Properties

Label 4-1344e2-1.1-c1e2-0-32
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 $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 5-s + 5·7-s + 11-s − 15-s + 8·17-s − 4·19-s − 5·21-s − 4·23-s + 5·25-s + 27-s + 10·29-s − 7·31-s − 33-s + 5·35-s + 8·37-s + 8·41-s − 20·43-s − 6·47-s + 18·49-s − 8·51-s − 53-s + 55-s + 4·57-s + 9·59-s − 2·61-s + 2·67-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 1.88·7-s + 0.301·11-s − 0.258·15-s + 1.94·17-s − 0.917·19-s − 1.09·21-s − 0.834·23-s + 25-s + 0.192·27-s + 1.85·29-s − 1.25·31-s − 0.174·33-s + 0.845·35-s + 1.31·37-s + 1.24·41-s − 3.04·43-s − 0.875·47-s + 18/7·49-s − 1.12·51-s − 0.137·53-s + 0.134·55-s + 0.529·57-s + 1.17·59-s − 0.256·61-s + 0.244·67-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: \(0\)
Selberg data: \((4,\ 1806336,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.998926650\)
\(L(\frac12)\) \(\approx\) \(2.998926650\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - 5 T + p T^{2} \)
good5$C_2^2$ \( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - T - 10 T^{2} - p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - 8 T + 47 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 4 T - 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
37$C_2^2$ \( 1 - 8 T + 27 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + T - 52 T^{2} + p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 9 T + 22 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 2 T - 57 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 2 T - 63 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 9 T + 2 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
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

−9.771575565859570022491580385872, −9.659518403218010228369853799927, −8.742246299390820731005239613668, −8.648337938869299425675133949218, −8.064522442701276376092597899293, −8.022795475321647779525766668366, −7.38342539216342759655630631574, −7.00829318203567106109216455654, −6.36985627647395589409330629028, −6.06543630531402533596344589126, −5.68887996800110533036229011674, −5.05290141194100084625959097186, −4.77944480205365351660651605344, −4.63989452383242343099588936017, −3.64120534326883049856286020436, −3.46389469796025996116386638006, −2.45067608633037747285680715120, −2.05058978641858857343839272505, −1.32647461576426298406969447795, −0.846239260396293073982028686291, 0.846239260396293073982028686291, 1.32647461576426298406969447795, 2.05058978641858857343839272505, 2.45067608633037747285680715120, 3.46389469796025996116386638006, 3.64120534326883049856286020436, 4.63989452383242343099588936017, 4.77944480205365351660651605344, 5.05290141194100084625959097186, 5.68887996800110533036229011674, 6.06543630531402533596344589126, 6.36985627647395589409330629028, 7.00829318203567106109216455654, 7.38342539216342759655630631574, 8.022795475321647779525766668366, 8.064522442701276376092597899293, 8.648337938869299425675133949218, 8.742246299390820731005239613668, 9.659518403218010228369853799927, 9.771575565859570022491580385872

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