Properties

Label 4-3822e2-1.1-c1e2-0-4
Degree $4$
Conductor $14607684$
Sign $1$
Analytic cond. $931.398$
Root an. cond. $5.52438$
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  − 2·3-s − 4-s + 3·9-s + 2·12-s + 6·13-s + 16-s − 6·17-s − 2·23-s + 25-s − 4·27-s + 10·29-s − 3·36-s − 12·39-s − 2·43-s − 2·48-s + 12·51-s − 6·52-s + 28·53-s + 6·61-s − 64-s + 6·68-s + 4·69-s − 2·75-s + 5·81-s − 20·87-s + 2·92-s − 100-s + ⋯
L(s)  = 1  − 1.15·3-s − 1/2·4-s + 9-s + 0.577·12-s + 1.66·13-s + 1/4·16-s − 1.45·17-s − 0.417·23-s + 1/5·25-s − 0.769·27-s + 1.85·29-s − 1/2·36-s − 1.92·39-s − 0.304·43-s − 0.288·48-s + 1.68·51-s − 0.832·52-s + 3.84·53-s + 0.768·61-s − 1/8·64-s + 0.727·68-s + 0.481·69-s − 0.230·75-s + 5/9·81-s − 2.14·87-s + 0.208·92-s − 0.0999·100-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 14607684 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14607684 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(14607684\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{4} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(931.398\)
Root analytic conductor: \(5.52438\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 14607684,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.814912666\)
\(L(\frac12)\) \(\approx\) \(1.814912666\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
3$C_1$ \( ( 1 + T )^{2} \)
7 \( 1 \)
13$C_2$ \( 1 - 6 T + p T^{2} \)
good5$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 3 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
19$C_2^2$ \( 1 - 37 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - p T^{2} )^{2} \)
43$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 + 78 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 42 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
97$C_2^2$ \( 1 - 190 T^{2} + p^{2} T^{4} \)
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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

−8.657125988312815435372014826257, −8.582577335381216807710173931707, −7.77840636651054930316797343792, −7.72782540541938451114662998913, −6.89803646769211186237488301309, −6.88492420145800632857061378353, −6.37673355583598373350023667169, −5.96849787516930691220951283719, −5.95788415156388779443713366608, −5.12365964625832896332202733507, −5.06168442479569709608310555560, −4.55996690636076979084861311869, −4.02355196865025184086112996245, −3.91434705374179076335939784827, −3.38779819560221011376225504792, −2.66683924418493550889596527714, −2.19555752122016092262001852524, −1.59272909353369140486450796570, −0.813828937525400171009416244286, −0.62448721802453955512396882995, 0.62448721802453955512396882995, 0.813828937525400171009416244286, 1.59272909353369140486450796570, 2.19555752122016092262001852524, 2.66683924418493550889596527714, 3.38779819560221011376225504792, 3.91434705374179076335939784827, 4.02355196865025184086112996245, 4.55996690636076979084861311869, 5.06168442479569709608310555560, 5.12365964625832896332202733507, 5.95788415156388779443713366608, 5.96849787516930691220951283719, 6.37673355583598373350023667169, 6.88492420145800632857061378353, 6.89803646769211186237488301309, 7.72782540541938451114662998913, 7.77840636651054930316797343792, 8.582577335381216807710173931707, 8.657125988312815435372014826257

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