Properties

Degree 4
Conductor $ 2^{4} \cdot 11^{2} \cdot 13^{2} $
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  + 2·3-s − 3·9-s − 4·13-s + 12·17-s − 6·23-s − 25-s − 14·27-s − 8·39-s − 20·43-s − 10·49-s + 24·51-s − 12·53-s − 8·61-s − 12·69-s − 2·75-s + 4·79-s − 4·81-s + 36·101-s + 16·103-s + 12·107-s − 30·113-s + 12·117-s + 121-s + 127-s − 40·129-s + 131-s + 137-s + ⋯
L(s)  = 1  + 1.15·3-s − 9-s − 1.10·13-s + 2.91·17-s − 1.25·23-s − 1/5·25-s − 2.69·27-s − 1.28·39-s − 3.04·43-s − 1.42·49-s + 3.36·51-s − 1.64·53-s − 1.02·61-s − 1.44·69-s − 0.230·75-s + 0.450·79-s − 4/9·81-s + 3.58·101-s + 1.57·103-s + 1.16·107-s − 2.82·113-s + 1.10·117-s + 1/11·121-s + 0.0887·127-s − 3.52·129-s + 0.0873·131-s + 0.0854·137-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 327184 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 327184 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(327184\)    =    \(2^{4} \cdot 11^{2} \cdot 13^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{327184} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(4,\ 327184,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;11,\;13\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;11,\;13\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2 \( 1 \)
11$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
13$C_2$ \( 1 + 4 T + p T^{2} \)
good3$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
5$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
23$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
37$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
61$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
71$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
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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

−8.314187785972322305192087673902, −8.132665807481574488057539990847, −7.66471435961055676315007320566, −7.60852684015633966980596452192, −6.63716263814174422745171385492, −6.11181518853748875342915860937, −5.69510294756229368768383854980, −5.11227132195889762323685903220, −4.77814882467244849114054024293, −3.68139626518860523217158413051, −3.27381985666804460223319703873, −3.10915852533965624890419491672, −2.22738900554088144481773298380, −1.58763508602526038578941432974, 0, 1.58763508602526038578941432974, 2.22738900554088144481773298380, 3.10915852533965624890419491672, 3.27381985666804460223319703873, 3.68139626518860523217158413051, 4.77814882467244849114054024293, 5.11227132195889762323685903220, 5.69510294756229368768383854980, 6.11181518853748875342915860937, 6.63716263814174422745171385492, 7.60852684015633966980596452192, 7.66471435961055676315007320566, 8.132665807481574488057539990847, 8.314187785972322305192087673902

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