Properties

Label 4-2e14-1.1-c1e2-0-3
Degree $4$
Conductor $16384$
Sign $1$
Analytic cond. $1.04465$
Root an. cond. $1.01098$
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  + 6·9-s − 4·17-s − 6·25-s + 20·41-s − 14·49-s + 12·73-s + 27·81-s − 20·89-s − 36·97-s − 28·113-s + 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 24·153-s + 157-s + 163-s + 167-s + 10·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  + 2·9-s − 0.970·17-s − 6/5·25-s + 3.12·41-s − 2·49-s + 1.40·73-s + 3·81-s − 2.11·89-s − 3.65·97-s − 2.63·113-s + 2·121-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 − 1.94·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.769·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯

Functional equation

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

Invariants

Degree: \(4\)
Conductor: \(16384\)    =    \(2^{14}\)
Sign: $1$
Analytic conductor: \(1.04465\)
Root analytic conductor: \(1.01098\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{128} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 16384,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.215372628\)
\(L(\frac12)\) \(\approx\) \(1.215372628\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - p T^{2} )^{2} \)
5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - p T^{2} )^{2} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - p T^{2} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
59$C_2$ \( ( 1 - p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
67$C_2$ \( ( 1 - p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 18 T + p T^{2} )^{2} \)
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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

−13.53783374545753250377939509110, −13.02824755955478473580174590461, −12.51293411533616317032473602234, −12.41930941597087059874329181038, −11.36090455256956921638940592955, −11.11911838669299626302245413709, −10.51740556902965316698962093181, −9.803680106827546855846298258804, −9.576316991150715064482865075460, −9.061941896594789261995051039125, −8.036557796973512610753317860725, −7.79814902851498306451719208502, −7.01704861305086744721648990389, −6.62189904477336838247127256571, −5.89147895045870161977741553862, −5.01237511108207605464128768470, −4.20531702035781344901108809042, −3.97199102139077686237606533667, −2.57482920414217144195647823211, −1.52019857626457482641329780176, 1.52019857626457482641329780176, 2.57482920414217144195647823211, 3.97199102139077686237606533667, 4.20531702035781344901108809042, 5.01237511108207605464128768470, 5.89147895045870161977741553862, 6.62189904477336838247127256571, 7.01704861305086744721648990389, 7.79814902851498306451719208502, 8.036557796973512610753317860725, 9.061941896594789261995051039125, 9.576316991150715064482865075460, 9.803680106827546855846298258804, 10.51740556902965316698962093181, 11.11911838669299626302245413709, 11.36090455256956921638940592955, 12.41930941597087059874329181038, 12.51293411533616317032473602234, 13.02824755955478473580174590461, 13.53783374545753250377939509110

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