Properties

Label 4-864e2-1.1-c2e2-0-1
Degree $4$
Conductor $746496$
Sign $1$
Analytic cond. $554.239$
Root an. cond. $4.85204$
Motivic weight $2$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s − 10·7-s − 10·11-s + 25·25-s + 100·29-s + 38·31-s + 20·35-s + 49·49-s + 94·53-s + 20·55-s + 20·59-s − 50·73-s + 100·77-s + 116·79-s + 134·83-s + 190·97-s + 20·103-s + 86·107-s + 121·121-s − 142·125-s + 127-s + 131-s + 137-s + 139-s − 200·145-s + 149-s + 151-s + ⋯
L(s)  = 1  − 2/5·5-s − 1.42·7-s − 0.909·11-s + 25-s + 3.44·29-s + 1.22·31-s + 4/7·35-s + 49-s + 1.77·53-s + 4/11·55-s + 0.338·59-s − 0.684·73-s + 1.29·77-s + 1.46·79-s + 1.61·83-s + 1.95·97-s + 0.194·103-s + 0.803·107-s + 121-s − 1.13·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s − 1.37·145-s + 0.00671·149-s + 0.00662·151-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 746496 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 746496 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(746496\)    =    \(2^{10} \cdot 3^{6}\)
Sign: $1$
Analytic conductor: \(554.239\)
Root analytic conductor: \(4.85204\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{864} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 746496,\ (\ :1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.894481310\)
\(L(\frac12)\) \(\approx\) \(1.894481310\)
\(L(2)\) 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 \( 1 \)
good5$C_2^2$ \( 1 + 2 T - 21 T^{2} + 2 p^{2} T^{3} + p^{4} T^{4} \)
7$C_2^2$ \( 1 + 10 T + 51 T^{2} + 10 p^{2} T^{3} + p^{4} T^{4} \)
11$C_2^2$ \( 1 + 10 T - 21 T^{2} + 10 p^{2} T^{3} + p^{4} T^{4} \)
13$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
17$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
19$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
23$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
29$C_2$ \( ( 1 - 50 T + p^{2} T^{2} )^{2} \)
31$C_2^2$ \( 1 - 38 T + 483 T^{2} - 38 p^{2} T^{3} + p^{4} T^{4} \)
37$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
41$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
43$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
47$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
53$C_2^2$ \( 1 - 94 T + 6027 T^{2} - 94 p^{2} T^{3} + p^{4} T^{4} \)
59$C_2$ \( ( 1 - 10 T + p^{2} T^{2} )^{2} \)
61$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
67$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
71$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
73$C_2^2$ \( 1 + 50 T - 2829 T^{2} + 50 p^{2} T^{3} + p^{4} T^{4} \)
79$C_2$ \( ( 1 - 58 T + p^{2} T^{2} )^{2} \)
83$C_2^2$ \( 1 - 134 T + 11067 T^{2} - 134 p^{2} T^{3} + p^{4} T^{4} \)
89$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
97$C_2^2$ \( 1 - 190 T + 26691 T^{2} - 190 p^{2} T^{3} + p^{4} T^{4} \)
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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

−10.27698690499847492894371733495, −10.06075186511197498786992640977, −9.280162239023777035467651422160, −8.817320071468195396354310990645, −8.627913734733030451301225583250, −8.011126648380145098618843864587, −7.74577637533978357897971955422, −7.02657822281629818148146991200, −6.77168657627613631206641639461, −6.28165334955427960098520591649, −6.04347984886334806470054103310, −5.26021288921134958447220360003, −4.63818243729398263996721538850, −4.62251208921023152475242741351, −3.57019412820927212081875870152, −3.33028297615826984181789921790, −2.56396859392918943898518990170, −2.46398985917455291624305097961, −0.974323119477032501061001207524, −0.57864916617339310231452718149, 0.57864916617339310231452718149, 0.974323119477032501061001207524, 2.46398985917455291624305097961, 2.56396859392918943898518990170, 3.33028297615826984181789921790, 3.57019412820927212081875870152, 4.62251208921023152475242741351, 4.63818243729398263996721538850, 5.26021288921134958447220360003, 6.04347984886334806470054103310, 6.28165334955427960098520591649, 6.77168657627613631206641639461, 7.02657822281629818148146991200, 7.74577637533978357897971955422, 8.011126648380145098618843864587, 8.627913734733030451301225583250, 8.817320071468195396354310990645, 9.280162239023777035467651422160, 10.06075186511197498786992640977, 10.27698690499847492894371733495

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