Properties

Label 4-2e20-1.1-c1e2-0-12
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: induced by $\chi_{1024} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1048576,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.639354847\)
\(L(\frac12)\) \(\approx\) \(5.639354847\)
\(L(\frac{3}{2})\) 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

−9.861267452902821142165346095531, −9.764615062572975740680316170028, −9.301512972911754736804095257523, −9.114660113549770594600217856637, −8.705849885589958955331077862596, −7.927297533229742347163524709624, −7.79368861092914191011954337892, −7.15776588247263638959807987753, −7.00205120601783704123358472604, −6.32642198332482874502793387980, −5.70331115426585224516004049735, −5.50051297921217524724396112020, −5.04754153100810527987393446089, −4.57552259330940661498552565175, −3.55320615002066118266616457283, −3.37334240331858666483117869211, −2.70611621944822741206070694626, −2.41235962402359011867809460134, −1.38050411544664894275956908000, −1.28232496085313326742765465011, 1.28232496085313326742765465011, 1.38050411544664894275956908000, 2.41235962402359011867809460134, 2.70611621944822741206070694626, 3.37334240331858666483117869211, 3.55320615002066118266616457283, 4.57552259330940661498552565175, 5.04754153100810527987393446089, 5.50051297921217524724396112020, 5.70331115426585224516004049735, 6.32642198332482874502793387980, 7.00205120601783704123358472604, 7.15776588247263638959807987753, 7.79368861092914191011954337892, 7.927297533229742347163524709624, 8.705849885589958955331077862596, 9.114660113549770594600217856637, 9.301512972911754736804095257523, 9.764615062572975740680316170028, 9.861267452902821142165346095531

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