Properties

Label 4-6e8-1.1-c1e2-0-14
Degree $4$
Conductor $1679616$
Sign $1$
Analytic cond. $107.093$
Root an. cond. $3.21692$
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  − 7-s − 5·13-s + 14·19-s + 5·25-s − 4·31-s + 22·37-s + 8·43-s + 7·49-s + 61-s + 5·67-s − 14·73-s + 17·79-s + 5·91-s + 19·97-s − 13·103-s + 4·109-s + 11·121-s + 127-s + 131-s − 14·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  − 0.377·7-s − 1.38·13-s + 3.21·19-s + 25-s − 0.718·31-s + 3.61·37-s + 1.21·43-s + 49-s + 0.128·61-s + 0.610·67-s − 1.63·73-s + 1.91·79-s + 0.524·91-s + 1.92·97-s − 1.28·103-s + 0.383·109-s + 121-s + 0.0887·127-s + 0.0873·131-s − 1.21·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1679616 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1679616 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1679616\)    =    \(2^{8} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(107.093\)
Root analytic conductor: \(3.21692\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1679616,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.340759355\)
\(L(\frac12)\) \(\approx\) \(2.340759355\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3 \( 1 \)
good5$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
11$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
17$C_2$ \( ( 1 + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
47$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
83$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 5 T + p T^{2} ) \)
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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

−9.637954757352023195353773225988, −9.436609579693224932524319866018, −9.323182531683686274546260719397, −8.812043376342046742750524825566, −7.990532556592202806296267877566, −7.79759446672018945723047628869, −7.32360233866093777830243244572, −7.26743801739296391238585073693, −6.59953042449922657315845705552, −6.08930148838864866561545969630, −5.48384101687474692311236093692, −5.44747665364545683146625030746, −4.65174669802628260665949362265, −4.49588486129804469509521990774, −3.65244167524193960820034231533, −3.24947773180374233624660994484, −2.57839106648213412992233862389, −2.45071826489332977874040543854, −1.19138592078966392478662435604, −0.74946665045970302009324785232, 0.74946665045970302009324785232, 1.19138592078966392478662435604, 2.45071826489332977874040543854, 2.57839106648213412992233862389, 3.24947773180374233624660994484, 3.65244167524193960820034231533, 4.49588486129804469509521990774, 4.65174669802628260665949362265, 5.44747665364545683146625030746, 5.48384101687474692311236093692, 6.08930148838864866561545969630, 6.59953042449922657315845705552, 7.26743801739296391238585073693, 7.32360233866093777830243244572, 7.79759446672018945723047628869, 7.990532556592202806296267877566, 8.812043376342046742750524825566, 9.323182531683686274546260719397, 9.436609579693224932524319866018, 9.637954757352023195353773225988

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