Properties

Label 4-9747-1.1-c1e2-0-2
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 $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 10·7-s + 9-s + 4·13-s − 4·16-s − 2·19-s + 10·21-s − 25-s − 27-s − 12·31-s − 4·39-s − 2·43-s + 4·48-s + 61·49-s + 2·57-s − 2·61-s − 10·63-s + 16·67-s − 22·73-s + 75-s + 32·79-s + 81-s − 40·91-s + 12·93-s − 20·97-s − 4·103-s + 8·109-s + ⋯
L(s)  = 1  − 0.577·3-s − 3.77·7-s + 1/3·9-s + 1.10·13-s − 16-s − 0.458·19-s + 2.18·21-s − 1/5·25-s − 0.192·27-s − 2.15·31-s − 0.640·39-s − 0.304·43-s + 0.577·48-s + 61/7·49-s + 0.264·57-s − 0.256·61-s − 1.25·63-s + 1.95·67-s − 2.57·73-s + 0.115·75-s + 3.60·79-s + 1/9·81-s − 4.19·91-s + 1.24·93-s − 2.03·97-s − 0.394·103-s + 0.766·109-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: \(1\)
Selberg data: \((4,\ 9747,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
7$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
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$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
61$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
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

−11.00179000913096952209865162760, −10.84385583714429238920836096770, −9.914070570189995801171942379165, −9.769325574936890193507770039971, −9.028755790922139382896168795846, −8.836427381447004052087018681826, −7.47848778675296589128891353641, −6.85073886350288365363080648917, −6.41173993136661837405396202601, −6.12241751609850681792173625996, −5.34668146252658900228619106104, −3.81591641511052084615077615666, −3.67840796781373175574076394185, −2.61687998381550726785939862479, 0, 2.61687998381550726785939862479, 3.67840796781373175574076394185, 3.81591641511052084615077615666, 5.34668146252658900228619106104, 6.12241751609850681792173625996, 6.41173993136661837405396202601, 6.85073886350288365363080648917, 7.47848778675296589128891353641, 8.836427381447004052087018681826, 9.028755790922139382896168795846, 9.769325574936890193507770039971, 9.914070570189995801171942379165, 10.84385583714429238920836096770, 11.00179000913096952209865162760

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