Properties

Label 4-84672-1.1-c1e2-0-17
Degree $4$
Conductor $84672$
Sign $-1$
Analytic cond. $5.39876$
Root an. cond. $1.52431$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 4·5-s + 9-s + 4·15-s + 4·17-s + 2·25-s − 27-s + 12·37-s − 12·41-s + 8·43-s − 4·45-s − 7·49-s − 4·51-s + 8·59-s − 8·67-s − 2·75-s − 16·79-s + 81-s − 8·83-s − 16·85-s − 12·89-s − 36·101-s − 4·109-s − 12·111-s − 6·121-s + 12·123-s + 28·125-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.78·5-s + 1/3·9-s + 1.03·15-s + 0.970·17-s + 2/5·25-s − 0.192·27-s + 1.97·37-s − 1.87·41-s + 1.21·43-s − 0.596·45-s − 49-s − 0.560·51-s + 1.04·59-s − 0.977·67-s − 0.230·75-s − 1.80·79-s + 1/9·81-s − 0.878·83-s − 1.73·85-s − 1.27·89-s − 3.58·101-s − 0.383·109-s − 1.13·111-s − 0.545·121-s + 1.08·123-s + 2.50·125-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84672 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84672 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(84672\)    =    \(2^{6} \cdot 3^{3} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(5.39876\)
Root analytic conductor: \(1.52431\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 84672,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 + T \)
7$C_2$ \( 1 + p T^{2} \)
good5$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
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$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
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

−9.570243823567071698489558383687, −8.882269440513519494115608595002, −8.098990694093691505068092710868, −8.070493008159816865665193495447, −7.48262505518830506099686358492, −7.01507823160614642101417797126, −6.42897107072744719896577758454, −5.68882761687599646139467960885, −5.31393935703508661243014911411, −4.26006602995316384735017711370, −4.25303028692796488061253338187, −3.44517807903584076206897070581, −2.73151126809740041559980819161, −1.33234813740225181579568980351, 0, 1.33234813740225181579568980351, 2.73151126809740041559980819161, 3.44517807903584076206897070581, 4.25303028692796488061253338187, 4.26006602995316384735017711370, 5.31393935703508661243014911411, 5.68882761687599646139467960885, 6.42897107072744719896577758454, 7.01507823160614642101417797126, 7.48262505518830506099686358492, 8.070493008159816865665193495447, 8.098990694093691505068092710868, 8.882269440513519494115608595002, 9.570243823567071698489558383687

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