Properties

Degree 4
Conductor $ 2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7^{4} $
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  − 4-s + 4·5-s − 9-s + 4·11-s + 16-s − 4·20-s + 11·25-s + 16·31-s + 36-s − 4·41-s − 4·44-s − 4·45-s + 16·55-s + 20·59-s − 4·61-s − 64-s + 24·71-s + 4·80-s + 81-s − 20·89-s − 4·99-s − 11·100-s + 16·101-s − 20·109-s − 10·121-s − 16·124-s + 24·125-s + ⋯
L(s)  = 1  − 1/2·4-s + 1.78·5-s − 1/3·9-s + 1.20·11-s + 1/4·16-s − 0.894·20-s + 11/5·25-s + 2.87·31-s + 1/6·36-s − 0.624·41-s − 0.603·44-s − 0.596·45-s + 2.15·55-s + 2.60·59-s − 0.512·61-s − 1/8·64-s + 2.84·71-s + 0.447·80-s + 1/9·81-s − 2.11·89-s − 0.402·99-s − 1.09·100-s + 1.59·101-s − 1.91·109-s − 0.909·121-s − 1.43·124-s + 2.14·125-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2160900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(2160900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{1470} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((4,\ 2160900,\ (\ :1/2, 1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(3.577460900\)
\(L(\frac12)\)  \(\approx\)  \(3.577460900\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_2$ \( 1 + T^{2} \)
3$C_2$ \( 1 + T^{2} \)
5$C_2$ \( 1 - 4 T + p T^{2} \)
7 \( 1 \)
good11$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 150 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
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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

−9.593148554494952185696762801018, −9.477712427087874597735074880976, −8.955552114257713804099780591386, −8.534156466492753092464422334564, −8.313272395432225578297641864524, −7.88094156907897375622247621512, −7.01389206298808619852366448788, −6.77572921875698048847123232792, −6.39644751423259458431223566533, −6.12271611463750799361807268494, −5.51024514190473386978132584240, −5.26316828814816728971688627562, −4.74496370413980754209600536122, −4.26985893026235069627115502598, −3.73917199616717111911288681286, −3.11398472898885359095516035133, −2.53322819024110257694410010583, −2.12487353228441756462978713389, −1.30261018380362140928162292080, −0.865571406357834911579609005348, 0.865571406357834911579609005348, 1.30261018380362140928162292080, 2.12487353228441756462978713389, 2.53322819024110257694410010583, 3.11398472898885359095516035133, 3.73917199616717111911288681286, 4.26985893026235069627115502598, 4.74496370413980754209600536122, 5.26316828814816728971688627562, 5.51024514190473386978132584240, 6.12271611463750799361807268494, 6.39644751423259458431223566533, 6.77572921875698048847123232792, 7.01389206298808619852366448788, 7.88094156907897375622247621512, 8.313272395432225578297641864524, 8.534156466492753092464422334564, 8.955552114257713804099780591386, 9.477712427087874597735074880976, 9.593148554494952185696762801018

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