Properties

Degree 4
Conductor $ 2^{7} \cdot 3^{8} $
Sign $-1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 1

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 8-s + 6·11-s + 16-s + 6·17-s − 2·19-s − 6·22-s − 10·25-s − 32-s − 6·34-s + 2·38-s − 18·41-s − 2·43-s + 6·44-s − 10·49-s + 10·50-s − 6·59-s + 64-s + 10·67-s + 6·68-s + 22·73-s − 2·76-s + 18·82-s − 24·83-s + 2·86-s − 6·88-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.353·8-s + 1.80·11-s + 1/4·16-s + 1.45·17-s − 0.458·19-s − 1.27·22-s − 2·25-s − 0.176·32-s − 1.02·34-s + 0.324·38-s − 2.81·41-s − 0.304·43-s + 0.904·44-s − 1.42·49-s + 1.41·50-s − 0.781·59-s + 1/8·64-s + 1.22·67-s + 0.727·68-s + 2.57·73-s − 0.229·76-s + 1.98·82-s − 2.63·83-s + 0.215·86-s − 0.639·88-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 839808 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 839808 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(839808\)    =    \(2^{7} \cdot 3^{8}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{839808} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(4,\ 839808,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_1$ \( 1 + T \)
3 \( 1 \)
good5$C_2$ \( ( 1 + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
59$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
83$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
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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

−7.930769578371338402756573985837, −7.80670854475898716623947697687, −7.03958314562926506761932301326, −6.68619713403218599725876916726, −6.38621009962525321192846394616, −5.86722110566278992511976681176, −5.34383976963704470462884839275, −4.86353492334106993174795804756, −4.05064804668395326172520142038, −3.65076060485342096132488694617, −3.35055085317237919419053143292, −2.42466756648709547485696737515, −1.60693251106342657124988453680, −1.34471597583379685537750572425, 0, 1.34471597583379685537750572425, 1.60693251106342657124988453680, 2.42466756648709547485696737515, 3.35055085317237919419053143292, 3.65076060485342096132488694617, 4.05064804668395326172520142038, 4.86353492334106993174795804756, 5.34383976963704470462884839275, 5.86722110566278992511976681176, 6.38621009962525321192846394616, 6.68619713403218599725876916726, 7.03958314562926506761932301326, 7.80670854475898716623947697687, 7.930769578371338402756573985837

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