Properties

Degree 4
Conductor $ 2^{4} \cdot 11^{2} \cdot 19^{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  − 2·2-s − 2·3-s + 2·4-s + 2·5-s + 4·6-s − 3·9-s − 4·10-s − 4·12-s − 4·15-s − 4·16-s − 4·17-s + 6·18-s + 4·20-s − 7·25-s + 14·27-s + 8·30-s + 14·31-s + 8·32-s + 8·34-s − 6·36-s − 6·45-s + 8·48-s − 10·49-s + 14·50-s + 8·51-s − 28·54-s + 10·59-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.15·3-s + 4-s + 0.894·5-s + 1.63·6-s − 9-s − 1.26·10-s − 1.15·12-s − 1.03·15-s − 16-s − 0.970·17-s + 1.41·18-s + 0.894·20-s − 7/5·25-s + 2.69·27-s + 1.46·30-s + 2.51·31-s + 1.41·32-s + 1.37·34-s − 36-s − 0.894·45-s + 1.15·48-s − 1.42·49-s + 1.97·50-s + 1.12·51-s − 3.81·54-s + 1.30·59-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 698896 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 698896 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(698896\)    =    \(2^{4} \cdot 11^{2} \cdot 19^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{698896} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(4,\ 698896,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;11,\;19\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;11,\;19\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_2$ \( 1 + p T + p T^{2} \)
11$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
19$C_2$ \( 1 + p T^{2} \)
good3$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
5$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \)
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

−8.449863588203762160392147957262, −7.71564009386520602031555979698, −7.26767839065965563915798103089, −6.55870218284239196343388663114, −6.36261389471308870138602900888, −6.05033687233339124910623524332, −5.45660481055156382794908917127, −5.04473970036247998468059015652, −4.52429902234882156392401341160, −3.86837030910736326609302269235, −2.79441437865048826367485081806, −2.49738876285848381191850982579, −1.74297906818372540900004970449, −0.836582978875406341548589152596, 0, 0.836582978875406341548589152596, 1.74297906818372540900004970449, 2.49738876285848381191850982579, 2.79441437865048826367485081806, 3.86837030910736326609302269235, 4.52429902234882156392401341160, 5.04473970036247998468059015652, 5.45660481055156382794908917127, 6.05033687233339124910623524332, 6.36261389471308870138602900888, 6.55870218284239196343388663114, 7.26767839065965563915798103089, 7.71564009386520602031555979698, 8.449863588203762160392147957262

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