Properties

Degree 4
Conductor $ 2^{2} \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  − 2·3-s + 4-s − 8·5-s − 3·9-s + 2·11-s − 2·12-s + 16·15-s + 16-s − 8·20-s − 2·23-s + 38·25-s + 14·27-s − 16·31-s − 4·33-s − 3·36-s − 4·37-s + 2·44-s + 24·45-s + 16·47-s − 2·48-s − 5·49-s − 2·53-s − 16·55-s + 30·59-s + 16·60-s + 64-s + 6·67-s + ⋯
L(s)  = 1  − 1.15·3-s + 1/2·4-s − 3.57·5-s − 9-s + 0.603·11-s − 0.577·12-s + 4.13·15-s + 1/4·16-s − 1.78·20-s − 0.417·23-s + 38/5·25-s + 2.69·27-s − 2.87·31-s − 0.696·33-s − 1/2·36-s − 0.657·37-s + 0.301·44-s + 3.57·45-s + 2.33·47-s − 0.288·48-s − 5/7·49-s − 0.274·53-s − 2.15·55-s + 3.90·59-s + 2.06·60-s + 1/8·64-s + 0.733·67-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 174724 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 174724 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(174724\)    =    \(2^{2} \cdot 11^{2} \cdot 19^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{174724} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(4,\ 174724,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
11$C_2$ \( 1 - 2 T + p T^{2} \)
19$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good3$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
5$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
13$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
17$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
23$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 15 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
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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

−8.713028473107534219835074143035, −8.408486909011940238981299566497, −7.979676687352676305303385977682, −7.25038064772684706053554512263, −7.24993124552845565031332433274, −6.68239009544393402711079299710, −5.98269892429734678737463344232, −5.28069074518443602737935578596, −5.02409024164795186610457056564, −3.99098546974644050534648424657, −3.86057015395193111701613139713, −3.39047289495238098925090461140, −2.52867986473958162980513935632, −0.76546142649705169453989550316, 0, 0.76546142649705169453989550316, 2.52867986473958162980513935632, 3.39047289495238098925090461140, 3.86057015395193111701613139713, 3.99098546974644050534648424657, 5.02409024164795186610457056564, 5.28069074518443602737935578596, 5.98269892429734678737463344232, 6.68239009544393402711079299710, 7.24993124552845565031332433274, 7.25038064772684706053554512263, 7.979676687352676305303385977682, 8.408486909011940238981299566497, 8.713028473107534219835074143035

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