Properties

Label 4-1680e2-1.1-c1e2-0-21
Degree $4$
Conductor $2822400$
Sign $1$
Analytic cond. $179.958$
Root an. cond. $3.66263$
Motivic weight $1$
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  − 4·5-s − 9-s + 8·11-s + 12·19-s + 11·25-s − 12·29-s + 4·31-s + 16·41-s + 4·45-s − 49-s − 32·55-s + 8·59-s + 28·61-s + 81-s − 16·89-s − 48·95-s − 8·99-s + 32·101-s + 28·109-s + 26·121-s − 24·125-s + 127-s + 131-s + 137-s + 139-s + 48·145-s + 149-s + ⋯
L(s)  = 1  − 1.78·5-s − 1/3·9-s + 2.41·11-s + 2.75·19-s + 11/5·25-s − 2.22·29-s + 0.718·31-s + 2.49·41-s + 0.596·45-s − 1/7·49-s − 4.31·55-s + 1.04·59-s + 3.58·61-s + 1/9·81-s − 1.69·89-s − 4.92·95-s − 0.804·99-s + 3.18·101-s + 2.68·109-s + 2.36·121-s − 2.14·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 3.98·145-s + 0.0819·149-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2822400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2822400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2822400\)    =    \(2^{8} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(179.958\)
Root analytic conductor: \(3.66263\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1680} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2822400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.323890274\)
\(L(\frac12)\) \(\approx\) \(2.323890274\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
5$C_2$ \( 1 + 4 T + p T^{2} \)
7$C_2$ \( 1 + T^{2} \)
good11$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 42 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 78 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 90 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 94 T^{2} + p^{2} T^{4} \)
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

−9.488183623370718288035170545107, −9.137980944463362249593741490558, −8.616106659845160174059244919632, −8.578180954717967574982679366391, −7.83696356226028951202082917396, −7.47284940877725177831588395027, −7.15798338761632556354860069909, −7.12963352568719928396862848136, −6.23263812359207440330252395833, −6.09743681843731010569910737818, −5.30397512914412908283084271049, −5.15449796315116488801900665728, −4.31160241874633658516906267280, −4.04747557179696972558413138932, −3.62796829751011359702272528238, −3.42031777145297627005271159277, −2.77274814962515678305357825915, −1.94256030229403833502821782546, −0.935188665223478661022326246369, −0.841417224320144161065720466590, 0.841417224320144161065720466590, 0.935188665223478661022326246369, 1.94256030229403833502821782546, 2.77274814962515678305357825915, 3.42031777145297627005271159277, 3.62796829751011359702272528238, 4.04747557179696972558413138932, 4.31160241874633658516906267280, 5.15449796315116488801900665728, 5.30397512914412908283084271049, 6.09743681843731010569910737818, 6.23263812359207440330252395833, 7.12963352568719928396862848136, 7.15798338761632556354860069909, 7.47284940877725177831588395027, 7.83696356226028951202082917396, 8.578180954717967574982679366391, 8.616106659845160174059244919632, 9.137980944463362249593741490558, 9.488183623370718288035170545107

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