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Properties

Label 4-91e2-1.1-c1e2-0-1
Degree $4$
Conductor $8281$
Sign $1$
Analytic cond. $0.528003$
Root an. cond. $0.852431$
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  − 2·4-s + 6·5-s + 2·7-s − 4·9-s − 2·13-s − 6·19-s − 12·20-s − 6·23-s + 19·25-s − 4·28-s + 6·29-s − 2·31-s + 12·35-s + 8·36-s − 4·37-s + 12·41-s − 10·43-s − 24·45-s + 6·47-s + 3·49-s + 4·52-s − 6·53-s + 12·59-s + 12·61-s − 8·63-s + 8·64-s − 12·65-s + ⋯
L(s)  = 1  − 4-s + 2.68·5-s + 0.755·7-s − 4/3·9-s − 0.554·13-s − 1.37·19-s − 2.68·20-s − 1.25·23-s + 19/5·25-s − 0.755·28-s + 1.11·29-s − 0.359·31-s + 2.02·35-s + 4/3·36-s − 0.657·37-s + 1.87·41-s − 1.52·43-s − 3.57·45-s + 0.875·47-s + 3/7·49-s + 0.554·52-s − 0.824·53-s + 1.56·59-s + 1.53·61-s − 1.00·63-s + 64-s − 1.48·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(8281\)    =    \(7^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(0.528003\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 8281,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.073208205\)
\(L(\frac12)\) \(\approx\) \(1.073208205\)
\(L(\frac{3}{2})\) 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)$
bad7$C_1$ \( ( 1 - T )^{2} \)
13$C_1$ \( ( 1 + T )^{2} \)
good2$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
3$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
5$C_2^2$ \( 1 - 6 T + 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 32 T^{2} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 6 T + 29 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 6 T + 47 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 6 T + 59 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 2 T + 45 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 4 T + 60 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 12 T + 110 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
43$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
47$D_{4}$ \( 1 - 6 T + 101 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 6 T + 107 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 12 T + 122 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
67$D_{4}$ \( 1 + 12 T + 98 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 12 T + 128 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 10 T + 153 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 14 T + 135 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 18 T + 229 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 6 T + 185 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 2 T + 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
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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

−14.19259351354014001633935862578, −13.91530704403553248137339920165, −13.32961004431274155578181993754, −13.18598525856379489294894562606, −12.31039705786425851174123193269, −11.81034674227379139892165090587, −10.92173925275991688173885509546, −10.43811267915924677562981769248, −9.956676790942336342764905946639, −9.453975216742547325282993143478, −8.720090509496169016660368964555, −8.710892224506815951873187739116, −7.78956849606875471561960327198, −6.55216153418865536823210553119, −6.15614991745813169223168872051, −5.46067764345373140398153217209, −5.09174108420603531238146202946, −4.20245225198679567841491642802, −2.59168801763164316544158957285, −1.98957828776268363334345914285, 1.98957828776268363334345914285, 2.59168801763164316544158957285, 4.20245225198679567841491642802, 5.09174108420603531238146202946, 5.46067764345373140398153217209, 6.15614991745813169223168872051, 6.55216153418865536823210553119, 7.78956849606875471561960327198, 8.710892224506815951873187739116, 8.720090509496169016660368964555, 9.453975216742547325282993143478, 9.956676790942336342764905946639, 10.43811267915924677562981769248, 10.92173925275991688173885509546, 11.81034674227379139892165090587, 12.31039705786425851174123193269, 13.18598525856379489294894562606, 13.32961004431274155578181993754, 13.91530704403553248137339920165, 14.19259351354014001633935862578

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