Properties

Label 4-172800-1.1-c1e2-0-16
Degree $4$
Conductor $172800$
Sign $1$
Analytic cond. $11.0178$
Root an. cond. $1.82189$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 8·7-s + 9-s + 4·13-s + 8·19-s − 8·21-s + 25-s − 27-s − 16·31-s + 4·37-s − 4·39-s + 8·43-s + 34·49-s − 8·57-s − 20·61-s + 8·63-s + 8·67-s + 4·73-s − 75-s − 16·79-s + 81-s + 32·91-s + 16·93-s + 4·97-s + 8·103-s − 20·109-s − 4·111-s + ⋯
L(s)  = 1  − 0.577·3-s + 3.02·7-s + 1/3·9-s + 1.10·13-s + 1.83·19-s − 1.74·21-s + 1/5·25-s − 0.192·27-s − 2.87·31-s + 0.657·37-s − 0.640·39-s + 1.21·43-s + 34/7·49-s − 1.05·57-s − 2.56·61-s + 1.00·63-s + 0.977·67-s + 0.468·73-s − 0.115·75-s − 1.80·79-s + 1/9·81-s + 3.35·91-s + 1.65·93-s + 0.406·97-s + 0.788·103-s − 1.91·109-s − 0.379·111-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: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 172800,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.241776282\)
\(L(\frac12)\) \(\approx\) \(2.241776282\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
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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

−9.206120678796429021467253404807, −8.713350404322888410914105544855, −7.965215938754459889106897012614, −7.88297907167905508023143292047, −7.38482725826507043429631860666, −6.96170559755127999142057961261, −5.85236328773314391596499497567, −5.76555679083117871916592673419, −4.97026786036012165020386374880, −4.95303891523898796353167575120, −4.08471431361169401081275954745, −3.62987230449253956202768748221, −2.50374962358163435732612030839, −1.44399343799308961081305068602, −1.36243712364701817507861978801, 1.36243712364701817507861978801, 1.44399343799308961081305068602, 2.50374962358163435732612030839, 3.62987230449253956202768748221, 4.08471431361169401081275954745, 4.95303891523898796353167575120, 4.97026786036012165020386374880, 5.76555679083117871916592673419, 5.85236328773314391596499497567, 6.96170559755127999142057961261, 7.38482725826507043429631860666, 7.88297907167905508023143292047, 7.965215938754459889106897012614, 8.713350404322888410914105544855, 9.206120678796429021467253404807

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