Properties

Label 4-448e2-1.1-c1e2-0-37
Degree $4$
Conductor $200704$
Sign $-1$
Analytic cond. $12.7970$
Root an. cond. $1.89137$
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  − 2·9-s + 8·13-s − 4·17-s − 10·25-s − 4·29-s − 20·37-s − 20·41-s + 49-s + 4·53-s + 16·61-s − 12·73-s − 5·81-s + 36·89-s − 4·97-s − 20·109-s + 12·113-s − 16·117-s − 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 8·153-s + 157-s + 163-s + ⋯
L(s)  = 1  − 2/3·9-s + 2.21·13-s − 0.970·17-s − 2·25-s − 0.742·29-s − 3.28·37-s − 3.12·41-s + 1/7·49-s + 0.549·53-s + 2.04·61-s − 1.40·73-s − 5/9·81-s + 3.81·89-s − 0.406·97-s − 1.91·109-s + 1.12·113-s − 1.47·117-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 + 0.646·153-s + 0.0798·157-s + 0.0783·163-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 200704 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200704 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(200704\)    =    \(2^{12} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(12.7970\)
Root analytic conductor: \(1.89137\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 200704,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
7$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
5$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 18 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
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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

−8.733630903561318471582211222656, −8.639094223849195400582656535158, −8.011213156558176860935097506339, −7.46301279752361470901499498374, −6.63420884366981729023986116142, −6.61600017498811391917873139111, −5.88709051867218789769739214518, −5.38712706105755021847422947230, −5.03645409422310771476414899480, −3.87524856765933661131099267326, −3.78995178946695161507197151620, −3.21551153951879944953165713886, −2.05991679989913379402995621149, −1.60688453692534518598007565808, 0, 1.60688453692534518598007565808, 2.05991679989913379402995621149, 3.21551153951879944953165713886, 3.78995178946695161507197151620, 3.87524856765933661131099267326, 5.03645409422310771476414899480, 5.38712706105755021847422947230, 5.88709051867218789769739214518, 6.61600017498811391917873139111, 6.63420884366981729023986116142, 7.46301279752361470901499498374, 8.011213156558176860935097506339, 8.639094223849195400582656535158, 8.733630903561318471582211222656

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