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 − 2·7-s − 3·8-s − 2·14-s − 16-s + 12·17-s − 6·25-s + 2·28-s + 5·32-s + 12·34-s − 4·41-s + 3·49-s − 6·50-s + 6·56-s + 7·64-s − 12·68-s − 12·73-s − 32·79-s − 4·82-s + 28·89-s + 36·97-s + 3·98-s + 6·100-s + 16·103-s + 2·112-s + 28·113-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s − 0.755·7-s − 1.06·8-s − 0.534·14-s − 1/4·16-s + 2.91·17-s − 6/5·25-s + 0.377·28-s + 0.883·32-s + 2.05·34-s − 0.624·41-s + 3/7·49-s − 0.848·50-s + 0.801·56-s + 7/8·64-s − 1.45·68-s − 1.40·73-s − 3.60·79-s − 0.441·82-s + 2.96·89-s + 3.65·97-s + 0.303·98-s + 3/5·100-s + 1.57·103-s + 0.188·112-s + 2.63·113-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  =  $(4,\ 254016,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $1.721513656$
$L(\frac12)$  $\approx$  $1.721513656$
$L(\frac{3}{2})$   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$C_1$ \( ( 1 + T )^{2} \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
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$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
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$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
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

−8.892272133899417496764084066801, −8.597097830639437040754263866908, −7.81295430367747747098376616892, −7.62195349072755325091614948497, −7.09250692106396557822287287333, −6.22411641525239374249127245960, −5.94607665445287604157300008429, −5.60865912599068037214391350294, −5.02709416296405066539370067618, −4.45044632349803314255794033962, −3.76538094002949404605955440773, −3.26988486528469799014112835839, −3.05422074105458389777226041971, −1.87090915832039374223761593463, −0.72892880515889973617029791676, 0.72892880515889973617029791676, 1.87090915832039374223761593463, 3.05422074105458389777226041971, 3.26988486528469799014112835839, 3.76538094002949404605955440773, 4.45044632349803314255794033962, 5.02709416296405066539370067618, 5.60865912599068037214391350294, 5.94607665445287604157300008429, 6.22411641525239374249127245960, 7.09250692106396557822287287333, 7.62195349072755325091614948497, 7.81295430367747747098376616892, 8.597097830639437040754263866908, 8.892272133899417496764084066801

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