Properties

Degree $4$
Conductor $4626801$
Sign $1$
Motivic weight $0$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 4-s + 2·5-s + 2·8-s − 4·10-s + 2·11-s − 4·16-s + 2·17-s + 2·20-s − 4·22-s + 25-s + 2·29-s − 2·31-s + 2·32-s − 4·34-s + 4·40-s + 2·44-s − 2·50-s + 4·55-s − 4·58-s + 2·61-s + 4·62-s + 3·64-s − 2·67-s + 2·68-s − 8·80-s − 2·83-s + ⋯
L(s)  = 1  − 2·2-s + 4-s + 2·5-s + 2·8-s − 4·10-s + 2·11-s − 4·16-s + 2·17-s + 2·20-s − 4·22-s + 25-s + 2·29-s − 2·31-s + 2·32-s − 4·34-s + 4·40-s + 2·44-s − 2·50-s + 4·55-s − 4·58-s + 2·61-s + 4·62-s + 3·64-s − 2·67-s + 2·68-s − 8·80-s − 2·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4626801 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4626801 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(4626801\)    =    \(3^{4} \cdot 239^{2}\)
Sign: $1$
Motivic weight: \(0\)
Character: induced by $\chi_{2151} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 4626801,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7209577906\)
\(L(\frac12)\) \(\approx\) \(0.7209577906\)
\(L(1)\) 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)$
bad3 \( 1 \)
239$C_1$ \( ( 1 - T )^{2} \)
good2$C_2$ \( ( 1 + T + T^{2} )^{2} \)
5$C_2$ \( ( 1 - T + T^{2} )^{2} \)
7$C_2^2$ \( 1 + T^{4} \)
11$C_2$ \( ( 1 - T + T^{2} )^{2} \)
13$C_2^2$ \( 1 + T^{4} \)
17$C_2$ \( ( 1 - T + T^{2} )^{2} \)
19$C_2^2$ \( 1 + T^{4} \)
23$C_2^2$ \( 1 + T^{4} \)
29$C_2$ \( ( 1 - T + T^{2} )^{2} \)
31$C_2$ \( ( 1 + T + T^{2} )^{2} \)
37$C_2^2$ \( 1 + T^{4} \)
41$C_2^2$ \( 1 + T^{4} \)
43$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
47$C_2^2$ \( 1 + T^{4} \)
53$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
59$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
61$C_2$ \( ( 1 - T + T^{2} )^{2} \)
67$C_2$ \( ( 1 + T + T^{2} )^{2} \)
71$C_2$ \( ( 1 + T^{2} )^{2} \)
73$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
79$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
83$C_2$ \( ( 1 + T + T^{2} )^{2} \)
89$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
97$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
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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

−9.403900991727383454971185836199, −9.208704971631530789653962510550, −8.735792119741634860765562326784, −8.613284827665564718178877719368, −8.134508540993616110825691166980, −7.58930044922002904632021011796, −7.26809148398465079991793981317, −6.96484129443832487962415621855, −6.28936765790551897519371521509, −6.16191576010009351777978057046, −5.42078580743045591351431096127, −5.38510191980781384452550227640, −4.67576576351142655721747271211, −4.15390296872720246988521497798, −3.75070727266357459269629251941, −3.12570361017273950383304364602, −2.25127978319250414000309853719, −1.78027265371229092511307052832, −1.28936431270954069317179779520, −1.07026718400937328860511981678, 1.07026718400937328860511981678, 1.28936431270954069317179779520, 1.78027265371229092511307052832, 2.25127978319250414000309853719, 3.12570361017273950383304364602, 3.75070727266357459269629251941, 4.15390296872720246988521497798, 4.67576576351142655721747271211, 5.38510191980781384452550227640, 5.42078580743045591351431096127, 6.16191576010009351777978057046, 6.28936765790551897519371521509, 6.96484129443832487962415621855, 7.26809148398465079991793981317, 7.58930044922002904632021011796, 8.134508540993616110825691166980, 8.613284827665564718178877719368, 8.735792119741634860765562326784, 9.208704971631530789653962510550, 9.403900991727383454971185836199

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