Properties

Degree $4$
Conductor $726$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 3·5-s − 4·7-s + 8-s − 2·9-s + 3·10-s + 2·11-s + 8·13-s + 4·14-s − 3·16-s − 4·17-s + 2·18-s − 3·20-s − 2·22-s − 7·23-s + 3·25-s − 8·26-s + 3·27-s − 4·28-s + 10·29-s − 31-s + 5·32-s + 4·34-s + 12·35-s − 2·36-s + 37-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 1.34·5-s − 1.51·7-s + 0.353·8-s − 2/3·9-s + 0.948·10-s + 0.603·11-s + 2.21·13-s + 1.06·14-s − 3/4·16-s − 0.970·17-s + 0.471·18-s − 0.670·20-s − 0.426·22-s − 1.45·23-s + 3/5·25-s − 1.56·26-s + 0.577·27-s − 0.755·28-s + 1.85·29-s − 0.179·31-s + 0.883·32-s + 0.685·34-s + 2.02·35-s − 1/3·36-s + 0.164·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 726 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 726 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(726\)    =    \(2 \cdot 3 \cdot 11^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{726} (1, \cdot )$
Sato-Tate group: $G_{3,3}$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 726,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3024817222\)
\(L(\frac12)\) \(\approx\) \(0.3024817222\)
\(L(\frac{3}{2})\) 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_1$$\times$$C_2$ \( ( 1 - T )( 1 + p T + p T^{2} ) \)
3$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + T + p T^{2} ) \)
11$C_1$ \( ( 1 - T )^{2} \)
good5$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
7$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + p T^{2} ) \)
31$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
43$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
47$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
53$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + p T^{2} ) \)
61$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
67$C_2$$\times$$C_2$ \( ( 1 + 7 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
71$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
73$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
83$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
89$C_2$$\times$$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 - 10 T + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
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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

−19.7801422519, −19.4348259844, −19.1857249719, −18.2200985458, −17.9414335735, −17.0336103204, −16.1831467776, −15.9261213942, −15.9140726033, −14.9306369221, −13.7992717180, −13.5686390571, −12.6398118567, −11.7777223349, −11.4512586103, −10.6319686571, −10.0355090972, −8.82266477585, −8.60353961929, −7.72339303053, −6.46917181705, −6.36261389471, −4.25842178548, −3.34833436263, 3.34833436263, 4.25842178548, 6.36261389471, 6.46917181705, 7.72339303053, 8.60353961929, 8.82266477585, 10.0355090972, 10.6319686571, 11.4512586103, 11.7777223349, 12.6398118567, 13.5686390571, 13.7992717180, 14.9306369221, 15.9140726033, 15.9261213942, 16.1831467776, 17.0336103204, 17.9414335735, 18.2200985458, 19.1857249719, 19.4348259844, 19.7801422519

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