Properties

Degree 4
Conductor $ 2^{2} \cdot 7^{3} \cdot 11^{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  + 4-s + 7-s − 2·9-s − 8·13-s + 16-s + 12·17-s + 4·19-s − 10·25-s + 28-s − 2·36-s + 4·37-s + 12·41-s + 49-s − 8·52-s + 12·53-s + 16·61-s − 2·63-s + 64-s − 8·67-s + 12·68-s + 4·73-s + 4·76-s − 5·81-s − 12·83-s − 8·91-s − 10·100-s + 112-s + ⋯
L(s)  = 1  + 1/2·4-s + 0.377·7-s − 2/3·9-s − 2.21·13-s + 1/4·16-s + 2.91·17-s + 0.917·19-s − 2·25-s + 0.188·28-s − 1/3·36-s + 0.657·37-s + 1.87·41-s + 1/7·49-s − 1.10·52-s + 1.64·53-s + 2.04·61-s − 0.251·63-s + 1/8·64-s − 0.977·67-s + 1.45·68-s + 0.468·73-s + 0.458·76-s − 5/9·81-s − 1.31·83-s − 0.838·91-s − 100-s + 0.0944·112-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 166012 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 166012 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(166012\)    =    \(2^{2} \cdot 7^{3} \cdot 11^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{166012} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((4,\ 166012,\ (\ :1/2, 1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(1.789507409\)
\(L(\frac12)\)  \(\approx\)  \(1.789507409\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;7,\;11\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
7$C_1$ \( 1 - T \)
11$C_2$ \( 1 + p T^{2} \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
5$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
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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

−9.609502670858404311814404044997, −8.590184015029907492289028892260, −8.138332901724243479464013903419, −7.57571100088867902110310233811, −7.44513617281958576804529864077, −7.09822342200141219854200263851, −5.99194604685105443423740465950, −5.57928681742950427486583645839, −5.52175284893141423110412623173, −4.70505559546438205265880319119, −3.97818544551430562677542015666, −3.26959289583375955783222233557, −2.68232544077048635906506868696, −2.07443948231294496741763787694, −0.901588970916622851028488806901, 0.901588970916622851028488806901, 2.07443948231294496741763787694, 2.68232544077048635906506868696, 3.26959289583375955783222233557, 3.97818544551430562677542015666, 4.70505559546438205265880319119, 5.52175284893141423110412623173, 5.57928681742950427486583645839, 5.99194604685105443423740465950, 7.09822342200141219854200263851, 7.44513617281958576804529864077, 7.57571100088867902110310233811, 8.138332901724243479464013903419, 8.590184015029907492289028892260, 9.609502670858404311814404044997

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