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Properties

Label 4-3267-1.1-c1e2-0-0
Degree $4$
Conductor $3267$
Sign $1$
Analytic cond. $0.208306$
Root an. cond. $0.675578$
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 − 3·4-s + 8·7-s + 9-s + 3·12-s − 4·13-s + 5·16-s − 8·21-s − 6·25-s − 27-s − 24·28-s − 16·31-s − 3·36-s + 12·37-s + 4·39-s − 5·48-s + 34·49-s + 12·52-s + 12·61-s + 8·63-s − 3·64-s − 8·67-s − 28·73-s + 6·75-s − 8·79-s + 81-s + 24·84-s + ⋯
L(s)  = 1  − 0.577·3-s − 3/2·4-s + 3.02·7-s + 1/3·9-s + 0.866·12-s − 1.10·13-s + 5/4·16-s − 1.74·21-s − 6/5·25-s − 0.192·27-s − 4.53·28-s − 2.87·31-s − 1/2·36-s + 1.97·37-s + 0.640·39-s − 0.721·48-s + 34/7·49-s + 1.66·52-s + 1.53·61-s + 1.00·63-s − 3/8·64-s − 0.977·67-s − 3.27·73-s + 0.692·75-s − 0.900·79-s + 1/9·81-s + 2.61·84-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: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 3267,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5921223399\)
\(L(\frac12)\) \(\approx\) \(0.5921223399\)
\(L(\frac{3}{2})\) 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_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
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$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
43$C_2$ \( ( 1 + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
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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

−12.97885466615636407297758144806, −11.86897475867739185827101909293, −11.61114553747330884538582571440, −11.06987917382067337680927181024, −10.39134307397473772625111872930, −9.637990407765945223554197990480, −8.975261543279013968212899152873, −8.389268452519734683124359977272, −7.60731726518247812897599595849, −7.46995816184461358823952634116, −5.64913594526162848677189952595, −5.31622482563829752699303340190, −4.52395871965258729630923756023, −4.20407903283329927801590362766, −1.82001999803537272093437549398, 1.82001999803537272093437549398, 4.20407903283329927801590362766, 4.52395871965258729630923756023, 5.31622482563829752699303340190, 5.64913594526162848677189952595, 7.46995816184461358823952634116, 7.60731726518247812897599595849, 8.389268452519734683124359977272, 8.975261543279013968212899152873, 9.637990407765945223554197990480, 10.39134307397473772625111872930, 11.06987917382067337680927181024, 11.61114553747330884538582571440, 11.86897475867739185827101909293, 12.97885466615636407297758144806

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