Properties

Degree 4
Conductor $ 3^{2} \cdot 7^{3} $
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  − 2·2-s − 4-s − 7-s + 8·8-s + 9-s + 8·11-s + 2·14-s − 7·16-s − 2·18-s − 16·22-s − 6·25-s + 28-s − 4·29-s − 14·32-s − 36-s + 12·37-s − 8·43-s − 8·44-s + 49-s + 12·50-s + 12·53-s − 8·56-s + 8·58-s − 63-s + 35·64-s + 8·67-s + 8·72-s + ⋯
L(s)  = 1  − 1.41·2-s − 1/2·4-s − 0.377·7-s + 2.82·8-s + 1/3·9-s + 2.41·11-s + 0.534·14-s − 7/4·16-s − 0.471·18-s − 3.41·22-s − 6/5·25-s + 0.188·28-s − 0.742·29-s − 2.47·32-s − 1/6·36-s + 1.97·37-s − 1.21·43-s − 1.20·44-s + 1/7·49-s + 1.69·50-s + 1.64·53-s − 1.06·56-s + 1.05·58-s − 0.125·63-s + 35/8·64-s + 0.977·67-s + 0.942·72-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 3087 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 3087 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(3087\)    =    \(3^{2} \cdot 7^{3}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{3087} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 3087,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.3258344524$
$L(\frac12)$  $\approx$  $0.3258344524$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;7\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{3,\;7\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad3$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
7$C_1$ \( 1 + T \)
good2$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
79$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
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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

−13.09697998821175856125894317781, −11.98659319295797721290412051260, −11.55595081754559805664353277943, −10.77789244508403227045944829665, −9.863214598051417092763914791206, −9.724661210726173973340366981146, −9.163045893798789171919999202805, −8.672365196683144779961098572501, −7.977686193886306753143677629185, −7.25047783802838427330028450541, −6.53484921474790582970192984243, −5.48510765590071063887274763802, −4.13559084050773741974089362056, −4.01671863219499060051377207353, −1.40786771277750203137263322761, 1.40786771277750203137263322761, 4.01671863219499060051377207353, 4.13559084050773741974089362056, 5.48510765590071063887274763802, 6.53484921474790582970192984243, 7.25047783802838427330028450541, 7.977686193886306753143677629185, 8.672365196683144779961098572501, 9.163045893798789171919999202805, 9.724661210726173973340366981146, 9.863214598051417092763914791206, 10.77789244508403227045944829665, 11.55595081754559805664353277943, 11.98659319295797721290412051260, 13.09697998821175856125894317781

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