Properties

Degree 4
Conductor $ 5^{2} \cdot 41 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s − 5-s − 4·9-s + 3·11-s + 5·19-s + 2·20-s − 4·25-s + 11·29-s − 6·31-s + 8·36-s − 41-s − 6·44-s + 4·45-s − 6·49-s − 3·55-s + 59-s − 2·61-s + 8·64-s − 3·71-s − 10·76-s − 15·79-s + 7·81-s − 9·89-s − 5·95-s − 12·99-s + 8·100-s + 15·101-s + ⋯
L(s)  = 1  − 4-s − 0.447·5-s − 4/3·9-s + 0.904·11-s + 1.14·19-s + 0.447·20-s − 4/5·25-s + 2.04·29-s − 1.07·31-s + 4/3·36-s − 0.156·41-s − 0.904·44-s + 0.596·45-s − 6/7·49-s − 0.404·55-s + 0.130·59-s − 0.256·61-s + 64-s − 0.356·71-s − 1.14·76-s − 1.68·79-s + 7/9·81-s − 0.953·89-s − 0.512·95-s − 1.20·99-s + 4/5·100-s + 1.49·101-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 1025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 1025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(1025\)    =    \(5^{2} \cdot 41\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{1025} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 1025,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.4222141590$
$L(\frac12)$  $\approx$  $0.4222141590$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{5,\;41\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{5,\;41\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad5$C_2$ \( 1 + T + p T^{2} \)
41$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + p T^{2} ) \)
good2$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
3$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
7$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
11$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
13$C_2^2$ \( 1 - 16 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
19$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + T + p T^{2} ) \)
23$C_2^2$ \( 1 - 19 T^{2} + p^{2} T^{4} \)
29$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
31$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
37$C_2^2$ \( 1 - 61 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 40 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 74 T^{2} + p^{2} T^{4} \)
59$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
61$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
67$C_2^2$ \( 1 + 88 T^{2} + p^{2} T^{4} \)
71$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
73$C_2^2$ \( 1 + 109 T^{2} + p^{2} T^{4} \)
79$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
83$C_2^2$ \( 1 - 79 T^{2} + p^{2} T^{4} \)
89$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
97$C_2^2$ \( 1 + 152 T^{2} + p^{2} T^{4} \)
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

−14.19454045717374451127091353927, −13.78836223859467724349513577428, −13.02132215935876782760497681759, −12.16321002560880394673395714560, −11.64675702466879730087811532096, −11.19128473596752981447430276521, −10.09738201918885273142298867748, −9.423661061919065844869674749372, −8.748731100307867456017587191214, −8.282454962036726379628037425237, −7.29437775775195108905884201252, −6.23530102487327827298160382161, −5.31271891550307177711333795025, −4.34036075847737870492735412284, −3.22692316031770542346454550404, 3.22692316031770542346454550404, 4.34036075847737870492735412284, 5.31271891550307177711333795025, 6.23530102487327827298160382161, 7.29437775775195108905884201252, 8.282454962036726379628037425237, 8.748731100307867456017587191214, 9.423661061919065844869674749372, 10.09738201918885273142298867748, 11.19128473596752981447430276521, 11.64675702466879730087811532096, 12.16321002560880394673395714560, 13.02132215935876782760497681759, 13.78836223859467724349513577428, 14.19454045717374451127091353927

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