Properties

Label 4-2016e2-1.1-c1e2-0-2
Degree $4$
Conductor $4064256$
Sign $1$
Analytic cond. $259.140$
Root an. cond. $4.01221$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 7-s − 2·11-s + 10·13-s − 2·17-s + 3·19-s − 2·23-s + 5·25-s − 16·29-s − 31-s + 5·37-s − 4·41-s − 14·43-s + 8·47-s − 6·49-s − 2·53-s − 10·59-s + 2·61-s − 11·67-s + 24·71-s + 3·73-s + 2·77-s + 17·79-s − 32·83-s + 12·89-s − 10·91-s − 28·97-s + 18·101-s + ⋯
L(s)  = 1  − 0.377·7-s − 0.603·11-s + 2.77·13-s − 0.485·17-s + 0.688·19-s − 0.417·23-s + 25-s − 2.97·29-s − 0.179·31-s + 0.821·37-s − 0.624·41-s − 2.13·43-s + 1.16·47-s − 6/7·49-s − 0.274·53-s − 1.30·59-s + 0.256·61-s − 1.34·67-s + 2.84·71-s + 0.351·73-s + 0.227·77-s + 1.91·79-s − 3.51·83-s + 1.27·89-s − 1.04·91-s − 2.84·97-s + 1.79·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4064256 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4064256 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(4064256\)    =    \(2^{10} \cdot 3^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(259.140\)
Root analytic conductor: \(4.01221\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2016} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 4064256,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.855855644\)
\(L(\frac12)\) \(\approx\) \(1.855855644\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_2$ \( 1 + T + p T^{2} \)
good5$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 3 T - 10 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 2 T - 19 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 + T - 30 T^{2} + p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 5 T - 12 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 8 T + 17 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 2 T - 49 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 10 T + 41 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 3 T - 64 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
83$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 12 T + 55 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
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

−9.321373833852006787912146243776, −8.999333564923491531527365074893, −8.534248896308258362089089932393, −8.087817976670662742418047574168, −8.048392526185724101371082483233, −7.35562504577457230625684515014, −6.93641609323326693702872909512, −6.62864834393501251245400976408, −6.10808034512077537490277602006, −5.74853529501354851062450096635, −5.56248223166213060823120987528, −4.89757100787011739868456377650, −4.48280947861078819784756692361, −3.75131724882290840631377320753, −3.54599306599638251290323240384, −3.27656771540192437820912824979, −2.52824361942447923571916898319, −1.74243995647585660642647000863, −1.45543627586309498611810570685, −0.49884445031552276713561240693, 0.49884445031552276713561240693, 1.45543627586309498611810570685, 1.74243995647585660642647000863, 2.52824361942447923571916898319, 3.27656771540192437820912824979, 3.54599306599638251290323240384, 3.75131724882290840631377320753, 4.48280947861078819784756692361, 4.89757100787011739868456377650, 5.56248223166213060823120987528, 5.74853529501354851062450096635, 6.10808034512077537490277602006, 6.62864834393501251245400976408, 6.93641609323326693702872909512, 7.35562504577457230625684515014, 8.048392526185724101371082483233, 8.087817976670662742418047574168, 8.534248896308258362089089932393, 8.999333564923491531527365074893, 9.321373833852006787912146243776

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