Properties

Label 4-387200-1.1-c1e2-0-11
Degree $4$
Conductor $387200$
Sign $-1$
Analytic cond. $24.6882$
Root an. cond. $2.22906$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·7-s + 2·9-s − 2·11-s + 25-s + 12·37-s + 6·43-s + 14·49-s − 12·63-s + 12·77-s − 5·81-s + 6·83-s − 12·89-s − 8·97-s − 4·99-s − 14·107-s − 7·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + 173-s + ⋯
L(s)  = 1  − 2.26·7-s + 2/3·9-s − 0.603·11-s + 1/5·25-s + 1.97·37-s + 0.914·43-s + 2·49-s − 1.51·63-s + 1.36·77-s − 5/9·81-s + 0.658·83-s − 1.27·89-s − 0.812·97-s − 0.402·99-s − 1.35·107-s − 0.636·121-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 + 0.769·169-s + 0.0760·173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 387200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(387200\)    =    \(2^{7} \cdot 5^{2} \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(24.6882\)
Root analytic conductor: \(2.22906\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 387200,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
5$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
11$C_2$ \( 1 + 2 T + p T^{2} \)
good3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
7$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
23$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
43$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
59$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
71$C_2^2$ \( 1 + 110 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
83$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \)
89$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
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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

−8.439180875938616727011940265770, −7.892170985343696563358425676700, −7.51666197094499066896034990634, −6.91132769902516708376002455266, −6.63365546151866683416204843590, −6.15707222007822272788714283325, −5.73680263696720981297183710116, −5.17520514953957125803195676558, −4.36164589017613056480387038807, −4.04053717803101629993754247115, −3.32255964088442333422663167124, −2.85069402772783293070255689544, −2.35511113042157275799689054493, −1.11620258833939670245722892847, 0, 1.11620258833939670245722892847, 2.35511113042157275799689054493, 2.85069402772783293070255689544, 3.32255964088442333422663167124, 4.04053717803101629993754247115, 4.36164589017613056480387038807, 5.17520514953957125803195676558, 5.73680263696720981297183710116, 6.15707222007822272788714283325, 6.63365546151866683416204843590, 6.91132769902516708376002455266, 7.51666197094499066896034990634, 7.892170985343696563358425676700, 8.439180875938616727011940265770

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