Properties

Label 4-432e2-1.1-c1e2-0-20
Degree $4$
Conductor $186624$
Sign $-1$
Analytic cond. $11.8993$
Root an. cond. $1.85729$
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  − 6·7-s + 8·13-s − 4·19-s − 9·25-s + 14·31-s − 12·37-s + 4·43-s + 13·49-s − 16·61-s + 20·67-s + 2·73-s − 32·79-s − 48·91-s − 2·97-s − 8·103-s + 20·109-s + 3·121-s + 127-s + 131-s + 24·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  − 2.26·7-s + 2.21·13-s − 0.917·19-s − 9/5·25-s + 2.51·31-s − 1.97·37-s + 0.609·43-s + 13/7·49-s − 2.04·61-s + 2.44·67-s + 0.234·73-s − 3.60·79-s − 5.03·91-s − 0.203·97-s − 0.788·103-s + 1.91·109-s + 3/11·121-s + 0.0887·127-s + 0.0873·131-s + 2.08·133-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 186624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 186624 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(186624\)    =    \(2^{8} \cdot 3^{6}\)
Sign: $-1$
Analytic conductor: \(11.8993\)
Root analytic conductor: \(1.85729\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 186624,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
good5$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
7$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
19$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
61$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
73$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
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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

−8.725756998247506784118845055879, −8.606452198182933509057251441889, −8.118827864475224879759848570443, −7.34035520144058455447847378661, −6.80823697240127279065674642746, −6.26453123148316406662568374097, −6.14352965397199809207231240604, −5.77986172759894960585934385055, −4.77743147789251747592373738616, −4.03096914392659909586507649705, −3.63205300189839305102641705028, −3.19407158132585076306422339349, −2.44435098491115322228983755215, −1.34144306055974384014082082012, 0, 1.34144306055974384014082082012, 2.44435098491115322228983755215, 3.19407158132585076306422339349, 3.63205300189839305102641705028, 4.03096914392659909586507649705, 4.77743147789251747592373738616, 5.77986172759894960585934385055, 6.14352965397199809207231240604, 6.26453123148316406662568374097, 6.80823697240127279065674642746, 7.34035520144058455447847378661, 8.118827864475224879759848570443, 8.606452198182933509057251441889, 8.725756998247506784118845055879

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