Properties

Degree $4$
Conductor $14745600$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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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

Degree: \(4\)
Conductor: \(14745600\)    =    \(2^{16} \cdot 3^{2} \cdot 5^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{3840} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 14745600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.238212068\)
\(L(\frac12)\) \(\approx\) \(1.238212068\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$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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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.734379806350938919729633373185, −8.325806178665803861551864609891, −7.908853449102611121118775583871, −7.46483876337736422533928003126, −7.23987004492467640026617087566, −6.84720043549624350358626348772, −6.22258883538268339401687359678, −6.14007842572282967512468526645, −5.62852528967901714807388011888, −5.60358478788117515773885525188, −4.97146855756344697124598267090, −4.25972264733339860129855668421, −4.11548539221463832232546862790, −3.82186515933564960511372617941, −3.35154896961385924792478773281, −2.99222608237856889134273527132, −1.93756372283623915499700052105, −1.60187568340872147871724595077, −1.03975617128410289720254036218, −0.42137465715308899645928942561, 0.42137465715308899645928942561, 1.03975617128410289720254036218, 1.60187568340872147871724595077, 1.93756372283623915499700052105, 2.99222608237856889134273527132, 3.35154896961385924792478773281, 3.82186515933564960511372617941, 4.11548539221463832232546862790, 4.25972264733339860129855668421, 4.97146855756344697124598267090, 5.60358478788117515773885525188, 5.62852528967901714807388011888, 6.14007842572282967512468526645, 6.22258883538268339401687359678, 6.84720043549624350358626348772, 7.23987004492467640026617087566, 7.46483876337736422533928003126, 7.908853449102611121118775583871, 8.325806178665803861551864609891, 8.734379806350938919729633373185

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