Properties

Label 4-60e3-1.1-c1e2-0-29
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  + 2·2-s − 3-s + 2·4-s − 5-s − 2·6-s + 9-s − 2·10-s − 2·12-s + 15-s − 4·16-s + 2·18-s − 2·20-s + 25-s − 27-s + 2·30-s − 20·31-s − 8·32-s + 2·36-s − 45-s + 4·48-s − 10·49-s + 2·50-s + 20·53-s − 2·54-s + 2·60-s − 40·62-s − 8·64-s + ⋯
L(s)  = 1  + 1.41·2-s − 0.577·3-s + 4-s − 0.447·5-s − 0.816·6-s + 1/3·9-s − 0.632·10-s − 0.577·12-s + 0.258·15-s − 16-s + 0.471·18-s − 0.447·20-s + 1/5·25-s − 0.192·27-s + 0.365·30-s − 3.59·31-s − 1.41·32-s + 1/3·36-s − 0.149·45-s + 0.577·48-s − 1.42·49-s + 0.282·50-s + 2.74·53-s − 0.272·54-s + 0.258·60-s − 5.08·62-s − 64-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$C_2$ \( 1 - p T + p T^{2} \)
3$C_1$ \( 1 + T \)
5$C_1$ \( 1 + T \)
good7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$ \( ( 1 + p T^{2} )^{2} \)
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2^2$ \( 1 - 78 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
61$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
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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

−8.796347544631119103074783647720, −8.385333539337096037266712335397, −7.47453418205941725695143179171, −7.27017097594245566397215443793, −6.84957657199386693405152566680, −6.16619991592203718868276685129, −5.68018452289742730764555051785, −5.34999487270269556130717190838, −4.87764075487564301030729914879, −4.11875786665161050620939976019, −3.85363731820904269766144976313, −3.26464349646293471940721383930, −2.45982269552428865640106975011, −1.61170310790947852080742475943, 0, 1.61170310790947852080742475943, 2.45982269552428865640106975011, 3.26464349646293471940721383930, 3.85363731820904269766144976313, 4.11875786665161050620939976019, 4.87764075487564301030729914879, 5.34999487270269556130717190838, 5.68018452289742730764555051785, 6.16619991592203718868276685129, 6.84957657199386693405152566680, 7.27017097594245566397215443793, 7.47453418205941725695143179171, 8.385333539337096037266712335397, 8.796347544631119103074783647720

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