Properties

Degree 4
Conductor $ 5 \cdot 101 \cdot 113 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 3

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·2-s − 4·3-s + 4·4-s − 3·5-s + 12·6-s − 5·7-s − 3·8-s + 7·9-s + 9·10-s − 5·11-s − 16·12-s − 6·13-s + 15·14-s + 12·15-s + 3·16-s − 5·17-s − 21·18-s − 5·19-s − 12·20-s + 20·21-s + 15·22-s − 9·23-s + 12·24-s + 2·25-s + 18·26-s − 4·27-s − 20·28-s + ⋯
L(s)  = 1  − 2.12·2-s − 2.30·3-s + 2·4-s − 1.34·5-s + 4.89·6-s − 1.88·7-s − 1.06·8-s + 7/3·9-s + 2.84·10-s − 1.50·11-s − 4.61·12-s − 1.66·13-s + 4.00·14-s + 3.09·15-s + 3/4·16-s − 1.21·17-s − 4.94·18-s − 1.14·19-s − 2.68·20-s + 4.36·21-s + 3.19·22-s − 1.87·23-s + 2.44·24-s + 2/5·25-s + 3.53·26-s − 0.769·27-s − 3.77·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 57065 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 57065 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(57065\)    =    \(5 \cdot 101 \cdot 113\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{57065} (1, \cdot )$
Sato-Tate  :  $\mathrm{USp}(4)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  3
Selberg data  =  $(4,\ 57065,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{5,\;101,\;113\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{5,\;101,\;113\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad5$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 2 T + p T^{2} ) \)
101$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 8 T + p T^{2} ) \)
113$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 2 T + p T^{2} ) \)
good2$C_2^2$ \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
3$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + p T + p T^{2} ) \)
7$D_{4}$ \( 1 + 5 T + 19 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + 5 T + 17 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 6 T + 32 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 5 T + 30 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 5 T + p T^{2} + 5 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 9 T + 62 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + T - 37 T^{2} + p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 3 T + 20 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 3 T - 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 2 T - 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 9 T + 77 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - T - 14 T^{2} - p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 9 T + 80 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 8 T + 13 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + T + 68 T^{2} + p T^{3} + p^{2} T^{4} \)
71$C_2$$\times$$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
73$D_{4}$ \( 1 + T - 8 T^{2} + p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 4 T + 36 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 2 T + 44 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + T - 7 T^{2} + p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 10 T + 100 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−15.7925168077, −15.169053215, −14.8586580823, −13.7077203752, −13.2374686275, −12.499788907, −12.3202414065, −12.1688978574, −11.3951539562, −11.0790922694, −10.5456543677, −10.2348811495, −9.84471422555, −9.50038309403, −8.7567553178, −8.08133446502, −7.88149582457, −7.17223572634, −6.70017185646, −6.18489563134, −5.76866971264, −4.86601576835, −4.42523313718, −3.33487380806, −2.3267203824, 0, 0, 0, 2.3267203824, 3.33487380806, 4.42523313718, 4.86601576835, 5.76866971264, 6.18489563134, 6.70017185646, 7.17223572634, 7.88149582457, 8.08133446502, 8.7567553178, 9.50038309403, 9.84471422555, 10.2348811495, 10.5456543677, 11.0790922694, 11.3951539562, 12.1688978574, 12.3202414065, 12.499788907, 13.2374686275, 13.7077203752, 14.8586580823, 15.169053215, 15.7925168077

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