, −7.85023832529147102938108374188, −7.72886987944764369103945800197, −6.97863353916608501655469369985, −6.83899886389135080126989629111, −6.21933245936249948878567538732, −5.99107627650993733301132069094, −5.00700533343344961201314165871, −4.50204448171704456851054176345, −3.88711225932820422659700164371, −3.68452738103680850149415273790, −3.40358533782168554823678821481, −2.77130085051307419705626505132, −2.14672615844771159328365190354, −2.02087816636786727949629038252, −0.48801766974882057292402732214, 0.48801766974882057292402732214, 2.02087816636786727949629038252, 2.14672615844771159328365190354, 2.77130085051307419705626505132, 3.40358533782168554823678821481, 3.68452738103680850149415273790, 3.88711225932820422659700164371, 4.50204448171704456851054176345, 5.00700533343344961201314165871, 5.99107627650993733301132069094, 6.21933245936249948878567538732, 6.83899886389135080126989629111, 6.97863353916608501655469369985, 7.72886987944764369103945800197, 7.85023832529147102938108374188, 8.385563852228643003862097865187, 8.584297267753593967619216433470, 9.126736827397984662897507742928, 9.413182050107134106519429533585, 9.973679911939139526748356336571

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