Properties

Label 4-172800-1.1-c1e2-0-8
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 $0$

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: \(0\)
Selberg data: \((4,\ 172800,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.000768501\)
\(L(\frac12)\) \(\approx\) \(2.000768501\)
\(L(\frac{3}{2})\) 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$ \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
59$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
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$ \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
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

−9.291298574402970679863122371973, −8.621666342891523369321526585402, −8.360951480912939394668117342589, −7.59137484122417926942388693646, −7.46466097862180301658442852635, −6.74701028543655247264600767887, −6.46317561196916208708331018309, −5.60216332901476637074044973637, −5.10776393049599874582799242162, −4.61517774336118835532603327204, −4.03845309884361087663949621816, −3.23381133409479097641702786283, −2.72724935346905167341287416802, −2.08564612966020829296841400230, −0.932124893814943855253891093379, 0.932124893814943855253891093379, 2.08564612966020829296841400230, 2.72724935346905167341287416802, 3.23381133409479097641702786283, 4.03845309884361087663949621816, 4.61517774336118835532603327204, 5.10776393049599874582799242162, 5.60216332901476637074044973637, 6.46317561196916208708331018309, 6.74701028543655247264600767887, 7.46466097862180301658442852635, 7.59137484122417926942388693646, 8.360951480912939394668117342589, 8.621666342891523369321526585402, 9.291298574402970679863122371973

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