Properties

Degree 4
Conductor $ 2^{4} \cdot 3^{8} \cdot 5^{2} $
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  + 5-s − 2·7-s − 2·13-s − 12·17-s − 8·19-s − 6·23-s − 6·29-s + 4·31-s − 2·35-s + 4·37-s − 6·41-s + 10·43-s + 6·47-s + 7·49-s − 12·53-s − 12·59-s − 2·61-s − 2·65-s − 2·67-s − 24·71-s + 4·73-s − 8·79-s − 6·83-s − 12·85-s − 12·89-s + 4·91-s − 8·95-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.755·7-s − 0.554·13-s − 2.91·17-s − 1.83·19-s − 1.25·23-s − 1.11·29-s + 0.718·31-s − 0.338·35-s + 0.657·37-s − 0.937·41-s + 1.52·43-s + 0.875·47-s + 49-s − 1.64·53-s − 1.56·59-s − 0.256·61-s − 0.248·65-s − 0.244·67-s − 2.84·71-s + 0.468·73-s − 0.900·79-s − 0.658·83-s − 1.30·85-s − 1.27·89-s + 0.419·91-s − 0.820·95-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2624400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2624400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(2624400\)    =    \(2^{4} \cdot 3^{8} \cdot 5^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{1620} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(2\)
Selberg data  =  \((4,\ 2624400,\ (\ :1/2, 1/2),\ 1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5$C_2$ \( 1 - T + T^{2} \)
good7$C_2^2$ \( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
17$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 10 T + 57 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 2 T - 57 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 6 T - 47 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 + 2 T - 93 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
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

−9.128268260075230639978810280679, −8.879609668633676560438179903739, −8.518701332677239511880295642056, −8.128006439029553491986246914026, −7.38160816732570281031336149602, −7.30574166128665362043917886705, −6.57545210475335878147183798094, −6.44791198724747539593326744519, −5.90277677478490330798174406784, −5.85851390667869065406076060278, −4.92223288312488907651471559014, −4.47514029994835495312362165577, −4.16105349651013886794664529191, −3.93944715685640249462758706971, −2.76324459854099422021748940251, −2.72807111839117201323872077617, −2.02989984762113154748685340570, −1.63169261711885065541736310622, 0, 0, 1.63169261711885065541736310622, 2.02989984762113154748685340570, 2.72807111839117201323872077617, 2.76324459854099422021748940251, 3.93944715685640249462758706971, 4.16105349651013886794664529191, 4.47514029994835495312362165577, 4.92223288312488907651471559014, 5.85851390667869065406076060278, 5.90277677478490330798174406784, 6.44791198724747539593326744519, 6.57545210475335878147183798094, 7.30574166128665362043917886705, 7.38160816732570281031336149602, 8.128006439029553491986246914026, 8.518701332677239511880295642056, 8.879609668633676560438179903739, 9.128268260075230639978810280679

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