Properties

Degree $4$
Conductor $1416$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 2·3-s + 4-s − 5-s − 2·6-s − 3·7-s + 8-s + 4·9-s − 10-s − 2·12-s − 6·13-s − 3·14-s + 2·15-s + 16-s + 4·17-s + 4·18-s − 3·19-s − 20-s + 6·21-s + 2·23-s − 2·24-s + 5·25-s − 6·26-s − 5·27-s − 3·28-s + 29-s + 2·30-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.447·5-s − 0.816·6-s − 1.13·7-s + 0.353·8-s + 4/3·9-s − 0.316·10-s − 0.577·12-s − 1.66·13-s − 0.801·14-s + 0.516·15-s + 1/4·16-s + 0.970·17-s + 0.942·18-s − 0.688·19-s − 0.223·20-s + 1.30·21-s + 0.417·23-s − 0.408·24-s + 25-s − 1.17·26-s − 0.962·27-s − 0.566·28-s + 0.185·29-s + 0.365·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1416 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1416\)    =    \(2^{3} \cdot 3 \cdot 59\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{1416} (1, \cdot )$
Sato-Tate group: $\mathrm{USp}(4)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1416,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5609835072\)
\(L(\frac12)\) \(\approx\) \(0.5609835072\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 - T \)
3$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + p T + p T^{2} ) \)
59$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 4 T + p T^{2} ) \)
good5$C_2^2$ \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
13$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_4$ \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 3 T + 26 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
23$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
29$D_{4}$ \( 1 - T - p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 6 T + 14 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 8 T + 62 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 5 T + 40 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
47$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$D_{4}$ \( 1 - 7 T + 4 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
73$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
79$D_{4}$ \( 1 + 5 T - 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 110 T^{2} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 16 T + 214 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 2 T - 142 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−19.3617505092, −19.0622522385, −18.4145277520, −17.5310713723, −17.1183484475, −16.5669433218, −16.1590318857, −15.6359187288, −14.9170188947, −14.5354233697, −13.5901194624, −12.8401958368, −12.4993479094, −12.0792742923, −11.4843478894, −10.5399974591, −10.1403169191, −9.53232461684, −8.31794782073, −7.12352074003, −6.95534344376, −5.98071958953, −5.09708930920, −4.35041306913, −3.02726240426, 3.02726240426, 4.35041306913, 5.09708930920, 5.98071958953, 6.95534344376, 7.12352074003, 8.31794782073, 9.53232461684, 10.1403169191, 10.5399974591, 11.4843478894, 12.0792742923, 12.4993479094, 12.8401958368, 13.5901194624, 14.5354233697, 14.9170188947, 15.6359187288, 16.1590318857, 16.5669433218, 17.1183484475, 17.5310713723, 18.4145277520, 19.0622522385, 19.3617505092

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