Properties

Label 4-21e4-1.1-c2e2-0-15
Degree $4$
Conductor $194481$
Sign $1$
Analytic cond. $144.393$
Root an. cond. $3.46646$
Motivic weight $2$
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  + 3·2-s + 4·4-s + 9·5-s + 9·8-s + 27·10-s + 15·11-s + 27·16-s + 18·17-s + 18·19-s + 36·20-s + 45·22-s + 29·25-s + 18·29-s + 21·31-s + 36·32-s + 54·34-s − 10·37-s + 54·38-s + 81·40-s − 148·43-s + 60·44-s + 87·50-s + 33·53-s + 135·55-s + 54·58-s + 27·59-s − 156·61-s + ⋯
L(s)  = 1  + 3/2·2-s + 4-s + 9/5·5-s + 9/8·8-s + 2.69·10-s + 1.36·11-s + 1.68·16-s + 1.05·17-s + 0.947·19-s + 9/5·20-s + 2.04·22-s + 1.15·25-s + 0.620·29-s + 0.677·31-s + 9/8·32-s + 1.58·34-s − 0.270·37-s + 1.42·38-s + 2.02·40-s − 3.44·43-s + 1.36·44-s + 1.73·50-s + 0.622·53-s + 2.45·55-s + 0.931·58-s + 0.457·59-s − 2.55·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(194481\)    =    \(3^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(144.393\)
Root analytic conductor: \(3.46646\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{441} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 194481,\ (\ :1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(10.11055786\)
\(L(\frac12)\) \(\approx\) \(10.11055786\)
\(L(2)\) 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)$
bad3 \( 1 \)
7 \( 1 \)
good2$C_2^2$ \( 1 - 3 T + 5 T^{2} - 3 p^{2} T^{3} + p^{4} T^{4} \)
5$C_2^2$ \( 1 - 9 T + 52 T^{2} - 9 p^{2} T^{3} + p^{4} T^{4} \)
11$C_2^2$ \( 1 - 15 T + 104 T^{2} - 15 p^{2} T^{3} + p^{4} T^{4} \)
13$C_2$ \( ( 1 - 22 T + p^{2} T^{2} )( 1 + 22 T + p^{2} T^{2} ) \)
17$C_2^2$ \( 1 - 18 T + 397 T^{2} - 18 p^{2} T^{3} + p^{4} T^{4} \)
19$C_2^2$ \( 1 - 18 T + 469 T^{2} - 18 p^{2} T^{3} + p^{4} T^{4} \)
23$C_2^2$ \( 1 - p^{2} T^{2} + p^{4} T^{4} \)
29$C_2$ \( ( 1 - 9 T + p^{2} T^{2} )^{2} \)
31$C_2^2$ \( 1 - 21 T + 1108 T^{2} - 21 p^{2} T^{3} + p^{4} T^{4} \)
37$C_2^2$ \( 1 + 10 T - 1269 T^{2} + 10 p^{2} T^{3} + p^{4} T^{4} \)
41$C_2^2$ \( 1 - 3254 T^{2} + p^{4} T^{4} \)
43$C_2$ \( ( 1 + 74 T + p^{2} T^{2} )^{2} \)
47$C_2$ \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \)
53$C_2^2$ \( 1 - 33 T - 1720 T^{2} - 33 p^{2} T^{3} + p^{4} T^{4} \)
59$C_2^2$ \( 1 - 27 T + 3724 T^{2} - 27 p^{2} T^{3} + p^{4} T^{4} \)
61$C_2^2$ \( 1 + 156 T + 11833 T^{2} + 156 p^{2} T^{3} + p^{4} T^{4} \)
67$C_2^2$ \( 1 - 76 T + 1287 T^{2} - 76 p^{2} T^{3} + p^{4} T^{4} \)
71$C_2$ \( ( 1 + 84 T + p^{2} T^{2} )^{2} \)
73$C_2^2$ \( 1 - 108 T + 9217 T^{2} - 108 p^{2} T^{3} + p^{4} T^{4} \)
79$C_2^2$ \( 1 - 43 T - 4392 T^{2} - 43 p^{2} T^{3} + p^{4} T^{4} \)
83$C_2^2$ \( 1 + 505 T^{2} + p^{4} T^{4} \)
89$C_2^2$ \( 1 + 126 T + 13213 T^{2} + 126 p^{2} T^{3} + p^{4} T^{4} \)
97$C_2^2$ \( 1 + 15529 T^{2} + p^{4} T^{4} \)
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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

−11.28239680886991323352204439331, −10.65184406160002211584628549957, −10.02819274721080405891246255996, −10.02089355902050954538393607182, −9.525546350238585802318926224962, −9.021821335528425036097809434405, −8.339846893183846490102728546834, −7.925207923471441567100627133405, −7.10813269465049303391730710493, −6.78706665317338901335409424432, −6.26736350532460625388042975350, −5.62549660869494719576336807405, −5.60668829887881067999572827009, −4.69408125065792332282838004798, −4.66743094418123392648240914773, −3.49244339944580300799412231899, −3.45177764469321416326660028116, −2.48541888439104750421038234984, −1.49820219931075731157811978334, −1.36584315168770142526077356479, 1.36584315168770142526077356479, 1.49820219931075731157811978334, 2.48541888439104750421038234984, 3.45177764469321416326660028116, 3.49244339944580300799412231899, 4.66743094418123392648240914773, 4.69408125065792332282838004798, 5.60668829887881067999572827009, 5.62549660869494719576336807405, 6.26736350532460625388042975350, 6.78706665317338901335409424432, 7.10813269465049303391730710493, 7.925207923471441567100627133405, 8.339846893183846490102728546834, 9.021821335528425036097809434405, 9.525546350238585802318926224962, 10.02089355902050954538393607182, 10.02819274721080405891246255996, 10.65184406160002211584628549957, 11.28239680886991323352204439331

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