Properties

Label 4-73008-1.1-c1e2-0-2
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  + 2-s − 3-s − 4-s − 6-s − 3·8-s + 9-s + 8·11-s + 12-s + 2·13-s − 16-s + 18-s + 8·22-s + 3·24-s − 6·25-s + 2·26-s − 27-s + 5·32-s − 8·33-s − 36-s − 4·37-s − 2·39-s − 8·44-s + 48-s + 2·49-s − 6·50-s − 2·52-s − 54-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.408·6-s − 1.06·8-s + 1/3·9-s + 2.41·11-s + 0.288·12-s + 0.554·13-s − 1/4·16-s + 0.235·18-s + 1.70·22-s + 0.612·24-s − 6/5·25-s + 0.392·26-s − 0.192·27-s + 0.883·32-s − 1.39·33-s − 1/6·36-s − 0.657·37-s − 0.320·39-s − 1.20·44-s + 0.144·48-s + 2/7·49-s − 0.848·50-s − 0.277·52-s − 0.136·54-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

\(L(1)\) \(\approx\) \(1.578274261\)
\(L(\frac12)\) \(\approx\) \(1.578274261\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + p T^{2} \)
3$C_1$ \( 1 + T \)
13$C_1$ \( ( 1 - T )^{2} \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
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$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
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

−9.952802346640542280811898231944, −9.192740554674130435254875131876, −8.921039436304358900231528874001, −8.484648143275568894036347393857, −7.73604447848719394517643815023, −6.99405970659997290412829823598, −6.54556545042783689480045128663, −6.08826329831047007406167672262, −5.65486780279705070512904066158, −4.94568196736683705255358366288, −4.31391709618999720844014512009, −3.71205447024015551816570926916, −3.55158930907206838697007657967, −2.10208151844778724001409156325, −1.00171028893857761930540835980, 1.00171028893857761930540835980, 2.10208151844778724001409156325, 3.55158930907206838697007657967, 3.71205447024015551816570926916, 4.31391709618999720844014512009, 4.94568196736683705255358366288, 5.65486780279705070512904066158, 6.08826329831047007406167672262, 6.54556545042783689480045128663, 6.99405970659997290412829823598, 7.73604447848719394517643815023, 8.484648143275568894036347393857, 8.921039436304358900231528874001, 9.192740554674130435254875131876, 9.952802346640542280811898231944

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