Properties

Degree 4
Conductor $ 2^{2} \cdot 7^{3} \cdot 11^{2} $
Sign $-1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 1

Origins

Origins of factors

Downloads

Learn more about

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  =  1
Selberg data  =  $(4,\ 166012,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;7,\;11\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]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$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
show more
show less
\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.888226006934558931566258496787, −8.624667079434487291694904422005, −8.145906800879012290031802042193, −7.55563465371540871894331307120, −6.82397964191468499058519060303, −6.42455331216463290634831658204, −6.12986038892916091786195943044, −5.76321350617402115136418955935, −4.85990751466979898816213085515, −4.09052365620160024762211243831, −3.87985229263682105648190826671, −3.03039761753313177141617611476, −2.25140513775369021989127014661, −1.67120553980965395706477292535, 0, 1.67120553980965395706477292535, 2.25140513775369021989127014661, 3.03039761753313177141617611476, 3.87985229263682105648190826671, 4.09052365620160024762211243831, 4.85990751466979898816213085515, 5.76321350617402115136418955935, 6.12986038892916091786195943044, 6.42455331216463290634831658204, 6.82397964191468499058519060303, 7.55563465371540871894331307120, 8.145906800879012290031802042193, 8.624667079434487291694904422005, 8.888226006934558931566258496787

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