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Properties

Label 4-896e2-1.1-c0e2-0-2
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  + 2·11-s + 2·29-s + 2·37-s − 2·43-s − 49-s − 2·53-s − 2·67-s − 81-s − 2·107-s − 2·109-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 + ⋯
L(s)  = 1  + 2·11-s + 2·29-s + 2·37-s − 2·43-s − 49-s − 2·53-s − 2·67-s − 81-s − 2·107-s − 2·109-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 + ⋯

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: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 802816,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.087605307\)
\(L(\frac12)\) \(\approx\) \(1.087605307\)
\(L(1)\) 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$C_2$ \( 1 + T^{2} \)
good3$C_2^2$ \( 1 + T^{4} \)
5$C_2^2$ \( 1 + T^{4} \)
11$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
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_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
29$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
31$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
37$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
41$C_2$ \( ( 1 + T^{2} )^{2} \)
43$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
47$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
53$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
59$C_2^2$ \( 1 + T^{4} \)
61$C_2^2$ \( 1 + T^{4} \)
67$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
71$C_2$ \( ( 1 + T^{2} )^{2} \)
73$C_2$ \( ( 1 + T^{2} )^{2} \)
79$C_2$ \( ( 1 + T^{2} )^{2} \)
83$C_2^2$ \( 1 + T^{4} \)
89$C_2$ \( ( 1 + T^{2} )^{2} \)
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

−10.43096756719523208032377652804, −10.12646156634763419906207487499, −9.466837397912958160309878311214, −9.457780361990515816240673801009, −8.898482236773261366700062761677, −8.482925147518149956285616867902, −7.906721637694024845984422229152, −7.81229612672820194244209303087, −6.83070471734107728099826645424, −6.74456999986489418142346205143, −6.23265098550158758680943743417, −6.05282413147094061700808480869, −5.08929932523714480970487642022, −4.83454051071575640101649654839, −4.07256301127090740614578750652, −4.01707793218408609635329984241, −2.96291420653715139077888087295, −2.87648605440487255235932315351, −1.62432983907612950363498786813, −1.28183308675642035097784966116, 1.28183308675642035097784966116, 1.62432983907612950363498786813, 2.87648605440487255235932315351, 2.96291420653715139077888087295, 4.01707793218408609635329984241, 4.07256301127090740614578750652, 4.83454051071575640101649654839, 5.08929932523714480970487642022, 6.05282413147094061700808480869, 6.23265098550158758680943743417, 6.74456999986489418142346205143, 6.83070471734107728099826645424, 7.81229612672820194244209303087, 7.906721637694024845984422229152, 8.482925147518149956285616867902, 8.898482236773261366700062761677, 9.457780361990515816240673801009, 9.466837397912958160309878311214, 10.12646156634763419906207487499, 10.43096756719523208032377652804

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