Properties

Degree 4
Conductor $ 2^{16} \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 − 2·5-s + 3·9-s + 12·13-s − 4·15-s − 25-s + 4·27-s + 16·31-s + 4·37-s + 24·39-s − 4·41-s − 8·43-s − 6·45-s + 10·49-s − 12·53-s − 24·65-s − 16·67-s + 24·71-s − 2·75-s + 5·81-s + 8·83-s − 20·89-s + 32·93-s − 24·107-s + 8·111-s + 36·117-s + 18·121-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.894·5-s + 9-s + 3.32·13-s − 1.03·15-s − 1/5·25-s + 0.769·27-s + 2.87·31-s + 0.657·37-s + 3.84·39-s − 0.624·41-s − 1.21·43-s − 0.894·45-s + 10/7·49-s − 1.64·53-s − 2.97·65-s − 1.95·67-s + 2.84·71-s − 0.230·75-s + 5/9·81-s + 0.878·83-s − 2.11·89-s + 3.31·93-s − 2.32·107-s + 0.759·111-s + 3.32·117-s + 1.63·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 14745600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14745600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(14745600\)    =    \(2^{16} \cdot 3^{2} \cdot 5^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{3840} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((4,\ 14745600,\ (\ :1/2, 1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(4.891059824\)
\(L(\frac12)\)  \(\approx\)  \(4.891059824\)
\(L(\frac{3}{2})\)   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$C_1$ \( ( 1 - T )^{2} \)
5$C_2$ \( 1 + 2 T + p T^{2} \)
good7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
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

−8.513275540089964195522773002856, −8.443956919788316230091552925038, −8.108102182958424751085592907644, −7.75466662178013593451115977926, −7.33152834929353575735137745207, −6.83856423514350708286463306376, −6.40954027438300541165022039644, −6.11728987448086918048478547519, −6.02668805031417311521054762534, −5.13410895862985464579805844317, −4.83117280368541482849724808824, −4.25471601595203104119118644008, −3.88503491778614909349831885469, −3.74271592482130219761233705900, −3.27607771599875131833892687277, −2.84530438744157926399235612444, −2.41968875467660309684656623631, −1.42825744328556809689928239227, −1.40458780056428103856653163408, −0.64980809887545010900785055522, 0.64980809887545010900785055522, 1.40458780056428103856653163408, 1.42825744328556809689928239227, 2.41968875467660309684656623631, 2.84530438744157926399235612444, 3.27607771599875131833892687277, 3.74271592482130219761233705900, 3.88503491778614909349831885469, 4.25471601595203104119118644008, 4.83117280368541482849724808824, 5.13410895862985464579805844317, 6.02668805031417311521054762534, 6.11728987448086918048478547519, 6.40954027438300541165022039644, 6.83856423514350708286463306376, 7.33152834929353575735137745207, 7.75466662178013593451115977926, 8.108102182958424751085592907644, 8.443956919788316230091552925038, 8.513275540089964195522773002856

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