Properties

Label 4-1216e2-1.1-c2e2-0-9
Degree $4$
Conductor $1478656$
Sign $1$
Analytic cond. $1097.83$
Root an. cond. $5.75617$
Motivic weight $2$
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  + 8·5-s + 2·7-s − 11·9-s + 28·11-s + 46·17-s + 20·19-s + 2·23-s − 2·25-s + 16·35-s + 136·43-s − 88·45-s − 52·47-s − 95·49-s + 224·55-s + 80·61-s − 22·63-s − 14·73-s + 56·77-s + 40·81-s + 64·83-s + 368·85-s + 160·95-s − 308·99-s − 28·101-s + 16·115-s + 92·119-s + 346·121-s + ⋯
L(s)  = 1  + 8/5·5-s + 2/7·7-s − 1.22·9-s + 2.54·11-s + 2.70·17-s + 1.05·19-s + 2/23·23-s − 0.0799·25-s + 0.457·35-s + 3.16·43-s − 1.95·45-s − 1.10·47-s − 1.93·49-s + 4.07·55-s + 1.31·61-s − 0.349·63-s − 0.191·73-s + 8/11·77-s + 0.493·81-s + 0.771·83-s + 4.32·85-s + 1.68·95-s − 3.11·99-s − 0.277·101-s + 0.139·115-s + 0.773·119-s + 2.85·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1478656\)    =    \(2^{12} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(1097.83\)
Root analytic conductor: \(5.75617\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1216} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1478656,\ (\ :1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(6.102580181\)
\(L(\frac12)\) \(\approx\) \(6.102580181\)
\(L(2)\) 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 \( 1 \)
19$C_2$ \( 1 - 20 T + p^{2} T^{2} \)
good3$C_2^2$ \( 1 + 11 T^{2} + p^{4} T^{4} \)
5$C_2$ \( ( 1 - 4 T + p^{2} T^{2} )^{2} \)
7$C_2$ \( ( 1 - T + p^{2} T^{2} )^{2} \)
11$C_2$ \( ( 1 - 14 T + p^{2} T^{2} )^{2} \)
13$C_2^2$ \( 1 - 77 T^{2} + p^{4} T^{4} \)
17$C_2$ \( ( 1 - 23 T + p^{2} T^{2} )^{2} \)
23$C_2$ \( ( 1 - T + p^{2} T^{2} )^{2} \)
29$C_2^2$ \( 1 + 23 p T^{2} + p^{4} T^{4} \)
31$C_2^2$ \( 1 - 878 T^{2} + p^{4} T^{4} \)
37$C_2^2$ \( 1 - 1694 T^{2} + p^{4} T^{4} \)
41$C_2^2$ \( 1 - 2318 T^{2} + p^{4} T^{4} \)
43$C_2$ \( ( 1 - 68 T + p^{2} T^{2} )^{2} \)
47$C_2$ \( ( 1 + 26 T + p^{2} T^{2} )^{2} \)
53$C_2^2$ \( 1 + 907 T^{2} + p^{4} T^{4} \)
59$C_2^2$ \( 1 - 6701 T^{2} + p^{4} T^{4} \)
61$C_2$ \( ( 1 - 40 T + p^{2} T^{2} )^{2} \)
67$C_2^2$ \( 1 - 8717 T^{2} + p^{4} T^{4} \)
71$C_2^2$ \( 1 - 9038 T^{2} + p^{4} T^{4} \)
73$C_2$ \( ( 1 + 7 T + p^{2} T^{2} )^{2} \)
79$C_2^2$ \( 1 - 3086 T^{2} + p^{4} T^{4} \)
83$C_2$ \( ( 1 - 32 T + p^{2} T^{2} )^{2} \)
89$C_2^2$ \( 1 + 862 T^{2} + p^{4} T^{4} \)
97$C_2^2$ \( 1 - 9422 T^{2} + p^{4} T^{4} \)
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

−9.849767427588763911175565021905, −9.322302678789353060981181437082, −9.260830292487217014082670146261, −8.522736283913833552330693976924, −8.249045386689032405343335985171, −7.54129496404459309833062785879, −7.47590301381111272033370804583, −6.65462470832823713973030884777, −6.24332934296524810959101456604, −5.98887171304563702153917503313, −5.57991657693264588891736896762, −5.35349135037206143523153771266, −4.71885420762624143477846122402, −3.90215098099036191626260203998, −3.61064707848301841295365696928, −3.08577152874536546932261570591, −2.48785799172814581395880074537, −1.70870053072043805899752136022, −1.30987901373179710315103427509, −0.818038274152329333782654862143, 0.818038274152329333782654862143, 1.30987901373179710315103427509, 1.70870053072043805899752136022, 2.48785799172814581395880074537, 3.08577152874536546932261570591, 3.61064707848301841295365696928, 3.90215098099036191626260203998, 4.71885420762624143477846122402, 5.35349135037206143523153771266, 5.57991657693264588891736896762, 5.98887171304563702153917503313, 6.24332934296524810959101456604, 6.65462470832823713973030884777, 7.47590301381111272033370804583, 7.54129496404459309833062785879, 8.249045386689032405343335985171, 8.522736283913833552330693976924, 9.260830292487217014082670146261, 9.322302678789353060981181437082, 9.849767427588763911175565021905

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