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  + 2-s − 2·4-s − 2·5-s − 2·7-s − 3·8-s − 2·10-s + 2·13-s − 2·14-s + 16-s + 6·17-s + 4·20-s + 2·23-s − 7·25-s + 2·26-s + 4·28-s + 10·29-s − 6·31-s + 2·32-s + 6·34-s + 4·35-s − 14·37-s + 6·40-s + 4·41-s + 2·43-s + 2·46-s + 4·47-s + 3·49-s + ⋯
L(s)  = 1  + 0.707·2-s − 4-s − 0.894·5-s − 0.755·7-s − 1.06·8-s − 0.632·10-s + 0.554·13-s − 0.534·14-s + 1/4·16-s + 1.45·17-s + 0.894·20-s + 0.417·23-s − 7/5·25-s + 0.392·26-s + 0.755·28-s + 1.85·29-s − 1.07·31-s + 0.353·32-s + 1.02·34-s + 0.676·35-s − 2.30·37-s + 0.948·40-s + 0.624·41-s + 0.304·43-s + 0.294·46-s + 0.583·47-s + 3/7·49-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$D_{4}$ \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \)
5$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
13$D_{4}$ \( 1 - 2 T + 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 6 T + 38 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 7 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
31$D_{4}$ \( 1 + 6 T + 66 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
41$D_{4}$ \( 1 - 4 T + 66 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 2 T + 42 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 4 T + 93 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 2 T - 18 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 7 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$D_{4}$ \( 1 + 24 T + 273 T^{2} + 24 p T^{3} + p^{2} T^{4} \)
71$C_4$ \( 1 + 4 T + 126 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 18 T + 207 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 20 T + 238 T^{2} - 20 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 18 T + 242 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 158 T^{2} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 6 T + 158 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
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

−7.50681152914112853247300145288, −7.40031438710040045426745298275, −7.25545466232050191809061493116, −6.44482349559888624452860590815, −6.17496929889396809620409417684, −6.04751281375309092309197997169, −5.39494460446182717767291438229, −5.30193137437481278032074178052, −4.82386464855389994609493442070, −4.47338777330616396383442114690, −4.06591202214995441206036166884, −3.69881020298103924638242649404, −3.43304295095258331453813440752, −3.29604547639311874100633429578, −2.67357873709595706672890516908, −2.13247712610977997763890786436, −1.34619014789795605092339838172, −0.995850504174931781794216194055, 0, 0, 0.995850504174931781794216194055, 1.34619014789795605092339838172, 2.13247712610977997763890786436, 2.67357873709595706672890516908, 3.29604547639311874100633429578, 3.43304295095258331453813440752, 3.69881020298103924638242649404, 4.06591202214995441206036166884, 4.47338777330616396383442114690, 4.82386464855389994609493442070, 5.30193137437481278032074178052, 5.39494460446182717767291438229, 6.04751281375309092309197997169, 6.17496929889396809620409417684, 6.44482349559888624452860590815, 7.25545466232050191809061493116, 7.40031438710040045426745298275, 7.50681152914112853247300145288

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