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Properties

Label 4-91e2-1.1-c1e2-0-5
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 $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 6·3-s + 2·4-s − 3·5-s + 6·6-s − 5·7-s − 5·8-s + 21·9-s + 3·10-s − 6·11-s − 12·12-s − 2·13-s + 5·14-s + 18·15-s + 5·16-s + 2·17-s − 21·18-s − 2·19-s − 6·20-s + 30·21-s + 6·22-s + 30·24-s + 5·25-s + 2·26-s − 54·27-s − 10·28-s − 7·29-s + ⋯
L(s)  = 1  − 0.707·2-s − 3.46·3-s + 4-s − 1.34·5-s + 2.44·6-s − 1.88·7-s − 1.76·8-s + 7·9-s + 0.948·10-s − 1.80·11-s − 3.46·12-s − 0.554·13-s + 1.33·14-s + 4.64·15-s + 5/4·16-s + 0.485·17-s − 4.94·18-s − 0.458·19-s − 1.34·20-s + 6.54·21-s + 1.27·22-s + 6.12·24-s + 25-s + 0.392·26-s − 10.3·27-s − 1.88·28-s − 1.29·29-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: \(2\)
Selberg data: \((4,\ 8281,\ (\ :1/2, 1/2),\ 1)\)

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)$
bad7$C_2$ \( 1 + 5 T + p T^{2} \)
13$C_2$ \( 1 + 2 T + p T^{2} \)
good2$C_2^2$ \( 1 + T - T^{2} + p T^{3} + p^{2} T^{4} \)
3$C_2$ \( ( 1 + p T + p T^{2} )^{2} \)
5$C_2^2$ \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 7 T + 20 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + 3 T - 22 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
41$C_2^2$ \( 1 + 3 T - 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 7 T + 6 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + T - 46 T^{2} + p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 3 T - 44 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 4 T - 43 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + 13 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
71$C_2^2$ \( 1 + 13 T + 98 T^{2} + 13 p T^{3} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 13 T + 96 T^{2} - 13 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 3 T - 70 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 19 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
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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

−13.17083072304890003935458042810, −12.87148941030843382217326914293, −12.35600926309536103741998797111, −12.08027994337324273952628204747, −11.82021533416337375398473062326, −11.03561694622216597655332642963, −10.61928492568430797996677418294, −10.54540591994848998103598344623, −9.710119850407055439006879556857, −9.180931829567947556896723749772, −7.74588984405759713913977021044, −7.37774941531169181062342851270, −6.70721235102570303894165474320, −6.33600266034391586792357201625, −5.59003972084021475716683567771, −5.38223469062928345016302182981, −4.19143906619076892162692347148, −3.04189453050271718744975784464, 0, 0, 3.04189453050271718744975784464, 4.19143906619076892162692347148, 5.38223469062928345016302182981, 5.59003972084021475716683567771, 6.33600266034391586792357201625, 6.70721235102570303894165474320, 7.37774941531169181062342851270, 7.74588984405759713913977021044, 9.180931829567947556896723749772, 9.710119850407055439006879556857, 10.54540591994848998103598344623, 10.61928492568430797996677418294, 11.03561694622216597655332642963, 11.82021533416337375398473062326, 12.08027994337324273952628204747, 12.35600926309536103741998797111, 12.87148941030843382217326914293, 13.17083072304890003935458042810

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