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Properties

Label 4-11979-1.1-c1e2-0-0
Degree $4$
Conductor $11979$
Sign $1$
Analytic cond. $0.763791$
Root an. cond. $0.934853$
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  − 2·3-s − 3·4-s − 4·5-s + 3·9-s + 11-s + 6·12-s + 8·15-s + 5·16-s + 12·20-s + 16·23-s + 2·25-s − 4·27-s − 16·31-s − 2·33-s − 9·36-s + 12·37-s − 3·44-s − 12·45-s + 16·47-s − 10·48-s + 2·49-s + 12·53-s − 4·55-s − 8·59-s − 24·60-s − 3·64-s − 8·67-s + ⋯
L(s)  = 1  − 1.15·3-s − 3/2·4-s − 1.78·5-s + 9-s + 0.301·11-s + 1.73·12-s + 2.06·15-s + 5/4·16-s + 2.68·20-s + 3.33·23-s + 2/5·25-s − 0.769·27-s − 2.87·31-s − 0.348·33-s − 3/2·36-s + 1.97·37-s − 0.452·44-s − 1.78·45-s + 2.33·47-s − 1.44·48-s + 2/7·49-s + 1.64·53-s − 0.539·55-s − 1.04·59-s − 3.09·60-s − 3/8·64-s − 0.977·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 11979 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11979 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(11979\)    =    \(3^{2} \cdot 11^{3}\)
Sign: $1$
Analytic conductor: \(0.763791\)
Root analytic conductor: \(0.934853\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 11979,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3092258069\)
\(L(\frac12)\) \(\approx\) \(0.3092258069\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
11$C_1$ \( 1 - T \)
good2$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
5$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
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

−11.21317067856663011790716323208, −11.06987917382067337680927181024, −10.45490550189802125561108883297, −9.528487026476181059669700861222, −8.975261543279013968212899152873, −8.829662012954888204160857557729, −7.67376126887882242787787369865, −7.46995816184461358823952634116, −6.85553577597705614902881023761, −5.66464594246847012421851082372, −5.31622482563829752699303340190, −4.33065348032451946577135661956, −4.20407903283329927801590362766, −3.31119773648628104035349562784, −0.71891279628917143719551815370, 0.71891279628917143719551815370, 3.31119773648628104035349562784, 4.20407903283329927801590362766, 4.33065348032451946577135661956, 5.31622482563829752699303340190, 5.66464594246847012421851082372, 6.85553577597705614902881023761, 7.46995816184461358823952634116, 7.67376126887882242787787369865, 8.829662012954888204160857557729, 8.975261543279013968212899152873, 9.528487026476181059669700861222, 10.45490550189802125561108883297, 11.06987917382067337680927181024, 11.21317067856663011790716323208

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