Properties

Degree 4
Conductor $ 3^{4} \cdot 7^{2} \cdot 11^{4} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 2·4-s − 2·7-s − 3·8-s + 4·13-s − 2·14-s + 16-s + 4·17-s − 6·19-s + 4·23-s − 5·25-s + 4·26-s + 4·28-s + 10·29-s + 12·31-s + 2·32-s + 4·34-s + 6·37-s − 6·38-s − 4·41-s + 8·43-s + 4·46-s − 10·47-s + 3·49-s − 5·50-s − 8·52-s − 4·53-s + ⋯
L(s)  = 1  + 0.707·2-s − 4-s − 0.755·7-s − 1.06·8-s + 1.10·13-s − 0.534·14-s + 1/4·16-s + 0.970·17-s − 1.37·19-s + 0.834·23-s − 25-s + 0.784·26-s + 0.755·28-s + 1.85·29-s + 2.15·31-s + 0.353·32-s + 0.685·34-s + 0.986·37-s − 0.973·38-s − 0.624·41-s + 1.21·43-s + 0.589·46-s − 1.45·47-s + 3/7·49-s − 0.707·50-s − 1.10·52-s − 0.549·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 58110129 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58110129 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(58110129\)    =    \(3^{4} \cdot 7^{2} \cdot 11^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{7623} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 58110129,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $2.879063364$
$L(\frac12)$  $\approx$  $2.879063364$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;7,\;11\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{3,\;7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad3 \( 1 \)
7$C_1$ \( ( 1 + T )^{2} \)
11 \( 1 \)
good2$D_{4}$ \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \)
5$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
13$D_{4}$ \( 1 - 4 T + 25 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
19$D_{4}$ \( 1 + 6 T + 27 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
31$D_{4}$ \( 1 - 12 T + 78 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 6 T + 63 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 4 T + 66 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 8 T + 82 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 10 T + 99 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 4 T + 90 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 2 T + 39 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
61$C_4$ \( 1 + 16 T + 166 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 4 T + 13 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
71$C_4$ \( 1 + 4 T + 126 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 8 T + 117 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 8 T + 154 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 24 T + 290 T^{2} - 24 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 8 T + 174 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
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

−8.068525426037856782283717373937, −7.81831357650664007853897358258, −7.37975144829213036071411583800, −6.82764814858202645336872175834, −6.51195704777086668620401400515, −6.16340696168307906343622535329, −5.89571483048926721200884005157, −5.88323085076054928860637490054, −4.89890866966525158300282345702, −4.87593648425912416663440445001, −4.58978580035627707297643587465, −4.17806742122661092475262955999, −3.74877569420850693272746386257, −3.44453051566755071249069025030, −2.93850217898377106423188940653, −2.82670851124901494302144256654, −2.08770197072324305690088884815, −1.46774441461409195078879817024, −0.840012123146977486997558457768, −0.48989743324104262001226516657, 0.48989743324104262001226516657, 0.840012123146977486997558457768, 1.46774441461409195078879817024, 2.08770197072324305690088884815, 2.82670851124901494302144256654, 2.93850217898377106423188940653, 3.44453051566755071249069025030, 3.74877569420850693272746386257, 4.17806742122661092475262955999, 4.58978580035627707297643587465, 4.87593648425912416663440445001, 4.89890866966525158300282345702, 5.88323085076054928860637490054, 5.89571483048926721200884005157, 6.16340696168307906343622535329, 6.51195704777086668620401400515, 6.82764814858202645336872175834, 7.37975144829213036071411583800, 7.81831357650664007853897358258, 8.068525426037856782283717373937

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