Properties

Label 4-1080e2-1.1-c1e2-0-28
Degree $4$
Conductor $1166400$
Sign $-1$
Analytic cond. $74.3706$
Root an. cond. $2.93663$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·19-s + 25-s − 10·31-s + 20·37-s − 12·43-s − 14·49-s + 10·61-s − 4·67-s + 12·73-s − 22·79-s + 16·97-s − 24·103-s + 18·109-s − 18·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 26·169-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  − 0.458·19-s + 1/5·25-s − 1.79·31-s + 3.28·37-s − 1.82·43-s − 2·49-s + 1.28·61-s − 0.488·67-s + 1.40·73-s − 2.47·79-s + 1.62·97-s − 2.36·103-s + 1.72·109-s − 1.63·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 2·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1166400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1166400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1166400\)    =    \(2^{6} \cdot 3^{6} \cdot 5^{2}\)
Sign: $-1$
Analytic conductor: \(74.3706\)
Root analytic conductor: \(2.93663\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 1166400,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
5$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good7$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
13$C_2$ \( ( 1 + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
19$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
31$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
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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

−7.88476640049411651853631457862, −7.35341216339073951945123749294, −7.00481999528092178116170950569, −6.46183002954720332042957563162, −6.09524055825850124940401922312, −5.67617773523253624389817414621, −5.07264494567311717754515914838, −4.68654110242436923100068599220, −4.17466568064428202291161107248, −3.62324019465866523110556351352, −3.12527269276681833472760842290, −2.47913934577997053460459890258, −1.88724812338382557411512712411, −1.13214773225246297442393721548, 0, 1.13214773225246297442393721548, 1.88724812338382557411512712411, 2.47913934577997053460459890258, 3.12527269276681833472760842290, 3.62324019465866523110556351352, 4.17466568064428202291161107248, 4.68654110242436923100068599220, 5.07264494567311717754515914838, 5.67617773523253624389817414621, 6.09524055825850124940401922312, 6.46183002954720332042957563162, 7.00481999528092178116170950569, 7.35341216339073951945123749294, 7.88476640049411651853631457862

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