Properties

Label 4-405e2-1.1-c3e2-0-7
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  − 4·2-s + 8·4-s − 5·5-s − 6·7-s − 32·8-s + 20·10-s + 32·11-s + 38·13-s + 24·14-s + 128·16-s − 52·17-s + 200·19-s − 40·20-s − 128·22-s − 78·23-s − 152·26-s − 48·28-s − 50·29-s + 108·31-s − 256·32-s + 208·34-s + 30·35-s + 532·37-s − 800·38-s + 160·40-s + 22·41-s − 442·43-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s − 0.447·5-s − 0.323·7-s − 1.41·8-s + 0.632·10-s + 0.877·11-s + 0.810·13-s + 0.458·14-s + 2·16-s − 0.741·17-s + 2.41·19-s − 0.447·20-s − 1.24·22-s − 0.707·23-s − 1.14·26-s − 0.323·28-s − 0.320·29-s + 0.625·31-s − 1.41·32-s + 1.04·34-s + 0.144·35-s + 2.36·37-s − 3.41·38-s + 0.632·40-s + 0.0838·41-s − 1.56·43-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

\(L(2)\) \(\approx\) \(0.9532927852\)
\(L(\frac12)\) \(\approx\) \(0.9532927852\)
\(L(\frac{5}{2})\) 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$C_2^2$ \( 1 + p^{2} T + p^{3} T^{2} + p^{5} T^{3} + p^{6} T^{4} \)
7$C_2^2$ \( 1 + 6 T - 307 T^{2} + 6 p^{3} T^{3} + p^{6} T^{4} \)
11$C_2^2$ \( 1 - 32 T - 307 T^{2} - 32 p^{3} T^{3} + p^{6} T^{4} \)
13$C_2^2$ \( 1 - 38 T - 753 T^{2} - 38 p^{3} T^{3} + p^{6} T^{4} \)
17$C_2$ \( ( 1 + 26 T + p^{3} T^{2} )^{2} \)
19$C_2$ \( ( 1 - 100 T + p^{3} T^{2} )^{2} \)
23$C_2^2$ \( 1 + 78 T - 6083 T^{2} + 78 p^{3} T^{3} + p^{6} T^{4} \)
29$C_2^2$ \( 1 + 50 T - 21889 T^{2} + 50 p^{3} T^{3} + p^{6} T^{4} \)
31$C_2^2$ \( 1 - 108 T - 18127 T^{2} - 108 p^{3} T^{3} + p^{6} T^{4} \)
37$C_2$ \( ( 1 - 266 T + p^{3} T^{2} )^{2} \)
41$C_2^2$ \( 1 - 22 T - 68437 T^{2} - 22 p^{3} T^{3} + p^{6} T^{4} \)
43$C_2^2$ \( 1 + 442 T + 115857 T^{2} + 442 p^{3} T^{3} + p^{6} T^{4} \)
47$C_2^2$ \( 1 + 514 T + 160373 T^{2} + 514 p^{3} T^{3} + p^{6} T^{4} \)
53$C_2$ \( ( 1 + 2 T + p^{3} T^{2} )^{2} \)
59$C_2^2$ \( 1 - 500 T + 44621 T^{2} - 500 p^{3} T^{3} + p^{6} T^{4} \)
61$C_2^2$ \( 1 - 518 T + 41343 T^{2} - 518 p^{3} T^{3} + p^{6} T^{4} \)
67$C_2^2$ \( 1 + 126 T - 284887 T^{2} + 126 p^{3} T^{3} + p^{6} T^{4} \)
71$C_2$ \( ( 1 + 412 T + p^{3} T^{2} )^{2} \)
73$C_2$ \( ( 1 + 878 T + p^{3} T^{2} )^{2} \)
79$C_2^2$ \( 1 + 600 T - 133039 T^{2} + 600 p^{3} T^{3} + p^{6} T^{4} \)
83$C_2^2$ \( 1 - 282 T - 492263 T^{2} - 282 p^{3} T^{3} + p^{6} T^{4} \)
89$C_2$ \( ( 1 - 150 T + p^{3} T^{2} )^{2} \)
97$C_2^2$ \( 1 + 386 T - 763677 T^{2} + 386 p^{3} T^{3} + p^{6} T^{4} \)
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

−11.24217720848927060810332846948, −10.28058560297404711309553622042, −9.926298345251979881918537188879, −9.811390665638521358375948076495, −9.077107714747055055248797142505, −8.936969366337035781244683183477, −8.425052020578346645661421239904, −7.906300266196755379791121207380, −7.55394054550925843426381606829, −6.90738431130613466228408742885, −6.50280654379135982273855093085, −5.85201848454111657624298825277, −5.63933536785303956169027539964, −4.58768982641429185714102578544, −4.03094424564287645589891679270, −3.11680274847789275431131654342, −3.11120197277054295144879417692, −1.85873732817549221747354452230, −1.04106971645300358767896784375, −0.51865190468414100090190551256, 0.51865190468414100090190551256, 1.04106971645300358767896784375, 1.85873732817549221747354452230, 3.11120197277054295144879417692, 3.11680274847789275431131654342, 4.03094424564287645589891679270, 4.58768982641429185714102578544, 5.63933536785303956169027539964, 5.85201848454111657624298825277, 6.50280654379135982273855093085, 6.90738431130613466228408742885, 7.55394054550925843426381606829, 7.906300266196755379791121207380, 8.425052020578346645661421239904, 8.936969366337035781244683183477, 9.077107714747055055248797142505, 9.811390665638521358375948076495, 9.926298345251979881918537188879, 10.28058560297404711309553622042, 11.24217720848927060810332846948

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