Properties

Label 4-405e2-1.1-c3e2-0-13
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

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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

\(L(2)\) \(\approx\) \(9.307531073\)
\(L(\frac12)\) \(\approx\) \(9.307531073\)
\(L(\frac{5}{2})\) 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$C_2^2$ \( 1 + 386 T - 763677 T^{2} + 386 p^{3} T^{3} + p^{6} T^{4} \)
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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

−10.96564691004052362523405092299, −10.86951685656738344899331292932, −10.03132147853888873729213535457, −9.957009281608892151805291097542, −9.430819919916220368927843657418, −8.758482121553877006308497796568, −7.929465598469521115816880658222, −7.911648146412266472245248909525, −7.11448364840591370913942027896, −6.90887239940080653372790351578, −5.93520628191959503933118557969, −5.59704928133762966125237090968, −5.44060499498308715617881966249, −4.63109012096907146754365110452, −4.30978486393932387358456080192, −3.46191836835597339796134806840, −3.06483354527067517422971676522, −2.52856591930502184954192232396, −1.34701853984755522017106908753, −0.940954114857839875114289992978, 0.940954114857839875114289992978, 1.34701853984755522017106908753, 2.52856591930502184954192232396, 3.06483354527067517422971676522, 3.46191836835597339796134806840, 4.30978486393932387358456080192, 4.63109012096907146754365110452, 5.44060499498308715617881966249, 5.59704928133762966125237090968, 5.93520628191959503933118557969, 6.90887239940080653372790351578, 7.11448364840591370913942027896, 7.911648146412266472245248909525, 7.929465598469521115816880658222, 8.758482121553877006308497796568, 9.430819919916220368927843657418, 9.957009281608892151805291097542, 10.03132147853888873729213535457, 10.86951685656738344899331292932, 10.96564691004052362523405092299

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