Properties

Label 4-7920e2-1.1-c1e2-0-8
Degree $4$
Conductor $62726400$
Sign $1$
Analytic cond. $3999.48$
Root an. cond. $7.95245$
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 + 2·7-s + 2·11-s + 2·13-s + 2·17-s − 8·23-s + 3·25-s − 4·29-s + 4·35-s + 12·37-s − 12·41-s − 2·43-s + 8·47-s + 6·49-s + 4·55-s + 8·59-s + 20·61-s + 4·65-s − 4·67-s + 4·71-s + 18·73-s + 4·77-s + 12·79-s − 2·83-s + 4·85-s + 16·89-s + 4·91-s + ⋯
L(s)  = 1  + 0.894·5-s + 0.755·7-s + 0.603·11-s + 0.554·13-s + 0.485·17-s − 1.66·23-s + 3/5·25-s − 0.742·29-s + 0.676·35-s + 1.97·37-s − 1.87·41-s − 0.304·43-s + 1.16·47-s + 6/7·49-s + 0.539·55-s + 1.04·59-s + 2.56·61-s + 0.496·65-s − 0.488·67-s + 0.474·71-s + 2.10·73-s + 0.455·77-s + 1.35·79-s − 0.219·83-s + 0.433·85-s + 1.69·89-s + 0.419·91-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 62726400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62726400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(62726400\)    =    \(2^{8} \cdot 3^{4} \cdot 5^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(3999.48\)
Root analytic conductor: \(7.95245\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 62726400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.963819284\)
\(L(\frac12)\) \(\approx\) \(5.963819284\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
11$C_1$ \( ( 1 - T )^{2} \)
good7$D_{4}$ \( 1 - 2 T - 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
13$C_4$ \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 2 T + 18 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$D_{4}$ \( 1 + 2 T + 70 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
53$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
67$D_{4}$ \( 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 18 T + 210 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 12 T + 126 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 2 T + 14 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 16 T + 174 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 8 T + 142 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
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

−7.949338894205320878089167246743, −7.79113151550828912787519103851, −7.31113726901051671291985953408, −6.99497560209724243211418413624, −6.46280100370158907927192273230, −6.34563277934750624058376727391, −5.91542824913304055342429138687, −5.63392272723616427125671875062, −5.24790158558533566102204383967, −4.96272650535946702272412748717, −4.50250032127343137300723910165, −4.03316665052340373805761375214, −3.64485949511265052374792476880, −3.55951301661589157295332010426, −2.80153591766526953788358795394, −2.20146145269512779888613878666, −2.05022394015728721080868867162, −1.70402871202403597406268872259, −0.847506695045480182788900706570, −0.74391906393527359044403466667, 0.74391906393527359044403466667, 0.847506695045480182788900706570, 1.70402871202403597406268872259, 2.05022394015728721080868867162, 2.20146145269512779888613878666, 2.80153591766526953788358795394, 3.55951301661589157295332010426, 3.64485949511265052374792476880, 4.03316665052340373805761375214, 4.50250032127343137300723910165, 4.96272650535946702272412748717, 5.24790158558533566102204383967, 5.63392272723616427125671875062, 5.91542824913304055342429138687, 6.34563277934750624058376727391, 6.46280100370158907927192273230, 6.99497560209724243211418413624, 7.31113726901051671291985953408, 7.79113151550828912787519103851, 7.949338894205320878089167246743

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