Properties

Degree 4
Conductor $ 2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 11^{2} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 3-s + 2·4-s + 5-s + 2·6-s − 4·7-s − 2·9-s − 2·10-s + 2·11-s − 2·12-s + 8·14-s − 15-s − 4·16-s − 4·17-s + 4·18-s + 2·20-s + 4·21-s − 4·22-s − 4·25-s + 5·27-s − 8·28-s + 2·30-s + 8·32-s − 2·33-s + 8·34-s − 4·35-s − 4·36-s + ⋯
L(s)  = 1  − 1.41·2-s − 0.577·3-s + 4-s + 0.447·5-s + 0.816·6-s − 1.51·7-s − 2/3·9-s − 0.632·10-s + 0.603·11-s − 0.577·12-s + 2.13·14-s − 0.258·15-s − 16-s − 0.970·17-s + 0.942·18-s + 0.447·20-s + 0.872·21-s − 0.852·22-s − 4/5·25-s + 0.962·27-s − 1.51·28-s + 0.365·30-s + 1.41·32-s − 0.348·33-s + 1.37·34-s − 0.676·35-s − 2/3·36-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 435600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 435600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(435600\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 11^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{435600} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 435600,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.2079652962$
$L(\frac12)$  $\approx$  $0.2079652962$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;5,\;11\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;5,\;11\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_2$ \( 1 + p T + p T^{2} \)
3$C_2$ \( 1 + T + p T^{2} \)
5$C_2$ \( 1 - T + p T^{2} \)
11$C_1$ \( ( 1 - T )^{2} \)
good7$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
43$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
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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

−8.603539619290756001226038948684, −8.399321207327632052846971537333, −7.79929959554017504493480383489, −7.11798617010190808321460327259, −6.75700481068273204827606843552, −6.36261389471308870138602900888, −6.16530520176828730622789022769, −5.41325790295005799624258361008, −4.92404269467477878351185887984, −4.19846804805603804873348477595, −3.55134596504967876675656965993, −2.91823833046493033163603826267, −2.22901314573716685967671256577, −1.47396545601602715073166640408, −0.32265848357517983832832095703, 0.32265848357517983832832095703, 1.47396545601602715073166640408, 2.22901314573716685967671256577, 2.91823833046493033163603826267, 3.55134596504967876675656965993, 4.19846804805603804873348477595, 4.92404269467477878351185887984, 5.41325790295005799624258361008, 6.16530520176828730622789022769, 6.36261389471308870138602900888, 6.75700481068273204827606843552, 7.11798617010190808321460327259, 7.79929959554017504493480383489, 8.399321207327632052846971537333, 8.603539619290756001226038948684

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