Properties

Label 4-405e2-1.1-c3e2-0-12
Degree $4$
Conductor $164025$
Sign $1$
Analytic cond. $571.007$
Root an. cond. $4.88833$
Motivic weight $3$
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  + 2·2-s + 8·4-s − 5·5-s + 40·8-s − 10·10-s − 10·11-s + 80·13-s + 80·16-s + 14·17-s − 226·19-s − 40·20-s − 20·22-s + 81·23-s + 160·26-s + 220·29-s + 189·31-s + 320·32-s + 28·34-s + 340·37-s − 452·38-s − 200·40-s + 130·41-s − 10·43-s − 80·44-s + 162·46-s − 160·47-s + 343·49-s + ⋯
L(s)  = 1  + 0.707·2-s + 4-s − 0.447·5-s + 1.76·8-s − 0.316·10-s − 0.274·11-s + 1.70·13-s + 5/4·16-s + 0.199·17-s − 2.72·19-s − 0.447·20-s − 0.193·22-s + 0.734·23-s + 1.20·26-s + 1.40·29-s + 1.09·31-s + 1.76·32-s + 0.141·34-s + 1.51·37-s − 1.92·38-s − 0.790·40-s + 0.495·41-s − 0.0354·43-s − 0.274·44-s + 0.519·46-s − 0.496·47-s + 49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 164025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164025 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(164025\)    =    \(3^{8} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(571.007\)
Root analytic conductor: \(4.88833\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{405} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 164025,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(6.128807786\)
\(L(\frac12)\) \(\approx\) \(6.128807786\)
\(L(\frac{5}{2})\) 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 \( 1 \)
5$C_2$ \( 1 + p T + p^{2} T^{2} \)
good2$C_2^2$ \( 1 - p T - p^{2} T^{2} - p^{4} T^{3} + p^{6} T^{4} \)
7$C_2^2$ \( 1 - p^{3} T^{2} + p^{6} T^{4} \)
11$C_2^2$ \( 1 + 10 T - 1231 T^{2} + 10 p^{3} T^{3} + p^{6} T^{4} \)
13$C_2^2$ \( 1 - 80 T + 4203 T^{2} - 80 p^{3} T^{3} + p^{6} T^{4} \)
17$C_2$ \( ( 1 - 7 T + p^{3} T^{2} )^{2} \)
19$C_2$ \( ( 1 + 113 T + p^{3} T^{2} )^{2} \)
23$C_2^2$ \( 1 - 81 T - 5606 T^{2} - 81 p^{3} T^{3} + p^{6} T^{4} \)
29$C_2^2$ \( 1 - 220 T + 24011 T^{2} - 220 p^{3} T^{3} + p^{6} T^{4} \)
31$C_2^2$ \( 1 - 189 T + 5930 T^{2} - 189 p^{3} T^{3} + p^{6} T^{4} \)
37$C_2$ \( ( 1 - 170 T + p^{3} T^{2} )^{2} \)
41$C_2^2$ \( 1 - 130 T - 52021 T^{2} - 130 p^{3} T^{3} + p^{6} T^{4} \)
43$C_2^2$ \( 1 + 10 T - 79407 T^{2} + 10 p^{3} T^{3} + p^{6} T^{4} \)
47$C_2^2$ \( 1 + 160 T - 78223 T^{2} + 160 p^{3} T^{3} + p^{6} T^{4} \)
53$C_2$ \( ( 1 - 631 T + p^{3} T^{2} )^{2} \)
59$C_2^2$ \( 1 - 560 T + 108221 T^{2} - 560 p^{3} T^{3} + p^{6} T^{4} \)
61$C_2^2$ \( 1 + 229 T - 174540 T^{2} + 229 p^{3} T^{3} + p^{6} T^{4} \)
67$C_2^2$ \( 1 + 750 T + 261737 T^{2} + 750 p^{3} T^{3} + p^{6} T^{4} \)
71$C_2$ \( ( 1 - 890 T + p^{3} T^{2} )^{2} \)
73$C_2$ \( ( 1 + 890 T + p^{3} T^{2} )^{2} \)
79$C_2^2$ \( 1 - 27 T - 492310 T^{2} - 27 p^{3} T^{3} + p^{6} T^{4} \)
83$C_2^2$ \( 1 + 429 T - 387746 T^{2} + 429 p^{3} T^{3} + p^{6} T^{4} \)
89$C_2$ \( ( 1 + 750 T + p^{3} T^{2} )^{2} \)
97$C_2^2$ \( 1 - 1480 T + 1277727 T^{2} - 1480 p^{3} T^{3} + p^{6} T^{4} \)
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

−11.14573202874587909315953686418, −10.57571434005830465618844711512, −10.29793437126315969448566650480, −10.19914843527716186449526683077, −8.991720875672659202206114653567, −8.673071436838610941183490654419, −8.210662595535854973225136133603, −7.908657646158297079983255914482, −7.13792582509322593793947896495, −6.82107133150000345965197748640, −6.17178170140818428417941749339, −6.06817113691749160782675534461, −5.16479484681170340555474660403, −4.55036436431716144292516809398, −4.00653498524167697162630630571, −3.91462561980045135461682604955, −2.60246916437341202100463681794, −2.52283878049274556591214733347, −1.39978303334069136820862239558, −0.797773692822905705911686711929, 0.797773692822905705911686711929, 1.39978303334069136820862239558, 2.52283878049274556591214733347, 2.60246916437341202100463681794, 3.91462561980045135461682604955, 4.00653498524167697162630630571, 4.55036436431716144292516809398, 5.16479484681170340555474660403, 6.06817113691749160782675534461, 6.17178170140818428417941749339, 6.82107133150000345965197748640, 7.13792582509322593793947896495, 7.908657646158297079983255914482, 8.210662595535854973225136133603, 8.673071436838610941183490654419, 8.991720875672659202206114653567, 10.19914843527716186449526683077, 10.29793437126315969448566650480, 10.57571434005830465618844711512, 11.14573202874587909315953686418

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