Properties

Label 4-1638e2-1.1-c1e2-0-40
Degree $4$
Conductor $2683044$
Sign $1$
Analytic cond. $171.073$
Root an. cond. $3.61655$
Motivic weight $1$
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-s + 3·5-s − 7-s − 8-s + 3·10-s + 3·11-s + 2·13-s − 14-s − 16-s + 6·17-s + 4·19-s + 3·22-s + 6·23-s + 5·25-s + 2·26-s + 18·29-s − 5·31-s + 6·34-s − 3·35-s + 4·37-s + 4·38-s − 3·40-s + 24·41-s − 8·43-s + 6·46-s − 12·47-s − 6·49-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.34·5-s − 0.377·7-s − 0.353·8-s + 0.948·10-s + 0.904·11-s + 0.554·13-s − 0.267·14-s − 1/4·16-s + 1.45·17-s + 0.917·19-s + 0.639·22-s + 1.25·23-s + 25-s + 0.392·26-s + 3.34·29-s − 0.898·31-s + 1.02·34-s − 0.507·35-s + 0.657·37-s + 0.648·38-s − 0.474·40-s + 3.74·41-s − 1.21·43-s + 0.884·46-s − 1.75·47-s − 6/7·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2683044 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2683044 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2683044\)    =    \(2^{2} \cdot 3^{4} \cdot 7^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(171.073\)
Root analytic conductor: \(3.61655\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2683044,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.832401519\)
\(L(\frac12)\) \(\approx\) \(5.832401519\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + T^{2} \)
3 \( 1 \)
7$C_2$ \( 1 + T + p T^{2} \)
13$C_1$ \( ( 1 - T )^{2} \)
good5$C_2^2$ \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 + 5 T - 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 4 T - 21 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 12 T + 97 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 9 T + 28 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 9 T + 22 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 14 T + 123 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - T - 78 T^{2} - p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
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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

−9.677286499342414638979079150218, −9.256641012689256851608678656865, −8.774867611674503932879902599317, −8.657381878749197627320158681074, −7.87264543978164493121379191625, −7.59049832618749718059399906596, −7.09623980941927222602735005639, −6.43074036615804138800314602138, −6.33032608211884014369495241872, −5.89779746362935082912521645757, −5.68381845720599514463471482957, −4.94117735804494608267835584105, −4.62321460050586810059880066670, −4.40262111055043961759429878924, −3.36747468089217585794957718983, −3.06127535050666841314399495950, −3.02667421671310850877187875558, −2.04132312773080753569056477495, −1.17240791147503843095174597719, −1.06755725453847286167940506226, 1.06755725453847286167940506226, 1.17240791147503843095174597719, 2.04132312773080753569056477495, 3.02667421671310850877187875558, 3.06127535050666841314399495950, 3.36747468089217585794957718983, 4.40262111055043961759429878924, 4.62321460050586810059880066670, 4.94117735804494608267835584105, 5.68381845720599514463471482957, 5.89779746362935082912521645757, 6.33032608211884014369495241872, 6.43074036615804138800314602138, 7.09623980941927222602735005639, 7.59049832618749718059399906596, 7.87264543978164493121379191625, 8.657381878749197627320158681074, 8.774867611674503932879902599317, 9.256641012689256851608678656865, 9.677286499342414638979079150218

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