Properties

Label 4-2e20-1.1-c1e2-0-5
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  + 4·7-s − 4·9-s + 4·17-s + 12·23-s − 8·25-s + 16·31-s + 16·47-s − 2·49-s − 16·63-s + 20·71-s + 8·73-s + 7·81-s + 8·89-s − 4·97-s − 12·103-s + 12·113-s + 16·119-s − 20·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 16·153-s + 157-s + 48·161-s + ⋯
L(s)  = 1  + 1.51·7-s − 4/3·9-s + 0.970·17-s + 2.50·23-s − 8/5·25-s + 2.87·31-s + 2.33·47-s − 2/7·49-s − 2.01·63-s + 2.37·71-s + 0.936·73-s + 7/9·81-s + 0.847·89-s − 0.406·97-s − 1.18·103-s + 1.12·113-s + 1.46·119-s − 1.81·121-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 + 3.78·161-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

\(L(1)\) \(\approx\) \(2.617194648\)
\(L(\frac12)\) \(\approx\) \(2.617194648\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
5$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
11$C_2^2$ \( 1 + 20 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 24 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
19$C_2^2$ \( 1 + 20 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
29$C_2^2$ \( 1 + 40 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 + 56 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2^2$ \( 1 + 36 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
53$C_2^2$ \( 1 + 56 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 100 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 40 T^{2} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 84 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 164 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
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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

−10.11746964753947292584623582371, −9.798366957221443305742266294404, −9.097610573816508318490204400983, −9.022895325882427049514167451752, −8.244032673042430475396139725529, −8.237407034588849020582938892720, −7.74767573012862956892103193782, −7.44739389036720333325609749594, −6.57374329894496280724980335626, −6.50387579953710679476530289783, −5.59763915045680867636494472590, −5.53945491454191232584588240670, −4.81289066944371836883804149335, −4.76582357539702087439548695148, −3.88620574767162200487105768768, −3.40623531880521545156415149512, −2.59462024857105447685902213317, −2.47948089871289773789243062662, −1.36707304681505534254321161287, −0.829238644358853302781468374646, 0.829238644358853302781468374646, 1.36707304681505534254321161287, 2.47948089871289773789243062662, 2.59462024857105447685902213317, 3.40623531880521545156415149512, 3.88620574767162200487105768768, 4.76582357539702087439548695148, 4.81289066944371836883804149335, 5.53945491454191232584588240670, 5.59763915045680867636494472590, 6.50387579953710679476530289783, 6.57374329894496280724980335626, 7.44739389036720333325609749594, 7.74767573012862956892103193782, 8.237407034588849020582938892720, 8.244032673042430475396139725529, 9.022895325882427049514167451752, 9.097610573816508318490204400983, 9.798366957221443305742266294404, 10.11746964753947292584623582371

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