Properties

Label 4-172800-1.1-c1e2-0-22
Degree $4$
Conductor $172800$
Sign $1$
Analytic cond. $11.0178$
Root an. cond. $1.82189$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3·3-s + 4-s − 3·5-s + 3·6-s − 7-s − 8-s + 6·9-s + 3·10-s − 3·12-s − 2·13-s + 14-s + 9·15-s + 16-s − 5·17-s − 6·18-s − 5·19-s − 3·20-s + 3·21-s − 8·23-s + 3·24-s + 4·25-s + 2·26-s − 9·27-s − 28-s + 4·29-s − 9·30-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.73·3-s + 1/2·4-s − 1.34·5-s + 1.22·6-s − 0.377·7-s − 0.353·8-s + 2·9-s + 0.948·10-s − 0.866·12-s − 0.554·13-s + 0.267·14-s + 2.32·15-s + 1/4·16-s − 1.21·17-s − 1.41·18-s − 1.14·19-s − 0.670·20-s + 0.654·21-s − 1.66·23-s + 0.612·24-s + 4/5·25-s + 0.392·26-s − 1.73·27-s − 0.188·28-s + 0.742·29-s − 1.64·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 172800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 172800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(172800\)    =    \(2^{8} \cdot 3^{3} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(11.0178\)
Root analytic conductor: \(1.82189\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 172800,\ (\ :1/2, 1/2),\ 1)\)

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)$
bad2$C_1$ \( 1 + T \)
3$C_2$ \( 1 + p T + p T^{2} \)
5$C_2$ \( 1 + 3 T + p T^{2} \)
good7$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
11$C_2^2$ \( 1 - 7 T^{2} + p^{2} T^{4} \)
13$C_4$ \( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
17$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
19$D_{4}$ \( 1 + 5 T + 18 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
29$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
31$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$D_{4}$ \( 1 + 10 T + 94 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - T + 6 T^{2} - p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 6 T + 2 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 5 T - 2 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 2 T + 66 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 7 T + 98 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 6 T - 22 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 14 T + 123 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 6 T + 94 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 9 T + 86 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - T - 84 T^{2} - p T^{3} + p^{2} T^{4} \)
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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

−14.0088665027, −13.3628348839, −12.7737890250, −12.4539273437, −12.2393640173, −11.7207970787, −11.4012893253, −11.0323451915, −10.6475347726, −10.1903528358, −9.82917301153, −9.27478100975, −8.60353727838, −8.06242082456, −7.93127524408, −6.98731957165, −6.80451348156, −6.43467492518, −5.90337122031, −5.07814562508, −4.74933509868, −4.03882601977, −3.65962170062, −2.50611097215, −1.60110997374, 0, 0, 1.60110997374, 2.50611097215, 3.65962170062, 4.03882601977, 4.74933509868, 5.07814562508, 5.90337122031, 6.43467492518, 6.80451348156, 6.98731957165, 7.93127524408, 8.06242082456, 8.60353727838, 9.27478100975, 9.82917301153, 10.1903528358, 10.6475347726, 11.0323451915, 11.4012893253, 11.7207970787, 12.2393640173, 12.4539273437, 12.7737890250, 13.3628348839, 14.0088665027

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