Properties

Degree $4$
Conductor $44100$
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-s − 3-s − 5-s + 6-s + 4·7-s + 8-s + 10-s + 11-s + 14·13-s − 4·14-s + 15-s − 16-s + 4·17-s − 19-s − 4·21-s − 22-s − 23-s − 24-s − 14·26-s + 27-s − 16·29-s − 30-s − 6·31-s − 33-s − 4·34-s − 4·35-s + 3·37-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s − 0.447·5-s + 0.408·6-s + 1.51·7-s + 0.353·8-s + 0.316·10-s + 0.301·11-s + 3.88·13-s − 1.06·14-s + 0.258·15-s − 1/4·16-s + 0.970·17-s − 0.229·19-s − 0.872·21-s − 0.213·22-s − 0.208·23-s − 0.204·24-s − 2.74·26-s + 0.192·27-s − 2.97·29-s − 0.182·30-s − 1.07·31-s − 0.174·33-s − 0.685·34-s − 0.676·35-s + 0.493·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 44100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(44100\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{210} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 44100,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.947887\)
\(L(\frac12)\) \(\approx\) \(0.947887\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T + T^{2} \)
3$C_2$ \( 1 + T + T^{2} \)
5$C_2$ \( 1 + T + T^{2} \)
7$C_2$ \( 1 - 4 T + p T^{2} \)
good11$C_2^2$ \( 1 - T - 10 T^{2} - p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - 4 T - T^{2} - 4 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
23$C_2^2$ \( 1 + T - 22 T^{2} + p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 + 6 T + 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 3 T - 28 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 3 T - 38 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - T - 52 T^{2} - p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 4 T - 45 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 12 T + 77 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 14 T + 123 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 17 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 2 T - 85 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
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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

−12.54222679626572656058241459616, −11.87709362304223388791814807686, −11.31436905617365287546044175032, −11.12932067639503526792264648730, −10.82982075231035922705162496598, −10.57486391943161221813996213901, −9.321348377552733244967478787206, −9.265336235159572918269579666828, −8.500901659901231653412929222067, −8.301127107896027728664376862286, −7.60354183261529834616723244201, −7.39747376377291974433566265062, −6.25500043333902438535148609731, −5.78941777267089190256945797876, −5.61158003553484384411560577567, −4.46917981207639980008355754257, −3.90167171031770626257004290166, −3.48712101176861286367841274171, −1.69377058548200865114807002463, −1.19813431854546906531279411642, 1.19813431854546906531279411642, 1.69377058548200865114807002463, 3.48712101176861286367841274171, 3.90167171031770626257004290166, 4.46917981207639980008355754257, 5.61158003553484384411560577567, 5.78941777267089190256945797876, 6.25500043333902438535148609731, 7.39747376377291974433566265062, 7.60354183261529834616723244201, 8.301127107896027728664376862286, 8.500901659901231653412929222067, 9.265336235159572918269579666828, 9.321348377552733244967478787206, 10.57486391943161221813996213901, 10.82982075231035922705162496598, 11.12932067639503526792264648730, 11.31436905617365287546044175032, 11.87709362304223388791814807686, 12.54222679626572656058241459616

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