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Properties

Label 4-11979-1.1-c1e2-0-3
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  + 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: \(0\)
Selberg data: \((4,\ 11979,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.963701228\)
\(L(\frac12)\) \(\approx\) \(2.963701228\)
\(L(\frac{3}{2})\) 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

−11.81631087306955963420628700213, −11.14136051051457299622132768480, −10.76517194615392112496312976161, −9.701226498296212058900989539920, −9.366222803316740054530803267641, −8.228854358433494441874279371750, −7.78789607043645765017922373874, −6.66503987304844816686807720890, −6.23870064789214498217427813253, −5.66407660024727617127120629681, −5.41175071356617719099292784896, −4.44574490854809981150981932393, −4.19085115174357929506261869806, −3.09000916592887094247460325417, −2.63044898935838963010520851422, 2.63044898935838963010520851422, 3.09000916592887094247460325417, 4.19085115174357929506261869806, 4.44574490854809981150981932393, 5.41175071356617719099292784896, 5.66407660024727617127120629681, 6.23870064789214498217427813253, 6.66503987304844816686807720890, 7.78789607043645765017922373874, 8.228854358433494441874279371750, 9.366222803316740054530803267641, 9.701226498296212058900989539920, 10.76517194615392112496312976161, 11.14136051051457299622132768480, 11.81631087306955963420628700213

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