Properties

Label 4-20e4-1.1-c1e2-0-7
Degree $4$
Conductor $160000$
Sign $1$
Analytic cond. $10.2017$
Root an. cond. $1.78718$
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  + 4·3-s + 6·9-s + 12·17-s − 8·19-s − 4·27-s + 12·41-s + 20·43-s − 10·49-s + 48·51-s − 32·57-s + 24·59-s − 4·67-s − 4·73-s − 37·81-s − 12·83-s − 12·89-s − 4·97-s + 12·107-s + 12·113-s − 22·121-s + 48·123-s + 127-s + 80·129-s + 131-s + 137-s + 139-s − 40·147-s + ⋯
L(s)  = 1  + 2.30·3-s + 2·9-s + 2.91·17-s − 1.83·19-s − 0.769·27-s + 1.87·41-s + 3.04·43-s − 1.42·49-s + 6.72·51-s − 4.23·57-s + 3.12·59-s − 0.488·67-s − 0.468·73-s − 4.11·81-s − 1.31·83-s − 1.27·89-s − 0.406·97-s + 1.16·107-s + 1.12·113-s − 2·121-s + 4.32·123-s + 0.0887·127-s + 7.04·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 3.29·147-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(160000\)    =    \(2^{8} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(10.2017\)
Root analytic conductor: \(1.78718\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 160000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.633451596\)
\(L(\frac12)\) \(\approx\) \(3.633451596\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
73$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
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

−9.249279365224539237566485048625, −8.550149092441660717391493732033, −8.414042470938792927705837581638, −7.950176007435402201982724906846, −7.38409000222510611898226618037, −7.33279621647751156873215628670, −6.14181160684855424693200797613, −5.85740317846722582719715899176, −5.25974231276140424950498810771, −4.19097847493160731891690631788, −3.90074996505307731038192172560, −3.32887958847832114233485788826, −2.58321256178540655780606724063, −2.41662881718811067600819256762, −1.26575059491370931038306222328, 1.26575059491370931038306222328, 2.41662881718811067600819256762, 2.58321256178540655780606724063, 3.32887958847832114233485788826, 3.90074996505307731038192172560, 4.19097847493160731891690631788, 5.25974231276140424950498810771, 5.85740317846722582719715899176, 6.14181160684855424693200797613, 7.33279621647751156873215628670, 7.38409000222510611898226618037, 7.950176007435402201982724906846, 8.414042470938792927705837581638, 8.550149092441660717391493732033, 9.249279365224539237566485048625

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