Properties

Label 4-2800e2-1.1-c1e2-0-10
Degree $4$
Conductor $7840000$
Sign $1$
Analytic cond. $499.885$
Root an. cond. $4.72843$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 2·7-s − 9-s + 4·11-s + 4·13-s − 17-s − 3·19-s + 2·21-s − 7·23-s + 11·29-s + 10·31-s − 4·33-s − 3·37-s − 4·39-s − 9·41-s − 9·43-s + 3·49-s + 51-s + 6·53-s + 3·57-s + 2·59-s + 14·61-s + 2·63-s − 24·67-s + 7·69-s + 9·71-s + 17·73-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.755·7-s − 1/3·9-s + 1.20·11-s + 1.10·13-s − 0.242·17-s − 0.688·19-s + 0.436·21-s − 1.45·23-s + 2.04·29-s + 1.79·31-s − 0.696·33-s − 0.493·37-s − 0.640·39-s − 1.40·41-s − 1.37·43-s + 3/7·49-s + 0.140·51-s + 0.824·53-s + 0.397·57-s + 0.260·59-s + 1.79·61-s + 0.251·63-s − 2.93·67-s + 0.842·69-s + 1.06·71-s + 1.98·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7840000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7840000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(7840000\)    =    \(2^{8} \cdot 5^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(499.885\)
Root analytic conductor: \(4.72843\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 7840000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.751834556\)
\(L(\frac12)\) \(\approx\) \(1.751834556\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_1$ \( ( 1 + T )^{2} \)
good3$D_{4}$ \( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 - 4 T + 9 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
17$D_{4}$ \( 1 + T + 30 T^{2} + p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 3 T + 36 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 7 T + 54 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 11 T + 84 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 10 T + 70 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 3 T - 30 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 9 T + 98 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 9 T + 102 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 6 T + 98 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 2 T - 34 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 14 T + 154 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 24 T + 261 T^{2} + 24 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 9 T + 124 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 17 T + 214 T^{2} - 17 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + T + 120 T^{2} + p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 15 T + 184 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + T + 72 T^{2} + p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
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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

−8.688683455387155613581403776537, −8.686836316681712630416726265822, −8.325998806098089460925302225855, −8.105989771355250711645574270655, −7.36539368178389640425103257355, −6.84711308537793359288502485619, −6.59692640585276807443928084474, −6.29369175410166031692703303917, −6.15939621118696828784396489361, −5.66305412199900832488074718763, −5.11869881094963377079997945892, −4.64812467797487582699702756328, −4.25109913236858075574713377850, −3.87092216329191383093929410183, −3.21307062290574471078958944884, −3.19803025316492481133890247660, −2.20026850763490384135452967619, −1.89629193556227010361176397411, −1.02595510733081615167410126626, −0.52673456447090455597037909987, 0.52673456447090455597037909987, 1.02595510733081615167410126626, 1.89629193556227010361176397411, 2.20026850763490384135452967619, 3.19803025316492481133890247660, 3.21307062290574471078958944884, 3.87092216329191383093929410183, 4.25109913236858075574713377850, 4.64812467797487582699702756328, 5.11869881094963377079997945892, 5.66305412199900832488074718763, 6.15939621118696828784396489361, 6.29369175410166031692703303917, 6.59692640585276807443928084474, 6.84711308537793359288502485619, 7.36539368178389640425103257355, 8.105989771355250711645574270655, 8.325998806098089460925302225855, 8.686836316681712630416726265822, 8.688683455387155613581403776537

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