Properties

Degree 4
Conductor $ 2^{14} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·3-s + 6·9-s − 4·11-s − 4·17-s + 4·19-s − 6·25-s − 4·27-s − 16·33-s − 12·41-s + 12·43-s + 2·49-s − 16·51-s + 16·57-s + 28·59-s + 20·67-s + 28·73-s − 24·75-s − 37·81-s − 12·83-s − 4·89-s − 4·97-s − 24·99-s − 4·107-s + 4·113-s − 10·121-s − 48·123-s + 127-s + ⋯
L(s)  = 1  + 2.30·3-s + 2·9-s − 1.20·11-s − 0.970·17-s + 0.917·19-s − 6/5·25-s − 0.769·27-s − 2.78·33-s − 1.87·41-s + 1.82·43-s + 2/7·49-s − 2.24·51-s + 2.11·57-s + 3.64·59-s + 2.44·67-s + 3.27·73-s − 2.77·75-s − 4.11·81-s − 1.31·83-s − 0.423·89-s − 0.406·97-s − 2.41·99-s − 0.386·107-s + 0.376·113-s − 0.909·121-s − 4.32·123-s + 0.0887·127-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

\( d \)  =  \(4\)
\( N \)  =  \(16384\)    =    \(2^{14}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{16384} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 16384,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $1.960380722$
$L(\frac12)$  $\approx$  $1.960380722$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 2$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p = 2$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2 \( 1 \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−11.15482415732823435746426270865, −10.22941768705266383849925633731, −9.742591354048855107896325616652, −9.385355573475711088007535446099, −8.728992519223720851294830795286, −8.173183686627627779120693205031, −8.098681382024412731658064042949, −7.33738117792255002879146989083, −6.78571919296342801026858814862, −5.64130249043000575312616744005, −5.16542238567969836254694300670, −3.91058908136761919842529082491, −3.56276783709589056726776122484, −2.46608091328452001690059553408, −2.30676446591740616355718333113, 2.30676446591740616355718333113, 2.46608091328452001690059553408, 3.56276783709589056726776122484, 3.91058908136761919842529082491, 5.16542238567969836254694300670, 5.64130249043000575312616744005, 6.78571919296342801026858814862, 7.33738117792255002879146989083, 8.098681382024412731658064042949, 8.173183686627627779120693205031, 8.728992519223720851294830795286, 9.385355573475711088007535446099, 9.742591354048855107896325616652, 10.22941768705266383849925633731, 11.15482415732823435746426270865

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