Properties

Label 4-546e2-1.1-c1e2-0-8
Degree $4$
Conductor $298116$
Sign $1$
Analytic cond. $19.0081$
Root an. cond. $2.08802$
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  − 3·3-s + 4-s − 2·5-s + 7-s + 6·9-s − 3·12-s + 6·15-s + 16-s − 6·17-s − 2·20-s − 3·21-s − 7·25-s − 9·27-s + 28-s − 2·35-s + 6·36-s + 6·37-s − 10·43-s − 12·45-s + 26·47-s − 3·48-s − 6·49-s + 18·51-s − 20·59-s + 6·60-s + 6·63-s + 64-s + ⋯
L(s)  = 1  − 1.73·3-s + 1/2·4-s − 0.894·5-s + 0.377·7-s + 2·9-s − 0.866·12-s + 1.54·15-s + 1/4·16-s − 1.45·17-s − 0.447·20-s − 0.654·21-s − 7/5·25-s − 1.73·27-s + 0.188·28-s − 0.338·35-s + 36-s + 0.986·37-s − 1.52·43-s − 1.78·45-s + 3.79·47-s − 0.433·48-s − 6/7·49-s + 2.52·51-s − 2.60·59-s + 0.774·60-s + 0.755·63-s + 1/8·64-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(298116\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(19.0081\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 298116,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5890106213\)
\(L(\frac12)\) \(\approx\) \(0.5890106213\)
\(L(\frac{3}{2})\) 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_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
3$C_2$ \( 1 + p T + p T^{2} \)
7$C_2$ \( 1 - T + p T^{2} \)
13$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good5$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
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$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 13 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
59$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
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

−8.905235966993525833388868947156, −8.225552760361780021957840571261, −7.62508756975103450239324980184, −7.45985560667588965978165906144, −6.92076425331888408807137767822, −6.38286220020503058850747043516, −5.88908450915949561478136129218, −5.73681320363144770957728323698, −4.71843544103535120742978284678, −4.61153453491182141362175201622, −4.05080151028369257637143221969, −3.36991235720395449589306296865, −2.34027315669446164505659313213, −1.64016380827058072854636802791, −0.49679880081187161703829014750, 0.49679880081187161703829014750, 1.64016380827058072854636802791, 2.34027315669446164505659313213, 3.36991235720395449589306296865, 4.05080151028369257637143221969, 4.61153453491182141362175201622, 4.71843544103535120742978284678, 5.73681320363144770957728323698, 5.88908450915949561478136129218, 6.38286220020503058850747043516, 6.92076425331888408807137767822, 7.45985560667588965978165906144, 7.62508756975103450239324980184, 8.225552760361780021957840571261, 8.905235966993525833388868947156

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