Properties

Label 4-840e2-1.1-c1e2-0-41
Degree $4$
Conductor $705600$
Sign $-1$
Analytic cond. $44.9896$
Root an. cond. $2.58987$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 2·7-s − 2·9-s + 2·13-s − 12·19-s + 2·21-s + 25-s + 5·27-s + 12·37-s − 2·39-s + 12·43-s + 3·49-s + 12·57-s − 8·61-s + 4·63-s − 8·67-s + 20·73-s − 75-s − 6·79-s + 81-s − 4·91-s + 14·97-s − 18·103-s − 22·109-s − 12·111-s − 4·117-s + 3·121-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.755·7-s − 2/3·9-s + 0.554·13-s − 2.75·19-s + 0.436·21-s + 1/5·25-s + 0.962·27-s + 1.97·37-s − 0.320·39-s + 1.82·43-s + 3/7·49-s + 1.58·57-s − 1.02·61-s + 0.503·63-s − 0.977·67-s + 2.34·73-s − 0.115·75-s − 0.675·79-s + 1/9·81-s − 0.419·91-s + 1.42·97-s − 1.77·103-s − 2.10·109-s − 1.13·111-s − 0.369·117-s + 3/11·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 705600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 705600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(705600\)    =    \(2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(44.9896\)
Root analytic conductor: \(2.58987\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{705600} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 705600,\ (\ :1/2, 1/2),\ -1)\)

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$C_2$ \( 1 + T + p T^{2} \)
5$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
7$C_1$ \( ( 1 + T )^{2} \)
good11$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
13$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
19$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
61$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
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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

−8.321037332106296729315848288758, −7.61068895852562797070369378134, −7.16216561796569421885219307291, −6.45877795748859351665331711034, −6.17106315121314565062659046983, −6.14206553924090701620723770176, −5.45734437575009036836059962399, −4.83171016164985859832688933282, −4.17514659999582833871651190557, −4.06182673812785905054132006247, −3.16093778472040229481013774454, −2.60467600555756229149551481858, −2.11101541760584806136071858440, −0.949618884734753047450408637076, 0, 0.949618884734753047450408637076, 2.11101541760584806136071858440, 2.60467600555756229149551481858, 3.16093778472040229481013774454, 4.06182673812785905054132006247, 4.17514659999582833871651190557, 4.83171016164985859832688933282, 5.45734437575009036836059962399, 6.14206553924090701620723770176, 6.17106315121314565062659046983, 6.45877795748859351665331711034, 7.16216561796569421885219307291, 7.61068895852562797070369378134, 8.321037332106296729315848288758

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