Properties

Label 4-231525-1.1-c1e2-0-8
Degree $4$
Conductor $231525$
Sign $1$
Analytic cond. $14.7622$
Root an. cond. $1.96014$
Motivic weight $1$
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  + 3-s − 3·4-s + 2·5-s + 7-s + 9-s − 3·12-s + 2·15-s + 5·16-s + 4·17-s − 6·20-s + 21-s + 3·25-s + 27-s − 3·28-s + 2·35-s − 3·36-s − 4·37-s − 12·41-s + 8·43-s + 2·45-s + 16·47-s + 5·48-s + 49-s + 4·51-s + 8·59-s − 6·60-s + 63-s + ⋯
L(s)  = 1  + 0.577·3-s − 3/2·4-s + 0.894·5-s + 0.377·7-s + 1/3·9-s − 0.866·12-s + 0.516·15-s + 5/4·16-s + 0.970·17-s − 1.34·20-s + 0.218·21-s + 3/5·25-s + 0.192·27-s − 0.566·28-s + 0.338·35-s − 1/2·36-s − 0.657·37-s − 1.87·41-s + 1.21·43-s + 0.298·45-s + 2.33·47-s + 0.721·48-s + 1/7·49-s + 0.560·51-s + 1.04·59-s − 0.774·60-s + 0.125·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 231525 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231525 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(231525\)    =    \(3^{3} \cdot 5^{2} \cdot 7^{3}\)
Sign: $1$
Analytic conductor: \(14.7622\)
Root analytic conductor: \(1.96014\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{231525} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 231525,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.875587480\)
\(L(\frac12)\) \(\approx\) \(1.875587480\)
\(L(\frac{3}{2})\) 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)$
bad3$C_1$ \( 1 - T \)
5$C_1$ \( ( 1 - T )^{2} \)
7$C_1$ \( 1 - T \)
good2$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
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

−9.158915116725801964521352538031, −8.447489066361171687346291758504, −8.275102727179600443330310060644, −7.82733728798443659692174656726, −6.97944382340517249810347443111, −6.82600380115164878858049675491, −5.81605182603986018580602603999, −5.37573342552736399524296425305, −5.25448844749250196408410433194, −4.33309058948755389799390879771, −4.05504421439464721652879145719, −3.34833988846054044432199186298, −2.63294756479994443235170715895, −1.79849192746842450415563444508, −0.888154170390839902230274080915, 0.888154170390839902230274080915, 1.79849192746842450415563444508, 2.63294756479994443235170715895, 3.34833988846054044432199186298, 4.05504421439464721652879145719, 4.33309058948755389799390879771, 5.25448844749250196408410433194, 5.37573342552736399524296425305, 5.81605182603986018580602603999, 6.82600380115164878858049675491, 6.97944382340517249810347443111, 7.82733728798443659692174656726, 8.275102727179600443330310060644, 8.447489066361171687346291758504, 9.158915116725801964521352538031

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