Properties

Degree $4$
Conductor $396900$
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 − 5-s + 4·7-s + 8-s + 10-s − 11-s + 2·13-s − 4·14-s − 16-s + 3·19-s + 22-s + 7·23-s − 2·26-s + 16·29-s + 2·31-s − 4·35-s − 11·37-s − 3·38-s − 40-s + 22·41-s + 16·43-s − 7·46-s − 5·47-s + 9·49-s − 11·53-s + 55-s + 4·56-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.447·5-s + 1.51·7-s + 0.353·8-s + 0.316·10-s − 0.301·11-s + 0.554·13-s − 1.06·14-s − 1/4·16-s + 0.688·19-s + 0.213·22-s + 1.45·23-s − 0.392·26-s + 2.97·29-s + 0.359·31-s − 0.676·35-s − 1.80·37-s − 0.486·38-s − 0.158·40-s + 3.43·41-s + 2.43·43-s − 1.03·46-s − 0.729·47-s + 9/7·49-s − 1.51·53-s + 0.134·55-s + 0.534·56-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 396900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 396900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(396900\)    =    \(2^{2} \cdot 3^{4} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{630} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 396900,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.61380\)
\(L(\frac12)\) \(\approx\) \(1.61380\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_2$ \( 1 + T + T^{2} \)
7$C_2$ \( 1 - 4 T + p T^{2} \)
good11$C_2^2$ \( 1 + T - 10 T^{2} + p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 3 T - 10 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 7 T + 26 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - 2 T - 27 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 5 T - 22 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 11 T + 68 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 4 T - 43 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 6 T - 37 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 8 T - 15 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 10 T + 11 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
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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

−10.72413962993683057602993174288, −10.70283156795102607203305589936, −9.762326456317124792704289626831, −9.579244728233134039099603337953, −8.938693709671863553690117432568, −8.613731162274783637990180096929, −8.065587700764027843232734547504, −8.057337288023372939254638159756, −7.35744104645292134380895109596, −7.06721290904608414378550394850, −6.41230963520796542499914952544, −5.84806422441936973010158090255, −5.10031771781455566574995644038, −4.94290701192483883303318433442, −4.29676874573128115283301916888, −3.87201001363945203039859530711, −2.82452359752624132613041189539, −2.54682111093608289868636870610, −1.26631601636406220354746746157, −0.988148358679682787051604993347, 0.988148358679682787051604993347, 1.26631601636406220354746746157, 2.54682111093608289868636870610, 2.82452359752624132613041189539, 3.87201001363945203039859530711, 4.29676874573128115283301916888, 4.94290701192483883303318433442, 5.10031771781455566574995644038, 5.84806422441936973010158090255, 6.41230963520796542499914952544, 7.06721290904608414378550394850, 7.35744104645292134380895109596, 8.057337288023372939254638159756, 8.065587700764027843232734547504, 8.613731162274783637990180096929, 8.938693709671863553690117432568, 9.579244728233134039099603337953, 9.762326456317124792704289626831, 10.70283156795102607203305589936, 10.72413962993683057602993174288

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