Properties

Label 4-508032-1.1-c1e2-0-43
Degree $4$
Conductor $508032$
Sign $-1$
Analytic cond. $32.3925$
Root an. cond. $2.38567$
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  − 8·19-s − 6·25-s − 12·29-s + 16·31-s + 12·37-s − 7·49-s + 4·53-s − 8·59-s + 8·83-s + 32·103-s − 4·109-s − 36·113-s − 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 22·169-s + 173-s + 179-s + 181-s + 191-s + ⋯
L(s)  = 1  − 1.83·19-s − 6/5·25-s − 2.22·29-s + 2.87·31-s + 1.97·37-s − 49-s + 0.549·53-s − 1.04·59-s + 0.878·83-s + 3.15·103-s − 0.383·109-s − 3.38·113-s − 0.545·121-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 − 1.69·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 508032 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 508032 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(508032\)    =    \(2^{7} \cdot 3^{4} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(32.3925\)
Root analytic conductor: \(2.38567\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{508032} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 508032,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
7$C_2$ \( 1 + p T^{2} \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
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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

−8.131315776481593009713606089546, −7.82263580546813636934054242173, −7.62403207119791885504595168937, −6.78395093605661555810316280628, −6.30966811163732211212778691110, −6.14969218484336320739052245813, −5.58667709375314371155679753614, −4.83326855247570334999188527992, −4.44811958782670156328485077225, −3.97597569667552498946447381576, −3.42103471864385914094720232253, −2.51659494204614894722398853037, −2.20947719434785265756306128762, −1.24950065754749508798543005172, 0, 1.24950065754749508798543005172, 2.20947719434785265756306128762, 2.51659494204614894722398853037, 3.42103471864385914094720232253, 3.97597569667552498946447381576, 4.44811958782670156328485077225, 4.83326855247570334999188527992, 5.58667709375314371155679753614, 6.14969218484336320739052245813, 6.30966811163732211212778691110, 6.78395093605661555810316280628, 7.62403207119791885504595168937, 7.82263580546813636934054242173, 8.131315776481593009713606089546

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