Properties

Label 4-172800-1.1-c1e2-0-25
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 − 8·13-s − 8·19-s − 25-s + 27-s + 8·31-s − 8·39-s − 8·43-s − 10·49-s − 8·57-s + 4·61-s + 8·67-s − 12·73-s − 75-s − 8·79-s + 81-s + 8·93-s − 20·97-s − 20·109-s − 8·117-s + 14·121-s + 127-s − 8·129-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/3·9-s − 2.21·13-s − 1.83·19-s − 1/5·25-s + 0.192·27-s + 1.43·31-s − 1.28·39-s − 1.21·43-s − 1.42·49-s − 1.05·57-s + 0.512·61-s + 0.977·67-s − 1.40·73-s − 0.115·75-s − 0.900·79-s + 1/9·81-s + 0.829·93-s − 2.03·97-s − 1.91·109-s − 0.739·117-s + 1.27·121-s + 0.0887·127-s − 0.704·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-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: $\chi_{172800} (1, \cdot )$
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$C_2$ \( 1 + T^{2} \)
good7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
13$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
61$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
67$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \)
71$C_2^2$ \( 1 - 66 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
79$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
83$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 66 T^{2} + p^{2} T^{4} \)
97$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
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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

−8.769573580069282186224148517204, −8.508141342699978244744630403662, −8.064795128067172028240885264787, −7.55052188118271805501039639427, −7.00207587787286223947283816459, −6.64034091715734973888481930790, −6.10258840514642787911062086441, −5.30895349319761409032047971051, −4.73824861562740587132212379808, −4.44219753532738549758621852603, −3.72375187627087732597125075546, −2.81408063292675100131590930973, −2.46498828777568616718336781231, −1.67664083378605742283005717605, 0, 1.67664083378605742283005717605, 2.46498828777568616718336781231, 2.81408063292675100131590930973, 3.72375187627087732597125075546, 4.44219753532738549758621852603, 4.73824861562740587132212379808, 5.30895349319761409032047971051, 6.10258840514642787911062086441, 6.64034091715734973888481930790, 7.00207587787286223947283816459, 7.55052188118271805501039639427, 8.064795128067172028240885264787, 8.508141342699978244744630403662, 8.769573580069282186224148517204

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