Properties

Label 4-60e3-1.1-c1e2-0-21
Degree $4$
Conductor $216000$
Sign $-1$
Analytic cond. $13.7723$
Root an. cond. $1.92642$
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-s − 2·4-s − 5-s + 9-s − 2·12-s − 2·13-s − 15-s + 4·16-s + 2·20-s + 25-s + 27-s − 8·31-s − 2·36-s + 10·37-s − 2·39-s − 18·41-s + 4·43-s − 45-s + 4·48-s + 2·49-s + 4·52-s − 12·53-s + 2·60-s − 8·64-s + 2·65-s − 8·67-s + 75-s + ⋯
L(s)  = 1  + 0.577·3-s − 4-s − 0.447·5-s + 1/3·9-s − 0.577·12-s − 0.554·13-s − 0.258·15-s + 16-s + 0.447·20-s + 1/5·25-s + 0.192·27-s − 1.43·31-s − 1/3·36-s + 1.64·37-s − 0.320·39-s − 2.81·41-s + 0.609·43-s − 0.149·45-s + 0.577·48-s + 2/7·49-s + 0.554·52-s − 1.64·53-s + 0.258·60-s − 64-s + 0.248·65-s − 0.977·67-s + 0.115·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(216000\)    =    \(2^{6} \cdot 3^{3} \cdot 5^{3}\)
Sign: $-1$
Analytic conductor: \(13.7723\)
Root analytic conductor: \(1.92642\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 216000,\ (\ :1/2, 1/2),\ -1)\)

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$C_2$ \( 1 + p T^{2} \)
3$C_1$ \( 1 - T \)
5$C_1$ \( 1 + T \)
good7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
13$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
37$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
41$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
43$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 98 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \)
79$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
89$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \)
97$C_2^2$ \( 1 - 170 T^{2} + p^{2} T^{4} \)
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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

−8.639234930650642765620751129680, −8.523775257728889674307296197910, −7.895646387548021642457581481954, −7.43218262021209478220184602720, −7.17125924843781199120084644241, −6.32168167241113410404567831150, −5.88169747075832064862554687099, −5.10907947613849939441179482998, −4.77290465581652273268316475331, −4.24331545329566705598520649279, −3.54614090615107725449531004688, −3.21991610284210338990303496611, −2.30332128407427443161963864401, −1.36328580534491248673091037460, 0, 1.36328580534491248673091037460, 2.30332128407427443161963864401, 3.21991610284210338990303496611, 3.54614090615107725449531004688, 4.24331545329566705598520649279, 4.77290465581652273268316475331, 5.10907947613849939441179482998, 5.88169747075832064862554687099, 6.32168167241113410404567831150, 7.17125924843781199120084644241, 7.43218262021209478220184602720, 7.895646387548021642457581481954, 8.523775257728889674307296197910, 8.639234930650642765620751129680

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