Properties

Degree 4
Conductor $ 2^{8} \cdot 3^{4} $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s + 2·7-s + 2·13-s + 4·16-s + 2·25-s − 4·28-s + 2·31-s + 14·37-s + 2·49-s − 4·52-s − 10·61-s − 8·64-s + 16·67-s − 22·79-s + 4·91-s − 4·100-s + 101-s + 103-s + 107-s + 109-s + 8·112-s + 113-s + 2·121-s − 4·124-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  − 4-s + 0.755·7-s + 0.554·13-s + 16-s + 2/5·25-s − 0.755·28-s + 0.359·31-s + 2.30·37-s + 2/7·49-s − 0.554·52-s − 1.28·61-s − 64-s + 1.95·67-s − 2.47·79-s + 0.419·91-s − 2/5·100-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.755·112-s + 0.0940·113-s + 2/11·121-s − 0.359·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 20736 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 20736 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(20736\)    =    \(2^{8} \cdot 3^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{20736} (1, \cdot )$
Sato-Tate  :  $J(E_4)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 20736,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $1.033342181$
$L(\frac12)$  $\approx$  $1.033342181$
$L(\frac{3}{2})$   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$C_2$ \( 1 + p T^{2} \)
3 \( 1 \)
good5$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
7$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \)
31$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
41$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 94 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
71$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 22 T + 242 T^{2} + 22 p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 82 T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - p T^{2} )^{2} \)
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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

−15.6573552799, −15.0493121231, −14.7989083334, −14.1265649677, −13.9792602007, −13.3186411952, −12.8699612441, −12.5414157211, −11.7734084601, −11.3403320356, −10.8796379876, −10.2435328648, −9.7256049107, −9.21164953256, −8.64977247819, −8.15104498432, −7.75993010545, −6.99953147484, −6.14051787937, −5.66287191424, −4.84339700748, −4.40255307769, −3.66989511421, −2.65205799832, −1.2184354835, 1.2184354835, 2.65205799832, 3.66989511421, 4.40255307769, 4.84339700748, 5.66287191424, 6.14051787937, 6.99953147484, 7.75993010545, 8.15104498432, 8.64977247819, 9.21164953256, 9.7256049107, 10.2435328648, 10.8796379876, 11.3403320356, 11.7734084601, 12.5414157211, 12.8699612441, 13.3186411952, 13.9792602007, 14.1265649677, 14.7989083334, 15.0493121231, 15.6573552799

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