Properties

Degree 4
Conductor $ 5^{4} \cdot 7^{2} \cdot 23^{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  + 2-s + 2·3-s − 2·4-s + 2·6-s + 2·7-s − 3·8-s − 3·9-s − 4·12-s + 4·13-s + 2·14-s + 16-s − 3·18-s − 10·19-s + 4·21-s + 2·23-s − 6·24-s + 4·26-s − 14·27-s − 4·28-s + 6·29-s − 18·31-s + 2·32-s + 6·36-s − 2·37-s − 10·38-s + 8·39-s + 4·42-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.15·3-s − 4-s + 0.816·6-s + 0.755·7-s − 1.06·8-s − 9-s − 1.15·12-s + 1.10·13-s + 0.534·14-s + 1/4·16-s − 0.707·18-s − 2.29·19-s + 0.872·21-s + 0.417·23-s − 1.22·24-s + 0.784·26-s − 2.69·27-s − 0.755·28-s + 1.11·29-s − 3.23·31-s + 0.353·32-s + 36-s − 0.328·37-s − 1.62·38-s + 1.28·39-s + 0.617·42-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16200625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16200625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(16200625\)    =    \(5^{4} \cdot 7^{2} \cdot 23^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{4025} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(2\)
Selberg data  =  \((4,\ 16200625,\ (\ :1/2, 1/2),\ 1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{5,\;7,\;23\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{5,\;7,\;23\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad5 \( 1 \)
7$C_1$ \( ( 1 - T )^{2} \)
23$C_1$ \( ( 1 - T )^{2} \)
good2$D_{4}$ \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \)
3$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
11$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
13$D_{4}$ \( 1 - 4 T + 25 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + p T^{2} )^{2} \)
19$D_{4}$ \( 1 + 10 T + 58 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 6 T + 47 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
37$D_{4}$ \( 1 + 2 T + 30 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
41$C_2^2$ \( 1 + 77 T^{2} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 4 T + 70 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 2 T + 75 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 18 T + 182 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 12 T + 134 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 16 T + 201 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
73$C_2^2$ \( 1 + 101 T^{2} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 10 T + 178 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 4 T + 150 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 98 T^{2} + p^{2} T^{4} \)
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

−8.350329933734952409859327177506, −8.125933760229292172296227731292, −7.59722754166431819530618122077, −7.39590964487500001541968724071, −6.70549368654592222642490328159, −6.23876918012109201011593600370, −5.86541483197722003307724598324, −5.77890108494033546614824547625, −4.98968771633073778616943756383, −4.98868513729720291122205823140, −4.31628866076734236043525363972, −4.08238891614911344155956974363, −3.61938829907485201539472834940, −3.30668076776429185708201866173, −2.89752439716216048790040016075, −2.32208641849278377881195589094, −1.82352050973926868699688820598, −1.39833687032560782469157297142, 0, 0, 1.39833687032560782469157297142, 1.82352050973926868699688820598, 2.32208641849278377881195589094, 2.89752439716216048790040016075, 3.30668076776429185708201866173, 3.61938829907485201539472834940, 4.08238891614911344155956974363, 4.31628866076734236043525363972, 4.98868513729720291122205823140, 4.98968771633073778616943756383, 5.77890108494033546614824547625, 5.86541483197722003307724598324, 6.23876918012109201011593600370, 6.70549368654592222642490328159, 7.39590964487500001541968724071, 7.59722754166431819530618122077, 8.125933760229292172296227731292, 8.350329933734952409859327177506

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