Properties

Label 4-980e2-1.1-c0e2-0-3
Degree $4$
Conductor $960400$
Sign $1$
Analytic cond. $0.239202$
Root an. cond. $0.699345$
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-s + 3-s + 5-s + 6-s − 8-s + 9-s + 10-s + 15-s − 16-s + 18-s − 23-s − 24-s + 2·27-s − 2·29-s + 30-s − 40-s + 2·41-s + 2·43-s + 45-s − 46-s − 2·47-s − 48-s + 2·54-s − 2·58-s − 61-s + 64-s − 67-s + ⋯
L(s)  = 1  + 2-s + 3-s + 5-s + 6-s − 8-s + 9-s + 10-s + 15-s − 16-s + 18-s − 23-s − 24-s + 2·27-s − 2·29-s + 30-s − 40-s + 2·41-s + 2·43-s + 45-s − 46-s − 2·47-s − 48-s + 2·54-s − 2·58-s − 61-s + 64-s − 67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(960400\)    =    \(2^{4} \cdot 5^{2} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(0.239202\)
Root analytic conductor: \(0.699345\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{980} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 960400,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.305322459\)
\(L(\frac12)\) \(\approx\) \(2.305322459\)
\(L(1)\) 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$C_2$ \( 1 - T + T^{2} \)
5$C_2$ \( 1 - T + T^{2} \)
7 \( 1 \)
good3$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
11$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
13$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
17$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
19$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
23$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 - T + T^{2} ) \)
29$C_2$ \( ( 1 + T + T^{2} )^{2} \)
31$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
37$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
41$C_2$ \( ( 1 - T + T^{2} )^{2} \)
43$C_2$ \( ( 1 - T + T^{2} )^{2} \)
47$C_2$ \( ( 1 + T + T^{2} )^{2} \)
53$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
59$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
61$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 - T + T^{2} ) \)
67$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 - T + T^{2} ) \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
73$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
79$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
83$C_2$ \( ( 1 + T + T^{2} )^{2} \)
89$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 - T + T^{2} ) \)
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

−10.27063485513183203739136769420, −9.927437613709668642929250081885, −9.367407458892367353981578338808, −9.306118320026644146129656688557, −8.920743982118600142009791349902, −8.386841777392394641761873334448, −7.78613230860870997643551956293, −7.61834630716422648512255728174, −6.99710191533280619868720604093, −6.28411092865483912116435593223, −6.23179800747396393729281022283, −5.43849516585984197964514009485, −5.42034426440017055741925048047, −4.50244504137645818918700393510, −4.17212403703495731685669257476, −3.86209325741397357243344498892, −3.01324797728851002423687673662, −2.75032645736142701288441007675, −2.07868888227772951163696460029, −1.44552514691148156794630113023, 1.44552514691148156794630113023, 2.07868888227772951163696460029, 2.75032645736142701288441007675, 3.01324797728851002423687673662, 3.86209325741397357243344498892, 4.17212403703495731685669257476, 4.50244504137645818918700393510, 5.42034426440017055741925048047, 5.43849516585984197964514009485, 6.23179800747396393729281022283, 6.28411092865483912116435593223, 6.99710191533280619868720604093, 7.61834630716422648512255728174, 7.78613230860870997643551956293, 8.386841777392394641761873334448, 8.920743982118600142009791349902, 9.306118320026644146129656688557, 9.367407458892367353981578338808, 9.927437613709668642929250081885, 10.27063485513183203739136769420

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