Properties

Degree 4
Conductor $ 2^{7} \cdot 5^{2} \cdot 11^{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  + 4·3-s + 7·9-s − 4·11-s − 4·13-s − 2·17-s − 4·19-s − 8·23-s − 5·25-s + 4·27-s − 16·33-s − 16·39-s + 8·47-s − 5·49-s − 8·51-s − 16·57-s + 8·67-s − 32·69-s − 12·73-s − 20·75-s + 8·79-s − 8·81-s + 14·89-s − 28·99-s − 16·103-s − 28·117-s + 5·121-s + 127-s + ⋯
L(s)  = 1  + 2.30·3-s + 7/3·9-s − 1.20·11-s − 1.10·13-s − 0.485·17-s − 0.917·19-s − 1.66·23-s − 25-s + 0.769·27-s − 2.78·33-s − 2.56·39-s + 1.16·47-s − 5/7·49-s − 1.12·51-s − 2.11·57-s + 0.977·67-s − 3.85·69-s − 1.40·73-s − 2.30·75-s + 0.900·79-s − 8/9·81-s + 1.48·89-s − 2.81·99-s − 1.57·103-s − 2.58·117-s + 5/11·121-s + 0.0887·127-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 387200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 387200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(387200\)    =    \(2^{7} \cdot 5^{2} \cdot 11^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{387200} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(4,\ 387200,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;5,\;11\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;5,\;11\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2 \( 1 \)
5$C_2$ \( 1 + p T^{2} \)
11$C_2$ \( 1 + 4 T + p T^{2} \)
good3$C_2$$\times$$C_2$ \( ( 1 - p T + p T^{2} )( 1 - T + p T^{2} ) \)
7$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
13$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
19$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
23$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
31$C_2^2$ \( 1 - 3 T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 19 T^{2} + p^{2} T^{4} \)
41$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
47$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \)
53$C_2^2$ \( 1 - 37 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 98 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 37 T^{2} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2^2$ \( 1 - 35 T^{2} + p^{2} T^{4} \)
73$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
79$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
83$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
89$C_2$$\times$$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 - 3 T + p T^{2} ) \)
97$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \)
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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

−8.356367122239688119465632523598, −8.013161623003913810526172289868, −7.75083754905669529596832792615, −7.41923409947571712417361584709, −6.73374282118455424420870009136, −6.15593493399353886403929655220, −5.55558341586249995711814231944, −4.99375321774290804247339921270, −4.21612644919682573004336207208, −3.99278789798550888415055709377, −3.22791246775205293569541102966, −2.72124716201087153697626339042, −2.13796367393343059861647482431, −2.02864381350513604826483179141, 0, 2.02864381350513604826483179141, 2.13796367393343059861647482431, 2.72124716201087153697626339042, 3.22791246775205293569541102966, 3.99278789798550888415055709377, 4.21612644919682573004336207208, 4.99375321774290804247339921270, 5.55558341586249995711814231944, 6.15593493399353886403929655220, 6.73374282118455424420870009136, 7.41923409947571712417361584709, 7.75083754905669529596832792615, 8.013161623003913810526172289868, 8.356367122239688119465632523598

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