Properties

Degree 4
Conductor $ 2^{6} \cdot 3^{4} \cdot 7^{2} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 4-s − 3·8-s − 8·11-s − 16-s + 12·17-s + 8·19-s − 8·22-s − 6·25-s + 5·32-s + 12·34-s + 8·38-s − 4·41-s − 8·43-s + 8·44-s + 49-s − 6·50-s − 24·59-s + 7·64-s + 8·67-s − 12·68-s − 12·73-s − 8·76-s − 4·82-s + 24·83-s − 8·86-s + 24·88-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s − 1.06·8-s − 2.41·11-s − 1/4·16-s + 2.91·17-s + 1.83·19-s − 1.70·22-s − 6/5·25-s + 0.883·32-s + 2.05·34-s + 1.29·38-s − 0.624·41-s − 1.21·43-s + 1.20·44-s + 1/7·49-s − 0.848·50-s − 3.12·59-s + 7/8·64-s + 0.977·67-s − 1.45·68-s − 1.40·73-s − 0.917·76-s − 0.441·82-s + 2.63·83-s − 0.862·86-s + 2.55·88-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 254016 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 254016 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(254016\)    =    \(2^{6} \cdot 3^{4} \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{254016} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 254016,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $1.625547847$
$L(\frac12)$  $\approx$  $1.625547847$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;7\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_2$ \( 1 - T + p T^{2} \)
3 \( 1 \)
7$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
11$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 18 T + p T^{2} )^{2} \)
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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

−9.042140676903253409137259496371, −8.229992153219430708166678625388, −7.81295430367747747098376616892, −7.70691183501011689426667743544, −7.32441936749744810989380005535, −6.14521652142582152085553341069, −5.94607665445287604157300008429, −5.34641745697560833577710874285, −5.02709416296405066539370067618, −4.80722452268725035282926398957, −3.62256199043071801163653681899, −3.22746506277116131252491518684, −3.05422074105458389777226041971, −1.93480858488759200478154145238, −0.69577924383102779401083388688, 0.69577924383102779401083388688, 1.93480858488759200478154145238, 3.05422074105458389777226041971, 3.22746506277116131252491518684, 3.62256199043071801163653681899, 4.80722452268725035282926398957, 5.02709416296405066539370067618, 5.34641745697560833577710874285, 5.94607665445287604157300008429, 6.14521652142582152085553341069, 7.32441936749744810989380005535, 7.70691183501011689426667743544, 7.81295430367747747098376616892, 8.229992153219430708166678625388, 9.042140676903253409137259496371

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