Properties

Degree 4
Conductor $ 3^{2} \cdot 5^{2} \cdot 17^{2} $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 4-s − 4·7-s + 9-s − 2·12-s − 3·16-s − 8·19-s − 8·21-s − 5·25-s − 4·27-s + 4·28-s − 36-s + 8·37-s − 6·48-s − 2·49-s − 16·57-s − 4·63-s + 7·64-s + 8·73-s − 10·75-s + 8·76-s − 11·81-s + 8·84-s − 16·97-s + 5·100-s + 4·108-s + 16·111-s + ⋯
L(s)  = 1  + 1.15·3-s − 1/2·4-s − 1.51·7-s + 1/3·9-s − 0.577·12-s − 3/4·16-s − 1.83·19-s − 1.74·21-s − 25-s − 0.769·27-s + 0.755·28-s − 1/6·36-s + 1.31·37-s − 0.866·48-s − 2/7·49-s − 2.11·57-s − 0.503·63-s + 7/8·64-s + 0.936·73-s − 1.15·75-s + 0.917·76-s − 1.22·81-s + 0.872·84-s − 1.62·97-s + 1/2·100-s + 0.384·108-s + 1.51·111-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 65025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 65025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(65025\)    =    \(3^{2} \cdot 5^{2} \cdot 17^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{65025} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(4,\ 65025,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;5,\;17\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{3,\;5,\;17\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad3$C_2$ \( 1 - 2 T + p T^{2} \)
5$C_2$ \( 1 + p T^{2} \)
17$C_2$ \( 1 + p T^{2} \)
good2$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
11$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - p T^{2} )^{2} \)
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$C_2^2$ \( 1 + 82 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
61$C_2$ \( ( 1 - p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
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\[\begin{aligned} L(s) = \prod_{\mathfrak{p}\ \mathrm{bad}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s})^{-1} \prod_{\mathfrak{p}\ \mathrm{good}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s} + (N\mathfrak{p})^{-2s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−9.390308195248213925262890437657, −9.296213617157272403992900715524, −8.715467295265622491630819838571, −8.142958304425255316228906608291, −7.87073308765623355773912108302, −7.00703267673792101031355011290, −6.50669597987628232679925335373, −6.13818345852553580176885032227, −5.39084355385318890088281032417, −4.40814951700777868244446600572, −4.05028937660463924524384788233, −3.37337138113178617546598395794, −2.67938102376015495844193066401, −2.04426910526792620981111038824, 0, 2.04426910526792620981111038824, 2.67938102376015495844193066401, 3.37337138113178617546598395794, 4.05028937660463924524384788233, 4.40814951700777868244446600572, 5.39084355385318890088281032417, 6.13818345852553580176885032227, 6.50669597987628232679925335373, 7.00703267673792101031355011290, 7.87073308765623355773912108302, 8.142958304425255316228906608291, 8.715467295265622491630819838571, 9.296213617157272403992900715524, 9.390308195248213925262890437657

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