Properties

Label 4-80e4-1.1-c1e2-0-33
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 $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 2·7-s + 4·11-s + 8·19-s + 4·21-s − 2·23-s + 2·27-s − 4·31-s − 8·33-s − 4·37-s + 4·41-s − 14·43-s − 10·47-s − 8·49-s + 16·53-s − 16·57-s + 8·59-s − 4·61-s − 18·67-s + 4·69-s − 4·71-s + 8·73-s − 8·77-s − 16·79-s − 81-s − 6·83-s − 4·89-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.755·7-s + 1.20·11-s + 1.83·19-s + 0.872·21-s − 0.417·23-s + 0.384·27-s − 0.718·31-s − 1.39·33-s − 0.657·37-s + 0.624·41-s − 2.13·43-s − 1.45·47-s − 8/7·49-s + 2.19·53-s − 2.11·57-s + 1.04·59-s − 0.512·61-s − 2.19·67-s + 0.481·69-s − 0.474·71-s + 0.936·73-s − 0.911·77-s − 1.80·79-s − 1/9·81-s − 0.658·83-s − 0.423·89-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: \(2\)
Selberg data: \((4,\ 40960000,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
5 \( 1 \)
good3$D_{4}$ \( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 + 2 T + 12 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 8 T + 42 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 2 T + 20 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$D_{4}$ \( 1 - 4 T + 74 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 14 T + 132 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 10 T + 116 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 16 T + 158 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 8 T + 122 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 18 T + 212 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 4 T + 134 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 8 T + 150 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 16 T + 174 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 6 T + 172 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 4 T + 134 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.51267124893288286738546843904, −7.48441769632462767461954643604, −7.16155534845415713616108572187, −6.68206735959554567566606648358, −6.25590751244613118552963330408, −6.24602930633120868417973549712, −5.65918875960848637895850258512, −5.53212893275684604920367679819, −4.93938191233125999326760709179, −4.84384424609689517875199023981, −4.25092979716496975445499045939, −3.67018408792319571140739612666, −3.41870124415684562841436515501, −3.23350412118007930517527162137, −2.55296280200935095951953355786, −1.98814002947627337760997010091, −1.28948747315570848740068437462, −1.13804597489230440655421950718, 0, 0, 1.13804597489230440655421950718, 1.28948747315570848740068437462, 1.98814002947627337760997010091, 2.55296280200935095951953355786, 3.23350412118007930517527162137, 3.41870124415684562841436515501, 3.67018408792319571140739612666, 4.25092979716496975445499045939, 4.84384424609689517875199023981, 4.93938191233125999326760709179, 5.53212893275684604920367679819, 5.65918875960848637895850258512, 6.24602930633120868417973549712, 6.25590751244613118552963330408, 6.68206735959554567566606648358, 7.16155534845415713616108572187, 7.48441769632462767461954643604, 7.51267124893288286738546843904

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