Properties

Degree 4
Conductor $ 2^{4} \cdot 3^{2} \cdot 5^{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  − 2·3-s + 4·7-s + 9-s + 4·13-s − 8·19-s − 8·21-s + 25-s + 4·27-s − 8·31-s + 4·37-s − 8·39-s − 20·43-s − 2·49-s + 16·57-s + 4·61-s + 4·63-s + 4·67-s + 4·73-s − 2·75-s + 16·79-s − 11·81-s + 16·91-s + 16·93-s + 4·97-s + 28·103-s + 4·109-s − 8·111-s + ⋯
L(s)  = 1  − 1.15·3-s + 1.51·7-s + 1/3·9-s + 1.10·13-s − 1.83·19-s − 1.74·21-s + 1/5·25-s + 0.769·27-s − 1.43·31-s + 0.657·37-s − 1.28·39-s − 3.04·43-s − 2/7·49-s + 2.11·57-s + 0.512·61-s + 0.503·63-s + 0.488·67-s + 0.468·73-s − 0.230·75-s + 1.80·79-s − 1.22·81-s + 1.67·91-s + 1.65·93-s + 0.406·97-s + 2.75·103-s + 0.383·109-s − 0.759·111-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(3600\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{3600} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((4,\ 3600,\ (\ :1/2, 1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(0.6180626677\)
\(L(\frac12)\)  \(\approx\)  \(0.6180626677\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

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

−12.64919926214782333798361971460, −11.66835822739294093715813029994, −11.50854453196989171707939391081, −10.83473950301065015207539180976, −10.66869217043212474435745329357, −9.749524176059453166876684221310, −8.552217550781204179646223792184, −8.536040373596851277524077678909, −7.64589778281564359800589548721, −6.57891116465648258947670054106, −6.21444115782927123192608967686, −5.19647848534431193907436500640, −4.78130792717525308450176413839, −3.71119793411816721603957340078, −1.81793015252092636076156145980, 1.81793015252092636076156145980, 3.71119793411816721603957340078, 4.78130792717525308450176413839, 5.19647848534431193907436500640, 6.21444115782927123192608967686, 6.57891116465648258947670054106, 7.64589778281564359800589548721, 8.536040373596851277524077678909, 8.552217550781204179646223792184, 9.749524176059453166876684221310, 10.66869217043212474435745329357, 10.83473950301065015207539180976, 11.50854453196989171707939391081, 11.66835822739294093715813029994, 12.64919926214782333798361971460

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