Properties

Label 4-252e2-1.1-c3e2-0-2
Degree $4$
Conductor $63504$
Sign $1$
Analytic cond. $221.071$
Root an. cond. $3.85596$
Motivic weight $3$
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  − 37·7-s + 178·13-s + 163·19-s + 125·25-s + 19·31-s + 433·37-s + 898·43-s + 1.02e3·49-s − 182·61-s − 1.00e3·67-s + 919·73-s − 503·79-s − 6.58e3·91-s − 2.66e3·97-s + 19·103-s + 1.56e3·109-s + 1.33e3·121-s + 127-s + 131-s − 6.03e3·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  − 1.99·7-s + 3.79·13-s + 1.96·19-s + 25-s + 0.110·31-s + 1.92·37-s + 3.18·43-s + 2.99·49-s − 0.382·61-s − 1.83·67-s + 1.47·73-s − 0.716·79-s − 7.58·91-s − 2.78·97-s + 0.0181·103-s + 1.37·109-s + 121-s + 0.000698·127-s + 0.000666·131-s − 3.93·133-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63504 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63504 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(63504\)    =    \(2^{4} \cdot 3^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(221.071\)
Root analytic conductor: \(3.85596\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 63504,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(2.899098620\)
\(L(\frac12)\) \(\approx\) \(2.899098620\)
\(L(\frac{5}{2})\) 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 \( 1 \)
7$C_2$ \( 1 + 37 T + p^{3} T^{2} \)
good5$C_2^2$ \( 1 - p^{3} T^{2} + p^{6} T^{4} \)
11$C_2^2$ \( 1 - p^{3} T^{2} + p^{6} T^{4} \)
13$C_2$ \( ( 1 - 89 T + p^{3} T^{2} )^{2} \)
17$C_2^2$ \( 1 - p^{3} T^{2} + p^{6} T^{4} \)
19$C_2$ \( ( 1 - 107 T + p^{3} T^{2} )( 1 - 56 T + p^{3} T^{2} ) \)
23$C_2^2$ \( 1 - p^{3} T^{2} + p^{6} T^{4} \)
29$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
31$C_2$ \( ( 1 - 308 T + p^{3} T^{2} )( 1 + 289 T + p^{3} T^{2} ) \)
37$C_2$ \( ( 1 - 323 T + p^{3} T^{2} )( 1 - 110 T + p^{3} T^{2} ) \)
41$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
43$C_2$ \( ( 1 - 449 T + p^{3} T^{2} )^{2} \)
47$C_2^2$ \( 1 - p^{3} T^{2} + p^{6} T^{4} \)
53$C_2^2$ \( 1 - p^{3} T^{2} + p^{6} T^{4} \)
59$C_2^2$ \( 1 - p^{3} T^{2} + p^{6} T^{4} \)
61$C_2$ \( ( 1 - 719 T + p^{3} T^{2} )( 1 + 901 T + p^{3} T^{2} ) \)
67$C_2$ \( ( 1 + 127 T + p^{3} T^{2} )( 1 + 880 T + p^{3} T^{2} ) \)
71$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
73$C_2$ \( ( 1 - 1190 T + p^{3} T^{2} )( 1 + 271 T + p^{3} T^{2} ) \)
79$C_2$ \( ( 1 - 884 T + p^{3} T^{2} )( 1 + 1387 T + p^{3} T^{2} ) \)
83$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
89$C_2^2$ \( 1 - p^{3} T^{2} + p^{6} T^{4} \)
97$C_2$ \( ( 1 + 1330 T + p^{3} T^{2} )^{2} \)
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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

−11.70590681576842493966553960138, −11.34378914664148130294773976324, −10.76707851053638031530109253656, −10.61359898254344370671068778424, −9.817189925845689694851714872995, −9.387745442176583721619123490349, −8.901575701749760013159730014988, −8.763820095806108690421341583587, −7.80222347106362064644519515669, −7.47715849709593328140660720171, −6.48122715099855660866464534333, −6.43275887400981311265324539650, −5.80619511458025153892959648891, −5.51690785068043219668600762510, −4.07502494686584631679024140890, −3.95433941747557553840356413728, −3.08530979194786457608277993893, −2.86341616661481952328288138261, −1.14117738000301919970041639552, −0.874151340891841604116570156248, 0.874151340891841604116570156248, 1.14117738000301919970041639552, 2.86341616661481952328288138261, 3.08530979194786457608277993893, 3.95433941747557553840356413728, 4.07502494686584631679024140890, 5.51690785068043219668600762510, 5.80619511458025153892959648891, 6.43275887400981311265324539650, 6.48122715099855660866464534333, 7.47715849709593328140660720171, 7.80222347106362064644519515669, 8.763820095806108690421341583587, 8.901575701749760013159730014988, 9.387745442176583721619123490349, 9.817189925845689694851714872995, 10.61359898254344370671068778424, 10.76707851053638031530109253656, 11.34378914664148130294773976324, 11.70590681576842493966553960138

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