Properties

Label 4-3267-1.1-c1e2-0-1
Degree $4$
Conductor $3267$
Sign $1$
Analytic cond. $0.208306$
Root an. cond. $0.675578$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 4-s + 9-s − 12-s − 3·16-s − 6·25-s − 27-s + 36-s − 4·37-s + 3·48-s − 6·49-s − 7·64-s + 24·67-s + 6·75-s + 81-s + 4·97-s − 6·100-s + 32·103-s − 108-s + 4·111-s − 11·121-s + 127-s + 131-s + 137-s + 139-s − 3·144-s + 6·147-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/2·4-s + 1/3·9-s − 0.288·12-s − 3/4·16-s − 6/5·25-s − 0.192·27-s + 1/6·36-s − 0.657·37-s + 0.433·48-s − 6/7·49-s − 7/8·64-s + 2.93·67-s + 0.692·75-s + 1/9·81-s + 0.406·97-s − 3/5·100-s + 3.15·103-s − 0.0962·108-s + 0.379·111-s − 121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1/4·144-s + 0.494·147-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3267\)    =    \(3^{3} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(0.208306\)
Root analytic conductor: \(0.675578\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 3267,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6903507470\)
\(L(\frac12)\) \(\approx\) \(0.6903507470\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 + T \)
11$C_2$ \( 1 + p T^{2} \)
good2$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \)
5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - p T^{2} )^{2} \)
17$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
29$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 66 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2^2$ \( 1 - 42 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
73$C_2^2$ \( 1 - 66 T^{2} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
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

−12.73421432487367820355377174588, −11.96687799800014069393294917976, −11.52120491201296223033915757551, −11.10136888088688663378683684506, −10.38270282240739998208866618081, −9.796638145426984715214907591659, −9.140706646002055672001290032296, −8.303707341281292782502033417558, −7.58519140543418276929455330472, −6.85368494998698565477316876144, −6.28555113180586181855575383934, −5.46521387846247693492728763599, −4.62045561058825591249668103433, −3.57633358779302502321554715678, −2.12508272083046916933008974996, 2.12508272083046916933008974996, 3.57633358779302502321554715678, 4.62045561058825591249668103433, 5.46521387846247693492728763599, 6.28555113180586181855575383934, 6.85368494998698565477316876144, 7.58519140543418276929455330472, 8.303707341281292782502033417558, 9.140706646002055672001290032296, 9.796638145426984715214907591659, 10.38270282240739998208866618081, 11.10136888088688663378683684506, 11.52120491201296223033915757551, 11.96687799800014069393294917976, 12.73421432487367820355377174588

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