Properties

Label 4-1368e2-1.1-c0e2-0-10
Degree $4$
Conductor $1871424$
Sign $1$
Analytic cond. $0.466107$
Root an. cond. $0.826269$
Motivic weight $0$
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  + 2·2-s + 3-s + 3·4-s + 2·6-s + 4·8-s + 3·12-s + 5·16-s − 3·17-s − 2·19-s + 4·24-s + 25-s − 27-s + 6·32-s − 6·34-s − 4·38-s − 41-s + 2·43-s + 5·48-s − 49-s + 2·50-s − 3·51-s − 2·54-s − 2·57-s + 59-s + 7·64-s − 9·68-s + 2·73-s + ⋯
L(s)  = 1  + 2·2-s + 3-s + 3·4-s + 2·6-s + 4·8-s + 3·12-s + 5·16-s − 3·17-s − 2·19-s + 4·24-s + 25-s − 27-s + 6·32-s − 6·34-s − 4·38-s − 41-s + 2·43-s + 5·48-s − 49-s + 2·50-s − 3·51-s − 2·54-s − 2·57-s + 59-s + 7·64-s − 9·68-s + 2·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1871424 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1871424 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1871424\)    =    \(2^{6} \cdot 3^{4} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(0.466107\)
Root analytic conductor: \(0.826269\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1871424,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(5.049609999\)
\(L(\frac12)\) \(\approx\) \(5.049609999\)
\(L(1)\) 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$C_1$ \( ( 1 - T )^{2} \)
3$C_2$ \( 1 - T + T^{2} \)
19$C_1$ \( ( 1 + T )^{2} \)
good5$C_2^2$ \( 1 - T^{2} + T^{4} \)
7$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
11$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
13$C_2$ \( ( 1 + T^{2} )^{2} \)
17$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T + T^{2} ) \)
23$C_2$ \( ( 1 + T^{2} )^{2} \)
29$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
31$C_2^2$ \( 1 - T^{2} + T^{4} \)
37$C_2$ \( ( 1 + T^{2} )^{2} \)
41$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 - T + T^{2} ) \)
43$C_2$ \( ( 1 - T + T^{2} )^{2} \)
47$C_2^2$ \( 1 - T^{2} + T^{4} \)
53$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
59$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
61$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
67$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
71$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
73$C_2$ \( ( 1 - T + T^{2} )^{2} \)
79$C_2$ \( ( 1 + T^{2} )^{2} \)
83$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T + T^{2} ) \)
89$C_2$ \( ( 1 + T + T^{2} )^{2} \)
97$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
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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

−10.05738317743343404393516529408, −9.645283325981158141300416177522, −9.035382161108632739797915950459, −8.603574794593597963786900524688, −8.375799794996604349316021392341, −8.022572272044741461334092658294, −7.31059235504777561464214283150, −6.89272122826084991012093978262, −6.56649810600448946036840716606, −6.49605851361768181737660137189, −5.61742885590298475158585292665, −5.51862397367057283459929811540, −4.54941426375695327588876820909, −4.47637269167217473955573615766, −4.13935682905790601537629219185, −3.60316751573377103670795345528, −2.90370896257695148726069880142, −2.52308499792742952572811717848, −2.19004075749737114168407515878, −1.67247916555918633457811860114, 1.67247916555918633457811860114, 2.19004075749737114168407515878, 2.52308499792742952572811717848, 2.90370896257695148726069880142, 3.60316751573377103670795345528, 4.13935682905790601537629219185, 4.47637269167217473955573615766, 4.54941426375695327588876820909, 5.51862397367057283459929811540, 5.61742885590298475158585292665, 6.49605851361768181737660137189, 6.56649810600448946036840716606, 6.89272122826084991012093978262, 7.31059235504777561464214283150, 8.022572272044741461334092658294, 8.375799794996604349316021392341, 8.603574794593597963786900524688, 9.035382161108632739797915950459, 9.645283325981158141300416177522, 10.05738317743343404393516529408

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