Properties

Label 4-420e2-1.1-c1e2-0-3
Degree $4$
Conductor $176400$
Sign $1$
Analytic cond. $11.2474$
Root an. cond. $1.83131$
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 − 2·4-s − 2·9-s − 6·11-s − 2·12-s + 10·13-s + 4·16-s − 12·23-s + 25-s − 5·27-s − 6·33-s + 4·36-s + 4·37-s + 10·39-s + 12·44-s + 18·47-s + 4·48-s + 49-s − 20·52-s + 16·61-s − 8·64-s − 12·69-s + 4·73-s + 75-s + 81-s + 24·83-s + 24·92-s + ⋯
L(s)  = 1  + 0.577·3-s − 4-s − 2/3·9-s − 1.80·11-s − 0.577·12-s + 2.77·13-s + 16-s − 2.50·23-s + 1/5·25-s − 0.962·27-s − 1.04·33-s + 2/3·36-s + 0.657·37-s + 1.60·39-s + 1.80·44-s + 2.62·47-s + 0.577·48-s + 1/7·49-s − 2.77·52-s + 2.04·61-s − 64-s − 1.44·69-s + 0.468·73-s + 0.115·75-s + 1/9·81-s + 2.63·83-s + 2.50·92-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(176400\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(11.2474\)
Root analytic conductor: \(1.83131\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{176400} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 176400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.283668432\)
\(L(\frac12)\) \(\approx\) \(1.283668432\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + p T^{2} \)
3$C_2$ \( 1 - T + p T^{2} \)
5$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
7$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good11$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
23$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
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$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
97$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
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

−9.034294939496608256127581309995, −8.679498853252053444462430241617, −8.153192339337884543589566310226, −7.980947296320422939728903060714, −7.67755521010834815156907872378, −6.60444721122988741193728663303, −6.06485042125071096226940610377, −5.54008340383010136479817620846, −5.47907564040983295563221596641, −4.41617775358459899843679315087, −3.82466491546191947603335262315, −3.61689003045514298236484607491, −2.72164418406817349261491719715, −2.03544754569172366495048437388, −0.72587888295584277318850692735, 0.72587888295584277318850692735, 2.03544754569172366495048437388, 2.72164418406817349261491719715, 3.61689003045514298236484607491, 3.82466491546191947603335262315, 4.41617775358459899843679315087, 5.47907564040983295563221596641, 5.54008340383010136479817620846, 6.06485042125071096226940610377, 6.60444721122988741193728663303, 7.67755521010834815156907872378, 7.980947296320422939728903060714, 8.153192339337884543589566310226, 8.679498853252053444462430241617, 9.034294939496608256127581309995

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