Properties

Label 4-2e20-1.1-c1e2-0-3
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

\(L(1)\) \(\approx\) \(1.850636083\)
\(L(\frac12)\) \(\approx\) \(1.850636083\)
\(L(\frac{3}{2})\) 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

−9.992676442534183611680202937412, −9.948975380079506745966139116837, −9.547914153327182136731585318217, −8.832670494873253177321195957629, −8.655161200073585778377237657365, −7.899411765043100827645955048993, −7.74883236945244734105833108280, −7.14775107391726688352507500281, −6.70344858586590378313709496923, −6.46078746563502339471693998299, −5.80543070102939722977492335971, −5.74531690973983054508349506842, −4.72812929748733416768286989422, −4.33846451309494687477940766460, −4.10166462469460812271823003049, −3.19994371502056489193105609228, −3.04721980727381412786029043622, −2.27686264849340236244693375021, −1.42605941539100853309724913155, −0.66311245126504812302628570019, 0.66311245126504812302628570019, 1.42605941539100853309724913155, 2.27686264849340236244693375021, 3.04721980727381412786029043622, 3.19994371502056489193105609228, 4.10166462469460812271823003049, 4.33846451309494687477940766460, 4.72812929748733416768286989422, 5.74531690973983054508349506842, 5.80543070102939722977492335971, 6.46078746563502339471693998299, 6.70344858586590378313709496923, 7.14775107391726688352507500281, 7.74883236945244734105833108280, 7.899411765043100827645955048993, 8.655161200073585778377237657365, 8.832670494873253177321195957629, 9.547914153327182136731585318217, 9.948975380079506745966139116837, 9.992676442534183611680202937412

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