Properties

Label 4-270000-1.1-c1e2-0-6
Degree $4$
Conductor $270000$
Sign $1$
Analytic cond. $17.2154$
Root an. cond. $2.03694$
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 − 2·7-s + 9-s + 10·13-s + 10·19-s − 2·21-s + 27-s − 2·31-s + 4·37-s + 10·39-s − 2·43-s − 11·49-s + 10·57-s − 26·61-s − 2·63-s + 22·67-s + 4·73-s + 16·79-s + 81-s − 20·91-s − 2·93-s − 14·97-s − 8·103-s − 14·109-s + 4·111-s + 10·117-s + 14·121-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.755·7-s + 1/3·9-s + 2.77·13-s + 2.29·19-s − 0.436·21-s + 0.192·27-s − 0.359·31-s + 0.657·37-s + 1.60·39-s − 0.304·43-s − 1.57·49-s + 1.32·57-s − 3.32·61-s − 0.251·63-s + 2.68·67-s + 0.468·73-s + 1.80·79-s + 1/9·81-s − 2.09·91-s − 0.207·93-s − 1.42·97-s − 0.788·103-s − 1.34·109-s + 0.379·111-s + 0.924·117-s + 1.27·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 270000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(270000\)    =    \(2^{4} \cdot 3^{3} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(17.2154\)
Root analytic conductor: \(2.03694\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{270000} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 270000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.495130817\)
\(L(\frac12)\) \(\approx\) \(2.495130817\)
\(L(\frac{3}{2})\) 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)$
bad2 \( 1 \)
3$C_1$ \( 1 - T \)
5 \( 1 \)
good7$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
61$C_2$ \( ( 1 + 13 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
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

−9.108883986383451182036456380552, −8.275919068288280643535776826565, −8.026266654629362330508166642807, −7.72957625844481983843839527768, −6.77234120616327244201676192765, −6.68901441807918237505792173786, −6.03058890823729073510214550931, −5.61334343701912201622834145502, −5.03310337383134036338192767807, −4.24382547629296535183762439094, −3.50841194957311191464253456169, −3.43647697037916365579697730592, −2.81121112269996318424524765777, −1.63891885984744417728938078317, −1.03773521644426615440225296994, 1.03773521644426615440225296994, 1.63891885984744417728938078317, 2.81121112269996318424524765777, 3.43647697037916365579697730592, 3.50841194957311191464253456169, 4.24382547629296535183762439094, 5.03310337383134036338192767807, 5.61334343701912201622834145502, 6.03058890823729073510214550931, 6.68901441807918237505792173786, 6.77234120616327244201676192765, 7.72957625844481983843839527768, 8.026266654629362330508166642807, 8.275919068288280643535776826565, 9.108883986383451182036456380552

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