Properties

Label 4-3240e2-1.1-c1e2-0-2
Degree $4$
Conductor $10497600$
Sign $1$
Analytic cond. $669.336$
Root an. cond. $5.08640$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 4·7-s + 6·13-s + 4·17-s + 8·19-s − 8·23-s − 6·29-s + 4·35-s − 12·37-s + 10·41-s + 4·43-s + 8·47-s + 7·49-s − 20·53-s − 6·61-s − 6·65-s + 4·67-s − 28·73-s − 16·79-s + 12·83-s − 4·85-s − 4·89-s − 24·91-s − 8·95-s − 2·97-s − 14·101-s − 4·103-s + ⋯
L(s)  = 1  − 0.447·5-s − 1.51·7-s + 1.66·13-s + 0.970·17-s + 1.83·19-s − 1.66·23-s − 1.11·29-s + 0.676·35-s − 1.97·37-s + 1.56·41-s + 0.609·43-s + 1.16·47-s + 49-s − 2.74·53-s − 0.768·61-s − 0.744·65-s + 0.488·67-s − 3.27·73-s − 1.80·79-s + 1.31·83-s − 0.433·85-s − 0.423·89-s − 2.51·91-s − 0.820·95-s − 0.203·97-s − 1.39·101-s − 0.394·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(10497600\)    =    \(2^{6} \cdot 3^{8} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(669.336\)
Root analytic conductor: \(5.08640\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 10497600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8690323702\)
\(L(\frac12)\) \(\approx\) \(0.8690323702\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_2$ \( 1 + T + T^{2} \)
good7$C_2$ \( ( 1 - T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
11$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 8 T + 41 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 10 T + 59 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 4 T - 27 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 8 T + 17 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 6 T - 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 + 16 T + 177 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 - 12 T + 61 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 + 2 T - 93 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
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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

−8.879165441675549819162479000339, −8.486890620499976171795781357785, −8.019354862040483284786881520762, −7.66709261725692101914901298622, −7.43256133003818657128492178788, −7.03743928303854781838466860022, −6.56385533080558562704276352189, −5.99508919014626089323380956792, −5.97533616059358178362659850069, −5.55916685728081394932689784937, −5.22156391908365394800213222384, −4.25774743919011393031473327542, −4.22526218927235616495563985378, −3.52020820957182122929711149374, −3.40951499354888937676908472563, −3.03827374883899310747033426176, −2.46445452097762539289244494149, −1.40537062787603658694890605059, −1.40383064709506987211565168120, −0.30027630621290779772511235307, 0.30027630621290779772511235307, 1.40383064709506987211565168120, 1.40537062787603658694890605059, 2.46445452097762539289244494149, 3.03827374883899310747033426176, 3.40951499354888937676908472563, 3.52020820957182122929711149374, 4.22526218927235616495563985378, 4.25774743919011393031473327542, 5.22156391908365394800213222384, 5.55916685728081394932689784937, 5.97533616059358178362659850069, 5.99508919014626089323380956792, 6.56385533080558562704276352189, 7.03743928303854781838466860022, 7.43256133003818657128492178788, 7.66709261725692101914901298622, 8.019354862040483284786881520762, 8.486890620499976171795781357785, 8.879165441675549819162479000339

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