Properties

Label 4-840e2-1.1-c1e2-0-51
Degree $4$
Conductor $705600$
Sign $1$
Analytic cond. $44.9896$
Root an. cond. $2.58987$
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  − 2·5-s + 9-s + 8·13-s + 12·17-s + 3·25-s − 4·29-s + 4·37-s + 20·41-s − 2·45-s + 49-s + 8·53-s + 12·61-s − 16·65-s − 24·73-s + 81-s − 24·85-s − 20·89-s + 16·97-s + 12·101-s + 36·109-s + 8·117-s − 18·121-s − 4·125-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  − 0.894·5-s + 1/3·9-s + 2.21·13-s + 2.91·17-s + 3/5·25-s − 0.742·29-s + 0.657·37-s + 3.12·41-s − 0.298·45-s + 1/7·49-s + 1.09·53-s + 1.53·61-s − 1.98·65-s − 2.80·73-s + 1/9·81-s − 2.60·85-s − 2.11·89-s + 1.62·97-s + 1.19·101-s + 3.44·109-s + 0.739·117-s − 1.63·121-s − 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 705600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 705600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(705600\)    =    \(2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(44.9896\)
Root analytic conductor: \(2.58987\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 705600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.428088104\)
\(L(\frac12)\) \(\approx\) \(2.428088104\)
\(L(\frac{3}{2})\) 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$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
5$C_1$ \( ( 1 + T )^{2} \)
7$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good11$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
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$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
73$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
89$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
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

−8.332271248013021932698556597271, −7.68317577599397836445076898677, −7.51396217146347047365173011116, −7.31185634397920351045683211126, −6.33243037687430168783386839948, −6.06002067218225743849105703999, −5.62881426763657287662958714868, −5.26529891888972619664454770587, −4.22625940543306930004754131404, −4.15883759269872432242069575338, −3.44474307442050975548787517711, −3.26028767051448009361622725203, −2.37434208740026948037764083228, −1.13225769705224512477609730978, −1.04831954899182159972970043573, 1.04831954899182159972970043573, 1.13225769705224512477609730978, 2.37434208740026948037764083228, 3.26028767051448009361622725203, 3.44474307442050975548787517711, 4.15883759269872432242069575338, 4.22625940543306930004754131404, 5.26529891888972619664454770587, 5.62881426763657287662958714868, 6.06002067218225743849105703999, 6.33243037687430168783386839948, 7.31185634397920351045683211126, 7.51396217146347047365173011116, 7.68317577599397836445076898677, 8.332271248013021932698556597271

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