Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 11-s − 4·16-s − 11·31-s + 9·41-s − 61-s + 19·71-s − 9·81-s + 101-s + 103-s + 107-s + 109-s + 113-s + 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 4·176-s + 179-s + 181-s + 191-s + ⋯
L(s)  = 1  − 0.301·11-s − 16-s − 1.97·31-s + 1.40·41-s − 0.128·61-s + 2.25·71-s − 81-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.0760·173-s + 0.301·176-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 3125 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 3125 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(3125\)    =    \(5^{5}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{3125} (1, \cdot )$
Sato-Tate  :  $F_{ac}$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 3125,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.7183136284$
$L(\frac12)$  $\approx$  $0.7183136284$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 5$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p = 5$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad5 \( 1 \)
good2$C_2$ \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \)
3$C_2^2$ \( 1 + p^{2} T^{4} \)
7$C_2^2$ \( 1 + p^{2} T^{4} \)
11$C_4$ \( 1 + T - 9 T^{2} + p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + p^{2} T^{4} \)
17$C_2^2$ \( 1 + p^{2} T^{4} \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + p^{2} T^{4} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_4$ \( 1 + 11 T + 61 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + p^{2} T^{4} \)
41$C_4$ \( 1 - 9 T + 71 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + p^{2} T^{4} \)
47$C_2^2$ \( 1 + p^{2} T^{4} \)
53$C_2^2$ \( 1 + p^{2} T^{4} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_4$ \( 1 + T + 91 T^{2} + p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + p^{2} T^{4} \)
71$C_4$ \( 1 - 19 T + 201 T^{2} - 19 p T^{3} + p^{2} T^{4} \)
73$C_2^2$ \( 1 + p^{2} T^{4} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + p^{2} T^{4} \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
97$C_2^2$ \( 1 + p^{2} T^{4} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−18.2513478698, −17.794030318, −17.0572865553, −16.6516421521, −15.9944202971, −15.6665478707, −15.0412242767, −14.3272056793, −14.0663480476, −13.2082005437, −12.8575532333, −12.2756492446, −11.4301570976, −11.07580254, −10.4664572499, −9.62470469199, −9.14526688174, −8.48931188846, −7.62285385181, −7.1021858939, −6.22989238233, −5.43292897511, −4.57292147662, −3.58578684104, −2.26074119601, 2.26074119601, 3.58578684104, 4.57292147662, 5.43292897511, 6.22989238233, 7.1021858939, 7.62285385181, 8.48931188846, 9.14526688174, 9.62470469199, 10.4664572499, 11.07580254, 11.4301570976, 12.2756492446, 12.8575532333, 13.2082005437, 14.0663480476, 14.3272056793, 15.0412242767, 15.6665478707, 15.9944202971, 16.6516421521, 17.0572865553, 17.794030318, 18.2513478698

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