Properties

Label 4-405e2-1.1-c3e2-0-1
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  − 3·2-s + 8·4-s + 5·5-s − 20·7-s − 45·8-s − 15·10-s + 24·11-s − 74·13-s + 60·14-s + 135·16-s + 108·17-s − 248·19-s + 40·20-s − 72·22-s + 120·23-s + 222·26-s − 160·28-s + 78·29-s − 200·31-s − 360·32-s − 324·34-s − 100·35-s − 140·37-s + 744·38-s − 225·40-s − 330·41-s − 92·43-s + ⋯
L(s)  = 1  − 1.06·2-s + 4-s + 0.447·5-s − 1.07·7-s − 1.98·8-s − 0.474·10-s + 0.657·11-s − 1.57·13-s + 1.14·14-s + 2.10·16-s + 1.54·17-s − 2.99·19-s + 0.447·20-s − 0.697·22-s + 1.08·23-s + 1.67·26-s − 1.07·28-s + 0.499·29-s − 1.15·31-s − 1.98·32-s − 1.63·34-s − 0.482·35-s − 0.622·37-s + 3.17·38-s − 0.889·40-s − 1.25·41-s − 0.326·43-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

\(L(2)\) \(\approx\) \(0.1746447683\)
\(L(\frac12)\) \(\approx\) \(0.1746447683\)
\(L(\frac{5}{2})\) 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$C_2^2$ \( 1 + 3 T + T^{2} + 3 p^{3} T^{3} + p^{6} T^{4} \)
7$C_2$ \( ( 1 - 17 T + p^{3} T^{2} )( 1 + 37 T + p^{3} T^{2} ) \)
11$C_2^2$ \( 1 - 24 T - 755 T^{2} - 24 p^{3} T^{3} + p^{6} T^{4} \)
13$C_2^2$ \( 1 + 74 T + 3279 T^{2} + 74 p^{3} T^{3} + p^{6} T^{4} \)
17$C_2$ \( ( 1 - 54 T + p^{3} T^{2} )^{2} \)
19$C_2$ \( ( 1 + 124 T + p^{3} T^{2} )^{2} \)
23$C_2^2$ \( 1 - 120 T + 2233 T^{2} - 120 p^{3} T^{3} + p^{6} T^{4} \)
29$C_2^2$ \( 1 - 78 T - 18305 T^{2} - 78 p^{3} T^{3} + p^{6} T^{4} \)
31$C_2^2$ \( 1 + 200 T + 10209 T^{2} + 200 p^{3} T^{3} + p^{6} T^{4} \)
37$C_2$ \( ( 1 + 70 T + p^{3} T^{2} )^{2} \)
41$C_2^2$ \( 1 + 330 T + 39979 T^{2} + 330 p^{3} T^{3} + p^{6} T^{4} \)
43$C_2^2$ \( 1 + 92 T - 71043 T^{2} + 92 p^{3} T^{3} + p^{6} T^{4} \)
47$C_2^2$ \( 1 - 24 T - 103247 T^{2} - 24 p^{3} T^{3} + p^{6} T^{4} \)
53$C_2$ \( ( 1 - 450 T + p^{3} T^{2} )^{2} \)
59$C_2^2$ \( 1 + 24 T - 204803 T^{2} + 24 p^{3} T^{3} + p^{6} T^{4} \)
61$C_2^2$ \( 1 - 322 T - 123297 T^{2} - 322 p^{3} T^{3} + p^{6} T^{4} \)
67$C_2^2$ \( 1 - 196 T - 262347 T^{2} - 196 p^{3} T^{3} + p^{6} T^{4} \)
71$C_2$ \( ( 1 + 288 T + p^{3} T^{2} )^{2} \)
73$C_2$ \( ( 1 + 430 T + p^{3} T^{2} )^{2} \)
79$C_2^2$ \( 1 - 520 T - 222639 T^{2} - 520 p^{3} T^{3} + p^{6} T^{4} \)
83$C_2^2$ \( 1 + 156 T - 547451 T^{2} + 156 p^{3} T^{3} + p^{6} T^{4} \)
89$C_2$ \( ( 1 - 1026 T + p^{3} T^{2} )^{2} \)
97$C_2^2$ \( 1 - 286 T - 830877 T^{2} - 286 p^{3} T^{3} + p^{6} T^{4} \)
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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

−10.64242540460174877212074903316, −10.50052817743630538045656477602, −10.17102159868620546083330937479, −9.589098157379380178436480150357, −9.321144161951093357792947208657, −8.769200870634848716516728820025, −8.602992565046768360185899042139, −7.939304051508737282863222268446, −7.05442837896064095777398280237, −7.03042738772054743014911304056, −6.44220325310688351622540790764, −6.00094508133569028804001667864, −5.48357609648767074543920883998, −4.86925689177323443874517668201, −3.78692971906753789824311146710, −3.45900330764592845205483565608, −2.50272903311968799131390611449, −2.34432299216424244253065808139, −1.23865926332527884056524629186, −0.17161051601563216432188918097, 0.17161051601563216432188918097, 1.23865926332527884056524629186, 2.34432299216424244253065808139, 2.50272903311968799131390611449, 3.45900330764592845205483565608, 3.78692971906753789824311146710, 4.86925689177323443874517668201, 5.48357609648767074543920883998, 6.00094508133569028804001667864, 6.44220325310688351622540790764, 7.03042738772054743014911304056, 7.05442837896064095777398280237, 7.939304051508737282863222268446, 8.602992565046768360185899042139, 8.769200870634848716516728820025, 9.321144161951093357792947208657, 9.589098157379380178436480150357, 10.17102159868620546083330937479, 10.50052817743630538045656477602, 10.64242540460174877212074903316

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