Properties

Label 4-73008-1.1-c1e2-0-8
Degree $4$
Conductor $73008$
Sign $1$
Analytic cond. $4.65505$
Root an. cond. $1.46886$
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 + 4·7-s + 9-s + 2·13-s + 4·19-s + 4·21-s − 10·25-s + 27-s + 4·31-s + 4·37-s + 2·39-s − 8·43-s − 2·49-s + 4·57-s + 4·61-s + 4·63-s − 20·67-s + 28·73-s − 10·75-s + 16·79-s + 81-s + 8·91-s + 4·93-s − 20·97-s − 32·103-s + 28·109-s + 4·111-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.51·7-s + 1/3·9-s + 0.554·13-s + 0.917·19-s + 0.872·21-s − 2·25-s + 0.192·27-s + 0.718·31-s + 0.657·37-s + 0.320·39-s − 1.21·43-s − 2/7·49-s + 0.529·57-s + 0.512·61-s + 0.503·63-s − 2.44·67-s + 3.27·73-s − 1.15·75-s + 1.80·79-s + 1/9·81-s + 0.838·91-s + 0.414·93-s − 2.03·97-s − 3.15·103-s + 2.68·109-s + 0.379·111-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 73008 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73008 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(73008\)    =    \(2^{4} \cdot 3^{3} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(4.65505\)
Root analytic conductor: \(1.46886\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 73008,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.174249526\)
\(L(\frac12)\) \(\approx\) \(2.174249526\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 - T \)
13$C_1$ \( ( 1 - T )^{2} \)
good5$C_2$ \( ( 1 + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
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$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
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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

−9.798663464890123710839719924879, −9.313358814272746303026263150835, −8.735814680045936069504206621070, −8.113228044243862719694405295069, −7.86871640882441783507560416572, −7.63405094423031110476573907993, −6.65843891177652373785895186115, −6.28264530689997275960320812665, −5.34550374972406957171983136173, −5.11454570698520102977754794044, −4.26811107337312343751504072959, −3.80067854107620357152276821608, −2.97113699960307297281361463262, −2.03881758362108754018941589683, −1.35316666911883856927466195548, 1.35316666911883856927466195548, 2.03881758362108754018941589683, 2.97113699960307297281361463262, 3.80067854107620357152276821608, 4.26811107337312343751504072959, 5.11454570698520102977754794044, 5.34550374972406957171983136173, 6.28264530689997275960320812665, 6.65843891177652373785895186115, 7.63405094423031110476573907993, 7.86871640882441783507560416572, 8.113228044243862719694405295069, 8.735814680045936069504206621070, 9.313358814272746303026263150835, 9.798663464890123710839719924879

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