Properties

Label 4-2e20-1.1-c1e2-0-1
Degree $4$
Conductor $1048576$
Sign $1$
Analytic cond. $66.8581$
Root an. cond. $2.85948$
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  − 2·3-s + 4·5-s + 2·9-s − 2·11-s − 4·13-s − 8·15-s + 8·17-s − 10·19-s + 8·25-s − 6·27-s + 12·29-s + 16·31-s + 4·33-s − 4·37-s + 8·39-s − 6·43-s + 8·45-s − 2·49-s − 16·51-s + 4·53-s − 8·55-s + 20·57-s + 6·59-s − 12·61-s − 16·65-s − 6·67-s − 16·75-s + ⋯
L(s)  = 1  − 1.15·3-s + 1.78·5-s + 2/3·9-s − 0.603·11-s − 1.10·13-s − 2.06·15-s + 1.94·17-s − 2.29·19-s + 8/5·25-s − 1.15·27-s + 2.22·29-s + 2.87·31-s + 0.696·33-s − 0.657·37-s + 1.28·39-s − 0.914·43-s + 1.19·45-s − 2/7·49-s − 2.24·51-s + 0.549·53-s − 1.07·55-s + 2.64·57-s + 0.781·59-s − 1.53·61-s − 1.98·65-s − 0.733·67-s − 1.84·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1048576 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1048576 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1048576\)    =    \(2^{20}\)
Sign: $1$
Analytic conductor: \(66.8581\)
Root analytic conductor: \(2.85948\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1048576,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.624804597\)
\(L(\frac12)\) \(\approx\) \(1.624804597\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good3$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
5$C_2^2$ \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
19$C_2^2$ \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
41$C_2^2$ \( 1 - 78 T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 126 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
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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

−10.18420862675541264162921550423, −9.859198925114695221524883110767, −9.818506001801546215291779492246, −8.771033550996432543368437619397, −8.681136416820765787373370717711, −7.972325435111813307463437604371, −7.77001272656745963491489573017, −6.96363348829148130727086511701, −6.54747492132804267053010412449, −6.27077741523261728201406524038, −5.97775275794457563797760139903, −5.34772799672872571407125926453, −5.18539599241483971308905331549, −4.56638589484403405670294575988, −4.30683599498965761890572279427, −3.15596283916248704529012133774, −2.73069511770774381683776601640, −2.15990742226220308509858522840, −1.51335161089839037117105340262, −0.63375534365157951667770666633, 0.63375534365157951667770666633, 1.51335161089839037117105340262, 2.15990742226220308509858522840, 2.73069511770774381683776601640, 3.15596283916248704529012133774, 4.30683599498965761890572279427, 4.56638589484403405670294575988, 5.18539599241483971308905331549, 5.34772799672872571407125926453, 5.97775275794457563797760139903, 6.27077741523261728201406524038, 6.54747492132804267053010412449, 6.96363348829148130727086511701, 7.77001272656745963491489573017, 7.972325435111813307463437604371, 8.681136416820765787373370717711, 8.771033550996432543368437619397, 9.818506001801546215291779492246, 9.859198925114695221524883110767, 10.18420862675541264162921550423

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