Properties

Label 4-1881e2-1.1-c1e2-0-7
Degree $4$
Conductor $3538161$
Sign $1$
Analytic cond. $225.596$
Root an. cond. $3.87554$
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  − 2·4-s + 2·5-s − 4·7-s + 2·11-s − 4·13-s − 4·17-s − 2·19-s − 4·20-s + 6·23-s − 7·25-s + 8·28-s + 4·29-s − 10·31-s − 8·35-s + 6·37-s − 8·41-s + 12·43-s − 4·44-s − 12·47-s + 8·52-s − 8·53-s + 4·55-s + 6·59-s − 8·61-s + 8·64-s − 8·65-s − 18·67-s + ⋯
L(s)  = 1  − 4-s + 0.894·5-s − 1.51·7-s + 0.603·11-s − 1.10·13-s − 0.970·17-s − 0.458·19-s − 0.894·20-s + 1.25·23-s − 7/5·25-s + 1.51·28-s + 0.742·29-s − 1.79·31-s − 1.35·35-s + 0.986·37-s − 1.24·41-s + 1.82·43-s − 0.603·44-s − 1.75·47-s + 1.10·52-s − 1.09·53-s + 0.539·55-s + 0.781·59-s − 1.02·61-s + 64-s − 0.992·65-s − 2.19·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3538161 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3538161 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3538161\)    =    \(3^{4} \cdot 11^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(225.596\)
Root analytic conductor: \(3.87554\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 3538161,\ (\ :1/2, 1/2),\ 1)\)

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)$
bad3 \( 1 \)
11$C_1$ \( ( 1 - T )^{2} \)
19$C_1$ \( ( 1 + T )^{2} \)
good2$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
5$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
7$D_{4}$ \( 1 + 4 T + 16 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 4 T + 36 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
29$D_{4}$ \( 1 - 4 T + 44 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 10 T + 85 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 6 T + 33 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 8 T + 66 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 12 T + 90 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 12 T + 122 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 8 T + 50 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 6 T + 125 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 8 T + 88 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 18 T + 213 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 22 T + 261 T^{2} - 22 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 8 T + 90 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 32 T + 412 T^{2} + 32 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 4 T + 168 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 10 T + 105 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 2 T + 193 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
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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

−9.091612033052995987677978546163, −9.029616056575936623640895012181, −8.288545594411779593161493154789, −7.960828268661814818842234474144, −7.32557306461599310872818866419, −7.02159082165025953566789189797, −6.53131322980631352966087170390, −6.30150604698211797407720559208, −5.91388422041459863675062062540, −5.36718718335809121550700037058, −4.85731195125410283319378659322, −4.65982749344289371084893169663, −3.90398743924686576664912597932, −3.77064372604746606096012529394, −2.97696773574229347793373809043, −2.61645048657172399571406500208, −1.99537281190882980669367239770, −1.34977452635185729534176938195, 0, 0, 1.34977452635185729534176938195, 1.99537281190882980669367239770, 2.61645048657172399571406500208, 2.97696773574229347793373809043, 3.77064372604746606096012529394, 3.90398743924686576664912597932, 4.65982749344289371084893169663, 4.85731195125410283319378659322, 5.36718718335809121550700037058, 5.91388422041459863675062062540, 6.30150604698211797407720559208, 6.53131322980631352966087170390, 7.02159082165025953566789189797, 7.32557306461599310872818866419, 7.960828268661814818842234474144, 8.288545594411779593161493154789, 9.029616056575936623640895012181, 9.091612033052995987677978546163

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