Properties

Label 4-172800-1.1-c1e2-0-19
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 9-s − 4·13-s − 8·23-s + 25-s − 27-s + 12·37-s + 4·39-s + 8·47-s + 2·49-s − 16·59-s − 4·61-s + 8·69-s − 16·71-s − 12·73-s − 75-s + 81-s − 8·83-s + 4·97-s + 8·107-s − 4·109-s − 12·111-s − 4·117-s − 6·121-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s − 1.10·13-s − 1.66·23-s + 1/5·25-s − 0.192·27-s + 1.97·37-s + 0.640·39-s + 1.16·47-s + 2/7·49-s − 2.08·59-s − 0.512·61-s + 0.963·69-s − 1.89·71-s − 1.40·73-s − 0.115·75-s + 1/9·81-s − 0.878·83-s + 0.406·97-s + 0.773·107-s − 0.383·109-s − 1.13·111-s − 0.369·117-s − 0.545·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-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: \(1\)
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 \( 1 \)
3$C_1$ \( 1 + T \)
5$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
29$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
31$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
37$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
47$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
59$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
61$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
73$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
83$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
97$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
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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

−9.078708301627855748942412226804, −8.461752277072843178252316260334, −7.79769161086611924559008207932, −7.53499221326312945395245645011, −7.13396987341472701822039150570, −6.28338669362981148011652990396, −6.04626156099497938177832826575, −5.57240268932539778004626159915, −4.77595227894881490913735587224, −4.43339993535207500187906625729, −3.89220098338467170975469010739, −2.91529246872248394447604670713, −2.36500969691943004667489432794, −1.38799300699998101566047980898, 0, 1.38799300699998101566047980898, 2.36500969691943004667489432794, 2.91529246872248394447604670713, 3.89220098338467170975469010739, 4.43339993535207500187906625729, 4.77595227894881490913735587224, 5.57240268932539778004626159915, 6.04626156099497938177832826575, 6.28338669362981148011652990396, 7.13396987341472701822039150570, 7.53499221326312945395245645011, 7.79769161086611924559008207932, 8.461752277072843178252316260334, 9.078708301627855748942412226804

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