Properties

Label 4-684e2-1.1-c1e2-0-22
Degree $4$
Conductor $467856$
Sign $1$
Analytic cond. $29.8309$
Root an. cond. $2.33704$
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  − 2·7-s + 7·13-s + 8·19-s + 5·25-s + 22·31-s − 2·37-s − 5·43-s − 11·49-s + 13·61-s − 5·67-s + 7·73-s + 13·79-s − 14·91-s − 14·97-s − 26·103-s − 2·109-s − 22·121-s + 127-s + 131-s − 16·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  − 0.755·7-s + 1.94·13-s + 1.83·19-s + 25-s + 3.95·31-s − 0.328·37-s − 0.762·43-s − 1.57·49-s + 1.66·61-s − 0.610·67-s + 0.819·73-s + 1.46·79-s − 1.46·91-s − 1.42·97-s − 2.56·103-s − 0.191·109-s − 2·121-s + 0.0887·127-s + 0.0873·131-s − 1.38·133-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 467856 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 467856 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(467856\)    =    \(2^{4} \cdot 3^{4} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(29.8309\)
Root analytic conductor: \(2.33704\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 467856,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.210440835\)
\(L(\frac12)\) \(\approx\) \(2.210440835\)
\(L(\frac{3}{2})\) 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 \( 1 \)
19$C_2$ \( 1 - 8 T + p T^{2} \)
good5$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
17$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
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$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
47$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + T + p T^{2} ) \)
67$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
71$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 19 T + p T^{2} ) \)
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

−10.58879529167237870064069907125, −10.29493081965397001916050512431, −9.738899619383556626914400116752, −9.548448162232928792631016333427, −9.002651724365164864793080192360, −8.451295298584189737520893350974, −8.009888937858998171785173582196, −8.006174992979102293291042449070, −6.86417724246588241096283935443, −6.77188411045087413249770336375, −6.31109773730481798330143354615, −5.92610215535546317499030056157, −5.05159542177268024693787578947, −5.01504930672454926815802982300, −4.03767499036298439738601663654, −3.66299249576563919500134900564, −2.92436734698993218876634694339, −2.78396680113423007614359943982, −1.38182999948179400056795663445, −0.931201664825024208370380336841, 0.931201664825024208370380336841, 1.38182999948179400056795663445, 2.78396680113423007614359943982, 2.92436734698993218876634694339, 3.66299249576563919500134900564, 4.03767499036298439738601663654, 5.01504930672454926815802982300, 5.05159542177268024693787578947, 5.92610215535546317499030056157, 6.31109773730481798330143354615, 6.77188411045087413249770336375, 6.86417724246588241096283935443, 8.006174992979102293291042449070, 8.009888937858998171785173582196, 8.451295298584189737520893350974, 9.002651724365164864793080192360, 9.548448162232928792631016333427, 9.738899619383556626914400116752, 10.29493081965397001916050512431, 10.58879529167237870064069907125

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