Properties

Label 4-9747-1.1-c1e2-0-1
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 6·7-s + 9-s − 12·13-s − 4·16-s − 2·19-s + 6·21-s − 9·25-s + 27-s + 4·31-s + 16·37-s − 12·39-s − 2·43-s − 4·48-s + 13·49-s − 2·57-s + 14·61-s + 6·63-s + 16·67-s − 22·73-s − 9·75-s + 81-s − 72·91-s + 4·93-s − 4·97-s + 28·103-s + 40·109-s + ⋯
L(s)  = 1  + 0.577·3-s + 2.26·7-s + 1/3·9-s − 3.32·13-s − 16-s − 0.458·19-s + 1.30·21-s − 9/5·25-s + 0.192·27-s + 0.718·31-s + 2.63·37-s − 1.92·39-s − 0.304·43-s − 0.577·48-s + 13/7·49-s − 0.264·57-s + 1.79·61-s + 0.755·63-s + 1.95·67-s − 2.57·73-s − 1.03·75-s + 1/9·81-s − 7.54·91-s + 0.414·93-s − 0.406·97-s + 2.75·103-s + 3.83·109-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

\(L(1)\) \(\approx\) \(1.255232601\)
\(L(\frac12)\) \(\approx\) \(1.255232601\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \)
5$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
7$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
13$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
43$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
73$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
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

−11.43002930912671853728259843597, −11.42285722847365155593370717983, −10.20230313909289126093125054009, −9.886397115513209246726221997818, −9.400588261603299107064903686230, −8.513296718878561122779255009178, −8.062798749262723709584757589645, −7.42319981131404835915630105083, −7.33179869161333338136909211767, −6.12831923497636436478284774657, −5.02993077790447449830764321556, −4.75227634922127063518685270995, −4.20883425493377341015603476651, −2.44031011733442394942262196334, −2.13307257724406321246824925475, 2.13307257724406321246824925475, 2.44031011733442394942262196334, 4.20883425493377341015603476651, 4.75227634922127063518685270995, 5.02993077790447449830764321556, 6.12831923497636436478284774657, 7.33179869161333338136909211767, 7.42319981131404835915630105083, 8.062798749262723709584757589645, 8.513296718878561122779255009178, 9.400588261603299107064903686230, 9.886397115513209246726221997818, 10.20230313909289126093125054009, 11.42285722847365155593370717983, 11.43002930912671853728259843597

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