Properties

Label 4-6384e2-1.1-c1e2-0-8
Degree $4$
Conductor $40755456$
Sign $1$
Analytic cond. $2598.60$
Root an. cond. $7.13978$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 2·5-s − 2·7-s + 3·9-s − 4·11-s − 2·13-s − 4·15-s − 4·17-s + 2·19-s + 4·21-s + 2·23-s − 2·25-s − 4·27-s + 6·31-s + 8·33-s − 4·35-s + 6·37-s + 4·39-s + 4·41-s + 12·43-s + 6·45-s − 6·47-s + 3·49-s + 8·51-s + 8·53-s − 8·55-s − 4·57-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.894·5-s − 0.755·7-s + 9-s − 1.20·11-s − 0.554·13-s − 1.03·15-s − 0.970·17-s + 0.458·19-s + 0.872·21-s + 0.417·23-s − 2/5·25-s − 0.769·27-s + 1.07·31-s + 1.39·33-s − 0.676·35-s + 0.986·37-s + 0.640·39-s + 0.624·41-s + 1.82·43-s + 0.894·45-s − 0.875·47-s + 3/7·49-s + 1.12·51-s + 1.09·53-s − 1.07·55-s − 0.529·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 40755456 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40755456 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(40755456\)    =    \(2^{8} \cdot 3^{2} \cdot 7^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(2598.60\)
Root analytic conductor: \(7.13978\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 40755456,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
7$C_1$ \( ( 1 + T )^{2} \)
19$C_1$ \( ( 1 - T )^{2} \)
good5$D_{4}$ \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
13$D_{4}$ \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 6 T + 66 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 6 T + 78 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
43$D_{4}$ \( 1 - 12 T + 102 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 6 T + 58 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_4$ \( 1 + 16 T + 166 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 4 T + 58 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
71$C_4$ \( 1 + 4 T + 126 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 12 T + 102 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 10 T + 138 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
97$D_{4}$ \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
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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

−7.73674041782893554023067898512, −7.36669059121803215735298991578, −6.98389143146022798285590448159, −6.94794024092435305733205871937, −6.14329579347987773826354802434, −6.11302098134814773929640547804, −5.71392206758388613238777061848, −5.61956195964614208478148994423, −4.93622076675538860290802181389, −4.75177302331980145895866119315, −4.28099331642809708863295151047, −4.07764284996089666391469507922, −3.28521293171775886302327201260, −2.85623194208018120783506231031, −2.38566800926723974513394845997, −2.34852319883669533654725245476, −1.34357066211476855673484280248, −1.09223177798281766589076648670, 0, 0, 1.09223177798281766589076648670, 1.34357066211476855673484280248, 2.34852319883669533654725245476, 2.38566800926723974513394845997, 2.85623194208018120783506231031, 3.28521293171775886302327201260, 4.07764284996089666391469507922, 4.28099331642809708863295151047, 4.75177302331980145895866119315, 4.93622076675538860290802181389, 5.61956195964614208478148994423, 5.71392206758388613238777061848, 6.11302098134814773929640547804, 6.14329579347987773826354802434, 6.94794024092435305733205871937, 6.98389143146022798285590448159, 7.36669059121803215735298991578, 7.73674041782893554023067898512

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