Properties

Label 4-20e4-1.1-c1e2-0-11
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 $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·7-s − 2·9-s + 12·17-s − 12·23-s − 8·31-s + 12·41-s + 12·47-s − 2·49-s + 8·63-s − 24·71-s − 4·73-s + 16·79-s − 5·81-s − 12·89-s − 4·97-s − 28·103-s + 12·113-s − 48·119-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 24·153-s + 157-s + ⋯
L(s)  = 1  − 1.51·7-s − 2/3·9-s + 2.91·17-s − 2.50·23-s − 1.43·31-s + 1.87·41-s + 1.75·47-s − 2/7·49-s + 1.00·63-s − 2.84·71-s − 0.468·73-s + 1.80·79-s − 5/9·81-s − 1.27·89-s − 0.406·97-s − 2.75·103-s + 1.12·113-s − 4.40·119-s − 2·121-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 − 1.94·153-s + 0.0798·157-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: \(1\)
Selberg data: \((4,\ 160000,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
71$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
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

−9.258669147089765462577284951282, −8.414042470938792927705837581638, −8.059368180384024597996443374210, −7.38409000222510611898226618037, −7.35539571990596814095408613512, −6.29701258427927054860013270677, −5.85740317846722582719715899176, −5.82400824359360078327530562400, −5.13587744668360471627377345602, −3.90074996505307731038192172560, −3.88729748658685945494863770486, −3.05243989682609854421294881882, −2.58321256178540655780606724063, −1.38798538048163665478930118728, 0, 1.38798538048163665478930118728, 2.58321256178540655780606724063, 3.05243989682609854421294881882, 3.88729748658685945494863770486, 3.90074996505307731038192172560, 5.13587744668360471627377345602, 5.82400824359360078327530562400, 5.85740317846722582719715899176, 6.29701258427927054860013270677, 7.35539571990596814095408613512, 7.38409000222510611898226618037, 8.059368180384024597996443374210, 8.414042470938792927705837581638, 9.258669147089765462577284951282

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