Properties

Label 4-60e3-1.1-c1e2-0-31
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-s + 3-s − 4-s + 5-s + 6-s − 3·8-s + 9-s + 10-s − 12-s + 15-s − 16-s + 18-s − 6·19-s − 20-s − 12·23-s − 3·24-s + 25-s + 27-s − 10·29-s + 30-s + 5·32-s − 36-s − 6·38-s − 3·40-s − 10·43-s + 45-s − 12·46-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.447·5-s + 0.408·6-s − 1.06·8-s + 1/3·9-s + 0.316·10-s − 0.288·12-s + 0.258·15-s − 1/4·16-s + 0.235·18-s − 1.37·19-s − 0.223·20-s − 2.50·23-s − 0.612·24-s + 1/5·25-s + 0.192·27-s − 1.85·29-s + 0.182·30-s + 0.883·32-s − 1/6·36-s − 0.973·38-s − 0.474·40-s − 1.52·43-s + 0.149·45-s − 1.76·46-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$C_2$ \( 1 - T + p T^{2} \)
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$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
19$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \)
23$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
29$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
37$C_2^2$ \( 1 + 24 T^{2} + p^{2} T^{4} \)
41$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
43$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \)
53$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
59$C_2^2$ \( 1 - 32 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
71$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
73$C_2$$\times$$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$C_2^2$ \( 1 - 94 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 74 T^{2} + p^{2} T^{4} \)
97$C_2$$\times$$C_2$ \( ( 1 + 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
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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

−8.841949077573873823775079160332, −8.310387580710565750282513871774, −8.008759299199905597127093432791, −7.45120708943997382225768151918, −6.67801778768693653434556586990, −6.35074992708740092614652664848, −5.66703120125243249571427201763, −5.49998553895872731415445036150, −4.64979802762485321002813298462, −4.04402494779224559503968925162, −3.89398238296327538196455237767, −3.09438497492793793598861161837, −2.27164495510320738928114371029, −1.76895553779176625944886599601, 0, 1.76895553779176625944886599601, 2.27164495510320738928114371029, 3.09438497492793793598861161837, 3.89398238296327538196455237767, 4.04402494779224559503968925162, 4.64979802762485321002813298462, 5.49998553895872731415445036150, 5.66703120125243249571427201763, 6.35074992708740092614652664848, 6.67801778768693653434556586990, 7.45120708943997382225768151918, 8.008759299199905597127093432791, 8.310387580710565750282513871774, 8.841949077573873823775079160332

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