Properties

Label 4-4018e2-1.1-c1e2-0-4
Degree $4$
Conductor $16144324$
Sign $1$
Analytic cond. $1029.37$
Root an. cond. $5.66426$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 2·3-s + 3·4-s + 4·6-s − 4·8-s − 3·9-s − 6·12-s − 4·13-s + 5·16-s − 6·17-s + 6·18-s + 8·19-s + 8·23-s + 8·24-s − 5·25-s + 8·26-s + 14·27-s + 8·29-s − 8·31-s − 6·32-s + 12·34-s − 9·36-s − 8·37-s − 16·38-s + 8·39-s + 2·41-s + 18·43-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.15·3-s + 3/2·4-s + 1.63·6-s − 1.41·8-s − 9-s − 1.73·12-s − 1.10·13-s + 5/4·16-s − 1.45·17-s + 1.41·18-s + 1.83·19-s + 1.66·23-s + 1.63·24-s − 25-s + 1.56·26-s + 2.69·27-s + 1.48·29-s − 1.43·31-s − 1.06·32-s + 2.05·34-s − 3/2·36-s − 1.31·37-s − 2.59·38-s + 1.28·39-s + 0.312·41-s + 2.74·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16144324 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16144324 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(16144324\)    =    \(2^{2} \cdot 7^{4} \cdot 41^{2}\)
Sign: $1$
Analytic conductor: \(1029.37\)
Root analytic conductor: \(5.66426\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 16144324,\ (\ :1/2, 1/2),\ 1)\)

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$C_1$ \( ( 1 + T )^{2} \)
7 \( 1 \)
41$C_1$ \( ( 1 - T )^{2} \)
good3$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
5$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 4 T + 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
23$D_{4}$ \( 1 - 8 T + 42 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 8 T + 69 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 8 T + 73 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
47$D_{4}$ \( 1 - 8 T + 30 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 61 T^{2} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 16 T + 162 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 16 T + 141 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 4 T - 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 137 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
79$D_{4}$ \( 1 - 8 T + 129 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 12 T + 122 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
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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

−8.144266088889297967355070265649, −7.979740385401723340553391357470, −7.51057284819970932617427979484, −7.21701023445518339067007334277, −6.81454413211159409078766547924, −6.63605297134577361262915198317, −5.98061143500565850461054530073, −5.79010819383386366042612811583, −5.28452212364952464536099234048, −5.25760794332877520231020417073, −4.47619137220741030084960298606, −4.29587990435804457632792463958, −3.21885174204827141239944519031, −3.11077197980668492156709920257, −2.54237906975774364018531823053, −2.28815566204990683242665584370, −1.33183159011323951243990733491, −0.978948362350664744835581604278, 0, 0, 0.978948362350664744835581604278, 1.33183159011323951243990733491, 2.28815566204990683242665584370, 2.54237906975774364018531823053, 3.11077197980668492156709920257, 3.21885174204827141239944519031, 4.29587990435804457632792463958, 4.47619137220741030084960298606, 5.25760794332877520231020417073, 5.28452212364952464536099234048, 5.79010819383386366042612811583, 5.98061143500565850461054530073, 6.63605297134577361262915198317, 6.81454413211159409078766547924, 7.21701023445518339067007334277, 7.51057284819970932617427979484, 7.979740385401723340553391357470, 8.144266088889297967355070265649

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