Properties

Degree 4
Conductor $ 23 \cdot 61 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 3·3-s + 2·4-s − 2·5-s + 6·6-s − 7-s − 4·8-s + 4·9-s + 4·10-s − 3·11-s − 6·12-s − 3·13-s + 2·14-s + 6·15-s + 8·16-s + 3·17-s − 8·18-s + 5·19-s − 4·20-s + 3·21-s + 6·22-s − 3·23-s + 12·24-s + 2·25-s + 6·26-s − 6·27-s − 2·28-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.73·3-s + 4-s − 0.894·5-s + 2.44·6-s − 0.377·7-s − 1.41·8-s + 4/3·9-s + 1.26·10-s − 0.904·11-s − 1.73·12-s − 0.832·13-s + 0.534·14-s + 1.54·15-s + 2·16-s + 0.727·17-s − 1.88·18-s + 1.14·19-s − 0.894·20-s + 0.654·21-s + 1.27·22-s − 0.625·23-s + 2.44·24-s + 2/5·25-s + 1.17·26-s − 1.15·27-s − 0.377·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 1403 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 1403 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(1403\)    =    \(23 \cdot 61\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{1403} (1, \cdot )$
Sato-Tate  :  $\mathrm{USp}(4)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(4,\ 1403,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{23,\;61\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{23,\;61\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad23$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 2 T + p T^{2} ) \)
61$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 6 T + p T^{2} ) \)
good2$C_2^2$ \( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} \)
3$C_4$ \( 1 + p T + 5 T^{2} + p^{2} T^{3} + p^{2} T^{4} \)
5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
7$D_{4}$ \( 1 + T + p T^{2} + p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + 3 T + 15 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 3 T + 21 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 3 T + 35 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 5 T + 22 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
29$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$D_{4}$ \( 1 + 8 T + 50 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + T - 56 T^{2} + p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 4 T + 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 3 T + 57 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - T + 18 T^{2} - p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - T + 13 T^{2} - p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 7 T + 32 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 9 T + 106 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 2 T + 30 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 6 T + 66 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 3 T - 3 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 17 T + 183 T^{2} - 17 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + T + 18 T^{2} + p T^{3} + p^{2} T^{4} \)
show more
show less
\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−19.351159, −19.0280637683, −18.3365526062, −18.047019368, −17.7126558399, −16.8509857956, −16.7002367046, −16.129821525, −15.5338339458, −15.0925819871, −14.2368020772, −13.1290129319, −12.2983097321, −12.0533221795, −11.5672886058, −10.8650909748, −10.2321024076, −9.69221054347, −9.00337710612, −7.84992525454, −7.61663888445, −6.56642311068, −5.70429225088, −5.08881624682, −3.33741085275, 0, 3.33741085275, 5.08881624682, 5.70429225088, 6.56642311068, 7.61663888445, 7.84992525454, 9.00337710612, 9.69221054347, 10.2321024076, 10.8650909748, 11.5672886058, 12.0533221795, 12.2983097321, 13.1290129319, 14.2368020772, 15.0925819871, 15.5338339458, 16.129821525, 16.7002367046, 16.8509857956, 17.7126558399, 18.047019368, 18.3365526062, 19.0280637683, 19.351159

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