Properties

Label 4-108e2-1.1-c1e2-0-1
Degree $4$
Conductor $11664$
Sign $1$
Analytic cond. $0.743706$
Root an. cond. $0.928646$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s + 10·13-s + 4·16-s − 10·25-s + 22·37-s − 13·49-s − 20·52-s − 2·61-s − 8·64-s − 14·73-s − 38·97-s + 20·100-s + 4·109-s − 22·121-s + 127-s + 131-s + 137-s + 139-s − 44·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 49·169-s + 173-s + 179-s + ⋯
L(s)  = 1  − 4-s + 2.77·13-s + 16-s − 2·25-s + 3.61·37-s − 1.85·49-s − 2.77·52-s − 0.256·61-s − 64-s − 1.63·73-s − 3.85·97-s + 2·100-s + 0.383·109-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 3.61·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 3.76·169-s + 0.0760·173-s + 0.0747·179-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 11664 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11664 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(11664\)    =    \(2^{4} \cdot 3^{6}\)
Sign: $1$
Analytic conductor: \(0.743706\)
Root analytic conductor: \(0.928646\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 11664,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9009586960\)
\(L(\frac12)\) \(\approx\) \(0.9009586960\)
\(L(\frac{3}{2})\) 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)$
bad2$C_2$ \( 1 + p T^{2} \)
3 \( 1 \)
good5$C_2$ \( ( 1 + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 17 T + p T^{2} ) \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 19 T + p T^{2} )^{2} \)
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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

−11.38842166918163472638466197964, −10.90872829298908742564232271224, −10.22208553622788501090350672882, −9.429199208210393651255146412199, −9.387437094399487619618753359356, −8.341275212548067016845372469550, −8.217650367462526737991465554229, −7.60583694658959035837478078570, −6.37318226789872439689553131647, −6.04893540000987436402649531985, −5.48198608682619476325367860976, −4.24977516365535388494352784618, −4.04304401379743272242521882180, −3.07984083787626508123182523604, −1.36018793518762280095521620239, 1.36018793518762280095521620239, 3.07984083787626508123182523604, 4.04304401379743272242521882180, 4.24977516365535388494352784618, 5.48198608682619476325367860976, 6.04893540000987436402649531985, 6.37318226789872439689553131647, 7.60583694658959035837478078570, 8.217650367462526737991465554229, 8.341275212548067016845372469550, 9.387437094399487619618753359356, 9.429199208210393651255146412199, 10.22208553622788501090350672882, 10.90872829298908742564232271224, 11.38842166918163472638466197964

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