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Properties

Label 4-471e2-1.1-c1e2-0-2
Degree $4$
Conductor $221841$
Sign $1$
Analytic cond. $14.1447$
Root an. cond. $1.93931$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 2·3-s − 2·4-s − 2·5-s − 2·6-s − 6·7-s + 3·8-s + 3·9-s + 2·10-s − 11-s − 4·12-s − 3·13-s + 6·14-s − 4·15-s + 16-s − 4·17-s − 3·18-s − 6·19-s + 4·20-s − 12·21-s + 22-s + 2·23-s + 6·24-s − 7·25-s + 3·26-s + 4·27-s + 12·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.15·3-s − 4-s − 0.894·5-s − 0.816·6-s − 2.26·7-s + 1.06·8-s + 9-s + 0.632·10-s − 0.301·11-s − 1.15·12-s − 0.832·13-s + 1.60·14-s − 1.03·15-s + 1/4·16-s − 0.970·17-s − 0.707·18-s − 1.37·19-s + 0.894·20-s − 2.61·21-s + 0.213·22-s + 0.417·23-s + 1.22·24-s − 7/5·25-s + 0.588·26-s + 0.769·27-s + 2.26·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 221841 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 221841 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(221841\)    =    \(3^{2} \cdot 157^{2}\)
Sign: $1$
Analytic conductor: \(14.1447\)
Root analytic conductor: \(1.93931\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 221841,\ (\ :1/2, 1/2),\ 1)\)

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)$
bad3$C_1$ \( ( 1 - T )^{2} \)
157$C_1$ \( ( 1 - T )^{2} \)
good2$D_{4}$ \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \)
5$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
7$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
11$D_{4}$ \( 1 + T + p T^{2} + p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 3 T + 27 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 4 T + 33 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 6 T + 42 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
29$D_{4}$ \( 1 + T + 57 T^{2} + p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 3 T + 3 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 6 T + 78 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 16 T + 141 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + 18 T + 162 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 17 T + 155 T^{2} - 17 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 2 T - 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 16 T + 181 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 3 T + 105 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + T + 131 T^{2} + p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 11 T + 75 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 33 T^{2} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 13 T + 147 T^{2} - 13 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - T - 33 T^{2} - p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 8 T + 205 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
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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

−10.34607045405439905895378463380, −10.16759540438945152242720266006, −9.829701499215968728345216306463, −9.337997364178872995170188002763, −9.023210657321562920308200135946, −8.728804736653406036315929622812, −8.067812507896881194574442826522, −7.966748585378495339576607537798, −7.10424288765836229380569836759, −6.91987607244354900359049799156, −6.33363263547950343226467025512, −5.67906741708941499102537177938, −4.65367776509828522209295816646, −4.48122194053465733931348056605, −3.58511136587275176807411632848, −3.54579743297336607009807234871, −2.70077815715092268142519268126, −1.94957010491445475261879710490, 0, 0, 1.94957010491445475261879710490, 2.70077815715092268142519268126, 3.54579743297336607009807234871, 3.58511136587275176807411632848, 4.48122194053465733931348056605, 4.65367776509828522209295816646, 5.67906741708941499102537177938, 6.33363263547950343226467025512, 6.91987607244354900359049799156, 7.10424288765836229380569836759, 7.966748585378495339576607537798, 8.067812507896881194574442826522, 8.728804736653406036315929622812, 9.023210657321562920308200135946, 9.337997364178872995170188002763, 9.829701499215968728345216306463, 10.16759540438945152242720266006, 10.34607045405439905895378463380

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