Properties

Label 4-1344e2-1.1-c1e2-0-48
Degree $4$
Conductor $1806336$
Sign $1$
Analytic cond. $115.173$
Root an. cond. $3.27595$
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  − 3-s − 6·5-s − 7-s − 6·11-s + 6·15-s + 5·19-s + 21-s − 12·23-s + 19·25-s + 27-s + 5·31-s + 6·33-s + 6·35-s − 11·37-s + 6·47-s − 6·49-s + 12·53-s + 36·55-s − 5·57-s + 12·59-s + 24·61-s + 15·67-s + 12·69-s − 9·73-s − 19·75-s + 6·77-s − 21·79-s + ⋯
L(s)  = 1  − 0.577·3-s − 2.68·5-s − 0.377·7-s − 1.80·11-s + 1.54·15-s + 1.14·19-s + 0.218·21-s − 2.50·23-s + 19/5·25-s + 0.192·27-s + 0.898·31-s + 1.04·33-s + 1.01·35-s − 1.80·37-s + 0.875·47-s − 6/7·49-s + 1.64·53-s + 4.85·55-s − 0.662·57-s + 1.56·59-s + 3.07·61-s + 1.83·67-s + 1.44·69-s − 1.05·73-s − 2.19·75-s + 0.683·77-s − 2.36·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1806336 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1806336 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1806336\)    =    \(2^{12} \cdot 3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(115.173\)
Root analytic conductor: \(3.27595\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 1806336,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
3$C_2$ \( 1 + T + T^{2} \)
7$C_2$ \( 1 + T + p T^{2} \)
good5$C_2^2$ \( 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
17$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 5 T + 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 12 T + 71 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - 5 T - 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 11 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 12 T + 91 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 12 T + 85 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 24 T + 253 T^{2} - 24 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 15 T + 142 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 + 9 T + 100 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 17 T + p T^{2} ) \)
83$C_2$ \( ( 1 + 18 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 12 T + 137 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 146 T^{2} + p^{2} T^{4} \)
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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

−9.550307273787446611262612100924, −8.658019908715100210787079927498, −8.387692047319383850943820322797, −8.176936513294763150945063077086, −7.933189806714337169012839346649, −7.31723081790695043049624195077, −6.99764078438675412653686669285, −6.92625656552735588390272610055, −5.94588989838265931498718555703, −5.50528408489397892493327127898, −5.31330435445147604513911683068, −4.68379880786666889134455322063, −4.00718989475808231916138934540, −3.95337768321702341242064277203, −3.43572577166968565327994385101, −2.75231337814086034669753156895, −2.33313057210721583348055068698, −1.03454827912494300851240189950, 0, 0, 1.03454827912494300851240189950, 2.33313057210721583348055068698, 2.75231337814086034669753156895, 3.43572577166968565327994385101, 3.95337768321702341242064277203, 4.00718989475808231916138934540, 4.68379880786666889134455322063, 5.31330435445147604513911683068, 5.50528408489397892493327127898, 5.94588989838265931498718555703, 6.92625656552735588390272610055, 6.99764078438675412653686669285, 7.31723081790695043049624195077, 7.933189806714337169012839346649, 8.176936513294763150945063077086, 8.387692047319383850943820322797, 8.658019908715100210787079927498, 9.550307273787446611262612100924

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