Properties

Label 4-11979-1.1-c1e2-0-1
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 $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·2-s − 3-s + 8·4-s + 4·6-s − 8·8-s − 2·9-s + 11-s − 8·12-s − 4·16-s − 4·17-s + 8·18-s − 4·22-s + 8·24-s − 9·25-s + 5·27-s + 14·31-s + 32·32-s − 33-s + 16·34-s − 16·36-s + 6·37-s − 16·41-s + 8·44-s + 4·48-s − 10·49-s + 36·50-s + 4·51-s + ⋯
L(s)  = 1  − 2.82·2-s − 0.577·3-s + 4·4-s + 1.63·6-s − 2.82·8-s − 2/3·9-s + 0.301·11-s − 2.30·12-s − 16-s − 0.970·17-s + 1.88·18-s − 0.852·22-s + 1.63·24-s − 9/5·25-s + 0.962·27-s + 2.51·31-s + 5.65·32-s − 0.174·33-s + 2.74·34-s − 8/3·36-s + 0.986·37-s − 2.49·41-s + 1.20·44-s + 0.577·48-s − 1.42·49-s + 5.09·50-s + 0.560·51-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: \(1\)
Selberg data: \((4,\ 11979,\ (\ :1/2, 1/2),\ -1)\)

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$C_2$ \( 1 + T + p T^{2} \)
11$C_1$ \( 1 - T \)
good2$C_2$ \( ( 1 + p T + p T^{2} )^{2} \)
5$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
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$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
83$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \)
97$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
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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

−10.90721695501259097313070074829, −10.08710379123756686578115682884, −10.03550909718107888433464868208, −9.461316816361475492652505870168, −8.603539619290756001226038948684, −8.571719581705043558005338128568, −7.927183238362238013907773101180, −7.36742097706937993051027630389, −6.42924768192751341652305216268, −6.36261389471308870138602900888, −5.01784321266906662288549345024, −4.27593882764012714082242634416, −2.65868050490226641918200240944, −1.51509492168516879981983515735, 0, 1.51509492168516879981983515735, 2.65868050490226641918200240944, 4.27593882764012714082242634416, 5.01784321266906662288549345024, 6.36261389471308870138602900888, 6.42924768192751341652305216268, 7.36742097706937993051027630389, 7.927183238362238013907773101180, 8.571719581705043558005338128568, 8.603539619290756001226038948684, 9.461316816361475492652505870168, 10.03550909718107888433464868208, 10.08710379123756686578115682884, 10.90721695501259097313070074829

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