Properties

Label 4-896e2-1.1-c0e2-0-0
Degree $4$
Conductor $802816$
Sign $1$
Analytic cond. $0.199954$
Root an. cond. $0.668701$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·7-s + 3·49-s + 4·71-s + 4·79-s − 81-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯
L(s)  = 1  − 2·7-s + 3·49-s + 4·71-s + 4·79-s − 81-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 802816 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 802816 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(802816\)    =    \(2^{14} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(0.199954\)
Root analytic conductor: \(0.668701\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{896} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 802816,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6803471836\)
\(L(\frac12)\) \(\approx\) \(0.6803471836\)
\(L(1)\) 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 \( 1 \)
7$C_1$ \( ( 1 + T )^{2} \)
good3$C_2^2$ \( 1 + T^{4} \)
5$C_2^2$ \( 1 + T^{4} \)
11$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
13$C_2^2$ \( 1 + T^{4} \)
17$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
19$C_2^2$ \( 1 + T^{4} \)
23$C_2$ \( ( 1 + T^{2} )^{2} \)
29$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
31$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
37$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
41$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
43$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
47$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
53$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
59$C_2^2$ \( 1 + T^{4} \)
61$C_2^2$ \( 1 + T^{4} \)
67$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
71$C_1$ \( ( 1 - T )^{4} \)
73$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
79$C_1$ \( ( 1 - T )^{4} \)
83$C_2^2$ \( 1 + T^{4} \)
89$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
97$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.60897733640590946990167281856, −9.905047453160838557555947178868, −9.627348474367187936248560614553, −9.527308881005127497001650019730, −8.874171836849939427887581816672, −8.591675587339170309325299746213, −7.918725979961247470488906846370, −7.61559460902891377258952431573, −6.88928402697298387294447091967, −6.77565619617616596172050781987, −6.15758523147932605603450862004, −6.03976651127089508104081777173, −5.14686233740989740911193548778, −5.01908424709103718454780659728, −3.99104710818803354934324698318, −3.73845839731350988791793931379, −3.26737214085796217190633366410, −2.59903864581965609251739485447, −2.11120570570552100753675427989, −0.826826663636152418274709147478, 0.826826663636152418274709147478, 2.11120570570552100753675427989, 2.59903864581965609251739485447, 3.26737214085796217190633366410, 3.73845839731350988791793931379, 3.99104710818803354934324698318, 5.01908424709103718454780659728, 5.14686233740989740911193548778, 6.03976651127089508104081777173, 6.15758523147932605603450862004, 6.77565619617616596172050781987, 6.88928402697298387294447091967, 7.61559460902891377258952431573, 7.918725979961247470488906846370, 8.591675587339170309325299746213, 8.874171836849939427887581816672, 9.527308881005127497001650019730, 9.627348474367187936248560614553, 9.905047453160838557555947178868, 10.60897733640590946990167281856

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