↑

Properties

Label 4-9747-1.1-c1e2-0-0
Degree $4$
Conductor $9747$
Sign $1$
Analytic cond. $0.621477$
Root an. cond. $0.887884$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 3·4-s + 9-s − 3·12-s + 12·13-s + 5·16-s − 2·19-s − 6·25-s + 27-s + 16·31-s − 3·36-s − 20·37-s + 12·39-s − 8·43-s + 5·48-s − 14·49-s − 36·52-s − 2·57-s − 4·61-s − 3·64-s − 8·67-s + 20·73-s − 6·75-s + 6·76-s + 81-s + 16·93-s + 20·97-s + ⋯
L(s)  = 1  + 0.577·3-s − 3/2·4-s + 1/3·9-s − 0.866·12-s + 3.32·13-s + 5/4·16-s − 0.458·19-s − 6/5·25-s + 0.192·27-s + 2.87·31-s − 1/2·36-s − 3.28·37-s + 1.92·39-s − 1.21·43-s + 0.721·48-s − 2·49-s − 4.99·52-s − 0.264·57-s − 0.512·61-s − 3/8·64-s − 0.977·67-s + 2.34·73-s − 0.692·75-s + 0.688·76-s + 1/9·81-s + 1.65·93-s + 2.03·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9747 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9747 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(9747\)    =    \(3^{3} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(0.621477\)
Root analytic conductor: \(0.887884\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 9747,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9424345359\)
\(L(\frac12)\) \(\approx\) \(0.9424345359\)
\(L(\frac{3}{2})\) 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)$
bad3$C_1$ \( 1 - T \)
19$C_1$ \( ( 1 + T )^{2} \)
good2$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
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$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.63017134596575611070760390300, −10.84155613367776600825623791063, −10.33061644135141849320149848866, −9.849882478281281439722427012273, −9.032116632714081125535639739797, −8.700339636508216566718463872719, −8.263513432862467471567669978548, −7.962026255736907116683171399764, −6.46787589567348650226556085189, −6.37820814688737049661034336254, −5.30285359031178015236684552035, −4.56991321080597356703354311172, −3.69904169102384607719527879903, −3.44365518362789223021999993571, −1.48488050469889056862198933364, 1.48488050469889056862198933364, 3.44365518362789223021999993571, 3.69904169102384607719527879903, 4.56991321080597356703354311172, 5.30285359031178015236684552035, 6.37820814688737049661034336254, 6.46787589567348650226556085189, 7.962026255736907116683171399764, 8.263513432862467471567669978548, 8.700339636508216566718463872719, 9.032116632714081125535639739797, 9.849882478281281439722427012273, 10.33061644135141849320149848866, 10.84155613367776600825623791063, 11.63017134596575611070760390300

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