Properties

Label 4-6e6-1.1-c1e2-0-17
Degree $4$
Conductor $46656$
Sign $-1$
Analytic cond. $2.97482$
Root an. cond. $1.31330$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·7-s − 2·13-s − 12·19-s − 25-s − 8·37-s − 49-s + 4·61-s + 6·67-s + 73-s + 6·79-s + 6·91-s − 5·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + 5·121-s + 127-s + 131-s + 36·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + ⋯
L(s)  = 1  − 1.13·7-s − 0.554·13-s − 2.75·19-s − 1/5·25-s − 1.31·37-s − 1/7·49-s + 0.512·61-s + 0.733·67-s + 0.117·73-s + 0.675·79-s + 0.628·91-s − 0.507·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + 5/11·121-s + 0.0887·127-s + 0.0873·131-s + 3.12·133-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 46656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46656 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(46656\)    =    \(2^{6} \cdot 3^{6}\)
Sign: $-1$
Analytic conductor: \(2.97482\)
Root analytic conductor: \(1.31330\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{46656} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 46656,\ (\ :1/2, 1/2),\ -1)\)

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)$
bad2 \( 1 \)
3 \( 1 \)
good5$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
7$C_2$ \( ( 1 - T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2^2$ \( 1 + 12 T + 67 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2^2$ \( 1 - 35 T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 8 T + 27 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
59$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 4 T - 45 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
71$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - T - 72 T^{2} - p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 6 T + 91 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 139 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 19 T + p T^{2} ) \)
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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

−15.0559880195, −14.5919626278, −14.2549266507, −13.5403697788, −13.1564486101, −12.8010146193, −12.3467441536, −12.0474341093, −11.2893292993, −10.7328948384, −10.4572390753, −9.87860021915, −9.45312304997, −8.86351546934, −8.40923040531, −7.91444477824, −7.08807760127, −6.62486579215, −6.33497410550, −5.59065884454, −4.88190504911, −4.15366942941, −3.61026247041, −2.69834567186, −1.96406416312, 0, 1.96406416312, 2.69834567186, 3.61026247041, 4.15366942941, 4.88190504911, 5.59065884454, 6.33497410550, 6.62486579215, 7.08807760127, 7.91444477824, 8.40923040531, 8.86351546934, 9.45312304997, 9.87860021915, 10.4572390753, 10.7328948384, 11.2893292993, 12.0474341093, 12.3467441536, 12.8010146193, 13.1564486101, 13.5403697788, 14.2549266507, 14.5919626278, 15.0559880195

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