Properties

Label 4-2e18-1.1-c1e2-0-5
Degree $4$
Conductor $262144$
Sign $1$
Analytic cond. $16.7145$
Root an. cond. $2.02196$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s − 10·13-s + 16·17-s + 2·25-s − 6·29-s + 14·37-s + 14·49-s − 18·53-s + 22·61-s − 20·65-s − 9·81-s + 32·85-s − 16·97-s − 18·101-s + 26·109-s − 28·113-s + 10·125-s + 127-s + 131-s + 137-s + 139-s − 12·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  + 0.894·5-s − 2.77·13-s + 3.88·17-s + 2/5·25-s − 1.11·29-s + 2.30·37-s + 2·49-s − 2.47·53-s + 2.81·61-s − 2.48·65-s − 81-s + 3.47·85-s − 1.62·97-s − 1.79·101-s + 2.49·109-s − 2.63·113-s + 0.894·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.996·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 262144 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 262144 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(262144\)    =    \(2^{18}\)
Sign: $1$
Analytic conductor: \(16.7145\)
Root analytic conductor: \(2.02196\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{512} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 262144,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.044004972\)
\(L(\frac12)\) \(\approx\) \(2.044004972\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 + p^{2} T^{4} \)
5$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2$ \( ( 1 - p T^{2} )^{2} \)
11$C_2^2$ \( 1 + p^{2} T^{4} \)
13$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
19$C_2^2$ \( 1 + p^{2} T^{4} \)
23$C_2$ \( ( 1 - p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
43$C_2^2$ \( 1 + p^{2} T^{4} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
59$C_2^2$ \( 1 + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 10 T + p T^{2} ) \)
67$C_2^2$ \( 1 + p^{2} T^{4} \)
71$C_2$ \( ( 1 - p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
97$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
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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

−11.06236775639795567100244089205, −10.52001374860430329285730054128, −9.896753913495995417310259114373, −9.758859936295986518668239176825, −9.714459441067935872421169197455, −9.196400020440862872369552590347, −8.233111594885068046670183703719, −7.908003555302383178461448082083, −7.45643673119267695566580024035, −7.26818589013450089758301648273, −6.54334900405889425656547510256, −5.77660705467302345136684631443, −5.38909910079939966506619296906, −5.36316779682565399323150483515, −4.51558208219417136938845621403, −3.85485476713265904517512018193, −2.97598496441294941622438905756, −2.69885519437037653529796435207, −1.85044050054771805168019821255, −0.892281956705227384142959632513, 0.892281956705227384142959632513, 1.85044050054771805168019821255, 2.69885519437037653529796435207, 2.97598496441294941622438905756, 3.85485476713265904517512018193, 4.51558208219417136938845621403, 5.36316779682565399323150483515, 5.38909910079939966506619296906, 5.77660705467302345136684631443, 6.54334900405889425656547510256, 7.26818589013450089758301648273, 7.45643673119267695566580024035, 7.908003555302383178461448082083, 8.233111594885068046670183703719, 9.196400020440862872369552590347, 9.714459441067935872421169197455, 9.758859936295986518668239176825, 9.896753913495995417310259114373, 10.52001374860430329285730054128, 11.06236775639795567100244089205

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