Properties

Degree $4$
Conductor $4955076$
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  + 4-s − 2·7-s + 9-s − 8·11-s + 12·13-s + 16-s + 4·17-s − 6·25-s − 2·28-s − 4·29-s + 36-s − 20·37-s − 8·43-s − 8·44-s + 3·49-s + 12·52-s + 6·53-s + 8·59-s − 2·63-s + 64-s + 4·68-s + 16·77-s + 81-s − 12·89-s − 24·91-s − 28·97-s − 8·99-s + ⋯
L(s)  = 1  + 1/2·4-s − 0.755·7-s + 1/3·9-s − 2.41·11-s + 3.32·13-s + 1/4·16-s + 0.970·17-s − 6/5·25-s − 0.377·28-s − 0.742·29-s + 1/6·36-s − 3.28·37-s − 1.21·43-s − 1.20·44-s + 3/7·49-s + 1.66·52-s + 0.824·53-s + 1.04·59-s − 0.251·63-s + 1/8·64-s + 0.485·68-s + 1.82·77-s + 1/9·81-s − 1.27·89-s − 2.51·91-s − 2.84·97-s − 0.804·99-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4955076 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4955076 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(4955076\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 53^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{4955076} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 4955076,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.659141855\)
\(L(\frac12)\) \(\approx\) \(1.659141855\)
\(L(\frac{3}{2})\) 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_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
3$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
7$C_1$ \( ( 1 + T )^{2} \)
53$C_2$ \( 1 - 6 T + p T^{2} \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
11$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
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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

−7.31618436492604863435902543640, −6.84552480602986516155667793601, −6.56225054608614269754758337655, −6.05857616111761188736923156685, −5.61588131716717336152713515261, −5.33985014787602837985094112170, −5.24124351541382054400781904279, −4.16551102118751694100907626234, −3.67268213755857288293135595950, −3.62482887081886485478101246558, −3.02877457374453947654735474964, −2.58459732524656909069587264431, −1.64795775077704015094055119225, −1.55929188669763310789284398036, −0.43629256145672715618319237912, 0.43629256145672715618319237912, 1.55929188669763310789284398036, 1.64795775077704015094055119225, 2.58459732524656909069587264431, 3.02877457374453947654735474964, 3.62482887081886485478101246558, 3.67268213755857288293135595950, 4.16551102118751694100907626234, 5.24124351541382054400781904279, 5.33985014787602837985094112170, 5.61588131716717336152713515261, 6.05857616111761188736923156685, 6.56225054608614269754758337655, 6.84552480602986516155667793601, 7.31618436492604863435902543640

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