Properties

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

Origins

Origins of factors

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 4-s + 4·5-s − 2·7-s − 2·13-s − 3·16-s − 2·17-s − 4·19-s + 4·20-s + 4·23-s + 2·25-s − 2·28-s + 8·29-s − 10·31-s − 8·35-s − 8·37-s − 18·41-s − 16·43-s − 10·47-s + 3·49-s − 2·52-s − 8·53-s − 2·59-s + 10·61-s − 7·64-s − 8·65-s + 20·67-s − 2·68-s + ⋯
L(s)  = 1  + 1/2·4-s + 1.78·5-s − 0.755·7-s − 0.554·13-s − 3/4·16-s − 0.485·17-s − 0.917·19-s + 0.894·20-s + 0.834·23-s + 2/5·25-s − 0.377·28-s + 1.48·29-s − 1.79·31-s − 1.35·35-s − 1.31·37-s − 2.81·41-s − 2.43·43-s − 1.45·47-s + 3/7·49-s − 0.277·52-s − 1.09·53-s − 0.260·59-s + 1.28·61-s − 7/8·64-s − 0.992·65-s + 2.44·67-s − 0.242·68-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  =  2
Selberg data  =  $(4,\ 58110129,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;7,\;11\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]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$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \)
5$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
13$D_{4}$ \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 2 T + 30 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
29$C_4$ \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 10 T + 82 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 18 T + 158 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
43$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
47$D_{4}$ \( 1 + 10 T + 114 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 2 T + 114 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 10 T + 142 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 20 T + 214 T^{2} - 20 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 12 T + 158 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 6 T + 150 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 78 T^{2} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 4 T - 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
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

−7.72862752904650541078578623544, −6.92485576817166123002217388980, −6.84178033878918086473019447893, −6.67615697970454416412119202383, −6.54585758853931192056088207643, −6.10091394869788754672873127488, −5.55162246720538219839015970625, −5.20127743241274905586898795999, −4.95672095849414624037505487683, −4.86843857665934763990730064093, −3.98621323110010483607957956695, −3.63534980479515226249206214833, −3.28395844464187692191426110187, −2.84620157227462634890446400260, −2.27704977660579191787807640991, −1.97058496082294878063569811738, −1.83697598738312793209164987775, −1.24665757722581832356087043882, 0, 0, 1.24665757722581832356087043882, 1.83697598738312793209164987775, 1.97058496082294878063569811738, 2.27704977660579191787807640991, 2.84620157227462634890446400260, 3.28395844464187692191426110187, 3.63534980479515226249206214833, 3.98621323110010483607957956695, 4.86843857665934763990730064093, 4.95672095849414624037505487683, 5.20127743241274905586898795999, 5.55162246720538219839015970625, 6.10091394869788754672873127488, 6.54585758853931192056088207643, 6.67615697970454416412119202383, 6.84178033878918086473019447893, 6.92485576817166123002217388980, 7.72862752904650541078578623544

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