Properties

Label 4-2e14-1.1-c2e2-0-0
Degree $4$
Conductor $16384$
Sign $1$
Analytic cond. $12.1643$
Root an. cond. $1.86755$
Motivic weight $2$
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  − 18·9-s + 60·17-s − 14·25-s + 36·41-s + 98·49-s + 220·73-s + 243·81-s + 156·89-s − 260·97-s − 60·113-s − 242·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 1.08e3·153-s + 157-s + 163-s + 167-s − 238·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  − 2·9-s + 3.52·17-s − 0.559·25-s + 0.878·41-s + 2·49-s + 3.01·73-s + 3·81-s + 1.75·89-s − 2.68·97-s − 0.530·113-s − 2·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s − 7.05·153-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 1.40·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16384 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16384 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(16384\)    =    \(2^{14}\)
Sign: $1$
Analytic conductor: \(12.1643\)
Root analytic conductor: \(1.86755\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 16384,\ (\ :1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.501830137\)
\(L(\frac12)\) \(\approx\) \(1.501830137\)
\(L(2)\) 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$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
5$C_2$ \( ( 1 - 6 T + p^{2} T^{2} )( 1 + 6 T + p^{2} T^{2} ) \)
7$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
11$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
13$C_2$ \( ( 1 - 10 T + p^{2} T^{2} )( 1 + 10 T + p^{2} T^{2} ) \)
17$C_2$ \( ( 1 - 30 T + p^{2} T^{2} )^{2} \)
19$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
23$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
29$C_2$ \( ( 1 - 42 T + p^{2} T^{2} )( 1 + 42 T + p^{2} T^{2} ) \)
31$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
37$C_2$ \( ( 1 - 70 T + p^{2} T^{2} )( 1 + 70 T + p^{2} T^{2} ) \)
41$C_2$ \( ( 1 - 18 T + p^{2} T^{2} )^{2} \)
43$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
47$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
53$C_2$ \( ( 1 - 90 T + p^{2} T^{2} )( 1 + 90 T + p^{2} T^{2} ) \)
59$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
61$C_2$ \( ( 1 - 22 T + p^{2} T^{2} )( 1 + 22 T + p^{2} T^{2} ) \)
67$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
71$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
73$C_2$ \( ( 1 - 110 T + p^{2} T^{2} )^{2} \)
79$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
83$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
89$C_2$ \( ( 1 - 78 T + p^{2} T^{2} )^{2} \)
97$C_2$ \( ( 1 + 130 T + p^{2} T^{2} )^{2} \)
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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

−13.77676078865330230683853503155, −12.60696087425977997847109937278, −12.20865951635231157041891798894, −11.96485602203701156075599325867, −11.39787022735936419902063736354, −10.76492532689609751239162206837, −10.33729263689225348720288237108, −9.584591927863906772345047956544, −9.294466469909227778540559662291, −8.481061015318365249472335382929, −7.83655585834435200093201307417, −7.82339865808919886715035214529, −6.77517785117455227179730627952, −5.77243802406384235237393007373, −5.71263577535222720521205607235, −5.11864648982598716729457242297, −3.79571626667575362714368958340, −3.26350088861955763918854450182, −2.47511284011557875413663458057, −0.885053195404485168075259896355, 0.885053195404485168075259896355, 2.47511284011557875413663458057, 3.26350088861955763918854450182, 3.79571626667575362714368958340, 5.11864648982598716729457242297, 5.71263577535222720521205607235, 5.77243802406384235237393007373, 6.77517785117455227179730627952, 7.82339865808919886715035214529, 7.83655585834435200093201307417, 8.481061015318365249472335382929, 9.294466469909227778540559662291, 9.584591927863906772345047956544, 10.33729263689225348720288237108, 10.76492532689609751239162206837, 11.39787022735936419902063736354, 11.96485602203701156075599325867, 12.20865951635231157041891798894, 12.60696087425977997847109937278, 13.77676078865330230683853503155

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