Properties

Degree 4
Conductor $ 2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 53^{2} $
Sign $-1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 1

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 4-s − 8·7-s + 9-s + 4·13-s + 16-s + 12·17-s + 25-s − 8·28-s − 12·29-s + 36-s + 4·37-s − 8·43-s + 34·49-s + 4·52-s − 6·53-s − 8·63-s + 64-s + 12·68-s + 81-s + 36·89-s − 32·91-s + 4·97-s + 100-s − 24·107-s − 8·112-s − 36·113-s − 12·116-s + ⋯
L(s)  = 1  + 1/2·4-s − 3.02·7-s + 1/3·9-s + 1.10·13-s + 1/4·16-s + 2.91·17-s + 1/5·25-s − 1.51·28-s − 2.22·29-s + 1/6·36-s + 0.657·37-s − 1.21·43-s + 34/7·49-s + 0.554·52-s − 0.824·53-s − 1.00·63-s + 1/8·64-s + 1.45·68-s + 1/9·81-s + 3.81·89-s − 3.35·91-s + 0.406·97-s + 1/10·100-s − 2.32·107-s − 0.755·112-s − 3.38·113-s − 1.11·116-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 2528100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 2528100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(2528100\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 53^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{2528100} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(4,\ 2528100,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;5,\;53\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;5,\;53\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
3$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
5$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
53$C_2$ \( 1 + 6 T + p T^{2} \)
good7$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 18 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
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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

−7.40299975179098898623930398115, −6.93165528247052272579764875319, −6.42217617652666865799421983967, −6.29699741617005267930031156094, −5.92460482285255092724220602337, −5.37985442259477011829539614630, −5.16033961728710776108999214836, −3.92189663406678546150391342843, −3.78458604271642354350170110042, −3.36585804145949552210873743484, −3.11065830387495411526913738527, −2.55325351537140642139443836766, −1.59750473666486892284412590277, −0.978287305185245428011662937257, 0, 0.978287305185245428011662937257, 1.59750473666486892284412590277, 2.55325351537140642139443836766, 3.11065830387495411526913738527, 3.36585804145949552210873743484, 3.78458604271642354350170110042, 3.92189663406678546150391342843, 5.16033961728710776108999214836, 5.37985442259477011829539614630, 5.92460482285255092724220602337, 6.29699741617005267930031156094, 6.42217617652666865799421983967, 6.93165528247052272579764875319, 7.40299975179098898623930398115

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