Properties

Label 4-80e4-1.1-c1e2-0-18
Degree $4$
Conductor $40960000$
Sign $1$
Analytic cond. $2611.64$
Root an. cond. $7.14872$
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  − 9-s + 8·13-s + 6·17-s + 8·29-s + 16·37-s + 10·41-s − 14·49-s + 8·53-s + 16·61-s + 18·73-s − 8·81-s + 30·89-s + 4·97-s + 2·113-s − 8·117-s − 17·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 6·153-s + 157-s + 163-s + 167-s + 22·169-s + ⋯
L(s)  = 1  − 1/3·9-s + 2.21·13-s + 1.45·17-s + 1.48·29-s + 2.63·37-s + 1.56·41-s − 2·49-s + 1.09·53-s + 2.04·61-s + 2.10·73-s − 8/9·81-s + 3.17·89-s + 0.406·97-s + 0.188·113-s − 0.739·117-s − 1.54·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 0.485·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.69·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 40960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40960000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(40960000\)    =    \(2^{16} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(2611.64\)
Root analytic conductor: \(7.14872\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{6400} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 40960000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.059062624\)
\(L(\frac12)\) \(\approx\) \(5.059062624\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
good3$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2^2$ \( 1 + 17 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
19$C_2^2$ \( 1 + 33 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 89 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 121 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 15 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
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

−8.178833739113274228790155030005, −7.87578198971099780593245180647, −7.71200258701202988527179133311, −7.18457468321643443753253307762, −6.56506660099144910712706362383, −6.42148271699315207010383670336, −6.04193685492397176513380800139, −5.95686343036384286121563316093, −5.26939069807350142130492209008, −5.12741908486885863128051115047, −4.60021140137269249809631041711, −4.08644891454800828178811114779, −3.67659336810190408368152696911, −3.62924183486464120899544522075, −2.89342311514597227421484031791, −2.67241019895291349767595549830, −2.13295121587361647972285050240, −1.39064664265563924031973611622, −0.892112337888599768436419517508, −0.78778378874472832130220495508, 0.78778378874472832130220495508, 0.892112337888599768436419517508, 1.39064664265563924031973611622, 2.13295121587361647972285050240, 2.67241019895291349767595549830, 2.89342311514597227421484031791, 3.62924183486464120899544522075, 3.67659336810190408368152696911, 4.08644891454800828178811114779, 4.60021140137269249809631041711, 5.12741908486885863128051115047, 5.26939069807350142130492209008, 5.95686343036384286121563316093, 6.04193685492397176513380800139, 6.42148271699315207010383670336, 6.56506660099144910712706362383, 7.18457468321643443753253307762, 7.71200258701202988527179133311, 7.87578198971099780593245180647, 8.178833739113274228790155030005

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