Properties

Label 4-273e2-1.1-c1e2-0-17
Degree $4$
Conductor $74529$
Sign $1$
Analytic cond. $4.75203$
Root an. cond. $1.47645$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s − 2·4-s + 4·7-s + 6·9-s − 6·12-s + 5·13-s + 6·19-s + 12·21-s − 10·25-s + 9·27-s − 8·28-s − 12·36-s − 10·37-s + 15·39-s − 13·43-s + 9·49-s − 10·52-s + 18·57-s − 15·61-s + 24·63-s + 8·64-s − 11·67-s − 30·75-s − 12·76-s − 26·79-s + 9·81-s − 24·84-s + ⋯
L(s)  = 1  + 1.73·3-s − 4-s + 1.51·7-s + 2·9-s − 1.73·12-s + 1.38·13-s + 1.37·19-s + 2.61·21-s − 2·25-s + 1.73·27-s − 1.51·28-s − 2·36-s − 1.64·37-s + 2.40·39-s − 1.98·43-s + 9/7·49-s − 1.38·52-s + 2.38·57-s − 1.92·61-s + 3.02·63-s + 64-s − 1.34·67-s − 3.46·75-s − 1.37·76-s − 2.92·79-s + 81-s − 2.61·84-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74529 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74529 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(74529\)    =    \(3^{2} \cdot 7^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(4.75203\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 74529,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.606837235\)
\(L(\frac12)\) \(\approx\) \(2.606837235\)
\(L(\frac{3}{2})\) 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)$
bad3$C_2$ \( 1 - p T + p T^{2} \)
7$C_2$ \( 1 - 4 T + p T^{2} \)
13$C_2$ \( 1 - 5 T + p T^{2} \)
good2$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
5$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + T + p T^{2} ) \)
23$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
37$C_2$ \( ( 1 - T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
41$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 + 5 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - p T^{2} )^{2} \)
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
71$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 17 T + p T^{2} ) \)
79$C_2$ \( ( 1 + 13 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 19 T + p T^{2} )( 1 - 14 T + p T^{2} ) \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.01775081598151894906884131328, −11.64649395896150823072218025337, −11.37501183017857877846891693132, −10.48381003860056307863465221018, −10.16641845627850680700476578894, −9.628603214583999606269110785913, −8.936973863531753034304449028783, −8.925950451305429702959682283376, −8.253558633798219654478234338012, −8.082943810695686673275596558862, −7.45513720265543218143103192689, −7.08623592823649647770118944379, −6.01882596893972613950782762000, −5.48773395311062042178303329598, −4.61041336590011580804505857709, −4.44984241155225366877211282197, −3.42907042790280078111511858902, −3.37554555061242399061086779178, −1.94807609762442698936696248302, −1.51455789003945511277452601152, 1.51455789003945511277452601152, 1.94807609762442698936696248302, 3.37554555061242399061086779178, 3.42907042790280078111511858902, 4.44984241155225366877211282197, 4.61041336590011580804505857709, 5.48773395311062042178303329598, 6.01882596893972613950782762000, 7.08623592823649647770118944379, 7.45513720265543218143103192689, 8.082943810695686673275596558862, 8.253558633798219654478234338012, 8.925950451305429702959682283376, 8.936973863531753034304449028783, 9.628603214583999606269110785913, 10.16641845627850680700476578894, 10.48381003860056307863465221018, 11.37501183017857877846891693132, 11.64649395896150823072218025337, 12.01775081598151894906884131328

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