Properties

Label 4-90e4-1.1-c1e2-0-2
Degree $4$
Conductor $65610000$
Sign $1$
Analytic cond. $4183.35$
Root an. cond. $8.04231$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 12·11-s − 4·19-s − 6·29-s − 8·31-s − 12·41-s + 10·49-s + 24·59-s + 10·61-s + 12·71-s + 20·79-s + 6·89-s + 12·101-s + 14·109-s + 86·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  − 3.61·11-s − 0.917·19-s − 1.11·29-s − 1.43·31-s − 1.87·41-s + 10/7·49-s + 3.12·59-s + 1.28·61-s + 1.42·71-s + 2.25·79-s + 0.635·89-s + 1.19·101-s + 1.34·109-s + 7.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1/13·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65610000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65610000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(65610000\)    =    \(2^{4} \cdot 3^{8} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(4183.35\)
Root analytic conductor: \(8.04231\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{8100} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 65610000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9885520064\)
\(L(\frac12)\) \(\approx\) \(0.9885520064\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
good7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 49 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 145 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 94 T^{2} + p^{2} T^{4} \)
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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

−8.043646670752233854528589348122, −7.51286835482754762533081633188, −7.39595517181118366384425852162, −7.21591532347231014218535267455, −6.52249938862100524283843083494, −6.39289644502238233384097934576, −5.69256116737658238253300917196, −5.45958955349260255145868380700, −5.23292382961246028370071152511, −5.04607003206315903452143038662, −4.66208884705078039334772759680, −3.94209099472952464764194416203, −3.63130951905797775119081344365, −3.43986437011666863051361522215, −2.69085462913392214965185512487, −2.31876788625713483086797959618, −2.25546324494438266553196219455, −1.77257783546612488084048496557, −0.70567478024749120046517004194, −0.31566750087458468002636643968, 0.31566750087458468002636643968, 0.70567478024749120046517004194, 1.77257783546612488084048496557, 2.25546324494438266553196219455, 2.31876788625713483086797959618, 2.69085462913392214965185512487, 3.43986437011666863051361522215, 3.63130951905797775119081344365, 3.94209099472952464764194416203, 4.66208884705078039334772759680, 5.04607003206315903452143038662, 5.23292382961246028370071152511, 5.45958955349260255145868380700, 5.69256116737658238253300917196, 6.39289644502238233384097934576, 6.52249938862100524283843083494, 7.21591532347231014218535267455, 7.39595517181118366384425852162, 7.51286835482754762533081633188, 8.043646670752233854528589348122

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