Properties

Degree $4$
Conductor $1920996$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3·5-s + 5·7-s + 8-s + 3·10-s − 11-s + 4·13-s − 5·14-s − 16-s + 3·17-s − 2·19-s + 22-s + 3·23-s + 5·25-s − 4·26-s + 12·29-s + 4·31-s − 3·34-s − 15·35-s − 2·37-s + 2·38-s − 3·40-s + 6·41-s + 4·43-s − 3·46-s − 9·47-s + 18·49-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.34·5-s + 1.88·7-s + 0.353·8-s + 0.948·10-s − 0.301·11-s + 1.10·13-s − 1.33·14-s − 1/4·16-s + 0.727·17-s − 0.458·19-s + 0.213·22-s + 0.625·23-s + 25-s − 0.784·26-s + 2.22·29-s + 0.718·31-s − 0.514·34-s − 2.53·35-s − 0.328·37-s + 0.324·38-s − 0.474·40-s + 0.937·41-s + 0.609·43-s − 0.442·46-s − 1.31·47-s + 18/7·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1920996 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920996 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1920996\)    =    \(2^{2} \cdot 3^{4} \cdot 7^{2} \cdot 11^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{1386} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1920996,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.580767092\)
\(L(\frac12)\) \(\approx\) \(1.580767092\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T + T^{2} \)
3 \( 1 \)
7$C_2$ \( 1 - 5 T + p T^{2} \)
11$C_2$ \( 1 + T + T^{2} \)
good5$C_2^2$ \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 3 T - 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
37$C_2^2$ \( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 9 T + 34 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 5 T - 36 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
71$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 16 T + 183 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
83$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 17 T + p T^{2} )^{2} \)
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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

−9.949354119307875869852054270915, −9.106582593319319140327175679174, −8.659725080321463544619447621246, −8.646188225349877601386588116476, −8.150354029420569066497710908951, −7.960369256014594624864973877121, −7.47037256122227930271297546017, −7.24094646244902765700639984972, −6.67322426898016020530935443680, −5.94864695055717207207822762676, −5.78715347848264763768852222171, −4.79876517842271712229745900127, −4.78046277483799343820175957142, −4.34670305938235612015213356893, −3.89553828636340115483740286384, −3.07480895586425640386497719543, −2.81638326254037990007750585255, −1.76966996698982048694031002079, −1.25211510885093977073377401536, −0.68392592750658741446396216996, 0.68392592750658741446396216996, 1.25211510885093977073377401536, 1.76966996698982048694031002079, 2.81638326254037990007750585255, 3.07480895586425640386497719543, 3.89553828636340115483740286384, 4.34670305938235612015213356893, 4.78046277483799343820175957142, 4.79876517842271712229745900127, 5.78715347848264763768852222171, 5.94864695055717207207822762676, 6.67322426898016020530935443680, 7.24094646244902765700639984972, 7.47037256122227930271297546017, 7.960369256014594624864973877121, 8.150354029420569066497710908951, 8.646188225349877601386588116476, 8.659725080321463544619447621246, 9.106582593319319140327175679174, 9.949354119307875869852054270915

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