Properties

Label 4-3072e2-1.1-c1e2-0-0
Degree $4$
Conductor $9437184$
Sign $1$
Analytic cond. $601.723$
Root an. cond. $4.95278$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s + 3·9-s − 8·11-s + 12·17-s − 4·19-s − 2·25-s + 4·27-s − 16·33-s − 20·41-s + 12·43-s + 4·49-s + 24·51-s − 8·57-s − 8·67-s + 32·73-s − 4·75-s + 5·81-s − 32·83-s + 28·89-s − 8·97-s − 24·99-s − 8·107-s − 12·113-s + 26·121-s − 40·123-s + 127-s + 24·129-s + ⋯
L(s)  = 1  + 1.15·3-s + 9-s − 2.41·11-s + 2.91·17-s − 0.917·19-s − 2/5·25-s + 0.769·27-s − 2.78·33-s − 3.12·41-s + 1.82·43-s + 4/7·49-s + 3.36·51-s − 1.05·57-s − 0.977·67-s + 3.74·73-s − 0.461·75-s + 5/9·81-s − 3.51·83-s + 2.96·89-s − 0.812·97-s − 2.41·99-s − 0.773·107-s − 1.12·113-s + 2.36·121-s − 3.60·123-s + 0.0887·127-s + 2.11·129-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9437184 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9437184 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(9437184\)    =    \(2^{20} \cdot 3^{2}\)
Sign: $1$
Analytic conductor: \(601.723\)
Root analytic conductor: \(4.95278\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3072} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 9437184,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.901585484\)
\(L(\frac12)\) \(\approx\) \(2.901585484\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
good5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
7$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + 44 T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 56 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 74 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2^2$ \( 1 + 104 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2^2$ \( 1 + 134 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 + 140 T^{2} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
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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

−8.635159546709504234667338907807, −8.510699356568066854635759986460, −7.974570990862250186745901337450, −7.948396338708538476838965701872, −7.51674537177975832348418944940, −7.33586840954896661299214907324, −6.68511301966452371287594349893, −6.32189295784899135292004637526, −5.63145989018020156642899237042, −5.42787690158571511922408982857, −5.14823138510250543370890819612, −4.71581312533343705134500115381, −3.95435075669513884682492463610, −3.75524652057486521148577619705, −3.13221744395784393588325089990, −2.91306687234717475144541813677, −2.44592567828117820293898493449, −1.92648141866716423382181490437, −1.33647034865909251669042359181, −0.48941152695802307824438879138, 0.48941152695802307824438879138, 1.33647034865909251669042359181, 1.92648141866716423382181490437, 2.44592567828117820293898493449, 2.91306687234717475144541813677, 3.13221744395784393588325089990, 3.75524652057486521148577619705, 3.95435075669513884682492463610, 4.71581312533343705134500115381, 5.14823138510250543370890819612, 5.42787690158571511922408982857, 5.63145989018020156642899237042, 6.32189295784899135292004637526, 6.68511301966452371287594349893, 7.33586840954896661299214907324, 7.51674537177975832348418944940, 7.948396338708538476838965701872, 7.974570990862250186745901337450, 8.510699356568066854635759986460, 8.635159546709504234667338907807

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