Properties

Degree 4
Conductor $ 7 \cdot 14303 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 2

Origins

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 3·2-s − 3-s + 4·4-s + 3·6-s − 3·7-s − 3·8-s − 3·9-s − 11-s − 4·12-s − 13-s + 9·14-s + 3·16-s − 4·17-s + 9·18-s − 3·19-s + 3·21-s + 3·22-s − 4·23-s + 3·24-s + 25-s + 3·26-s + 4·27-s − 12·28-s − 31-s − 6·32-s + 33-s + 12·34-s + ⋯
L(s)  = 1  − 2.12·2-s − 0.577·3-s + 2·4-s + 1.22·6-s − 1.13·7-s − 1.06·8-s − 9-s − 0.301·11-s − 1.15·12-s − 0.277·13-s + 2.40·14-s + 3/4·16-s − 0.970·17-s + 2.12·18-s − 0.688·19-s + 0.654·21-s + 0.639·22-s − 0.834·23-s + 0.612·24-s + 1/5·25-s + 0.588·26-s + 0.769·27-s − 2.26·28-s − 0.179·31-s − 1.06·32-s + 0.174·33-s + 2.05·34-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 100121 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 100121 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(100121\)    =    \(7 \cdot 14303\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{100121} (1, \cdot )$
Sato-Tate  :  $\mathrm{USp}(4)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  2
Selberg data  =  $(4,\ 100121,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{7,\;14303\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{7,\;14303\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad7$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 4 T + p T^{2} ) \)
14303$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 170 T + p T^{2} ) \)
good2$V_4$ \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
3$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
5$V_4$ \( 1 - T^{2} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + T - 5 T^{2} + p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + T + 15 T^{2} + p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 4 T + 32 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 3 T + 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 4 T + 21 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
29$V_4$ \( 1 - 32 T^{2} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + T + 12 T^{2} + p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + T + 55 T^{2} + p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 14 T + 105 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 8 T + 97 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 4 T + 60 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
53$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \)
59$D_{4}$ \( 1 - 6 T + 2 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 8 T + 68 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 + 3 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
71$D_{4}$ \( 1 + 6 T + 48 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
73$V_4$ \( 1 - 94 T^{2} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 4 T + 66 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 12 T + 112 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 14 T + 171 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 25 T + 345 T^{2} + 25 p T^{3} + p^{2} T^{4} \)
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

−14.7706469544, −13.9370107649, −13.5431774477, −13.2077106818, −12.5423592213, −12.1030651912, −11.7625322305, −11.179977311, −10.778879429, −10.2233909024, −10.0445417434, −9.60053715731, −8.9693018007, −8.61923043099, −8.44799012975, −7.79876663782, −7.17687310436, −6.61381146466, −6.29480109232, −5.63515366861, −5.08595632053, −4.17698504246, −3.29562345941, −2.64889963602, −1.65922345034, 0, 0, 1.65922345034, 2.64889963602, 3.29562345941, 4.17698504246, 5.08595632053, 5.63515366861, 6.29480109232, 6.61381146466, 7.17687310436, 7.79876663782, 8.44799012975, 8.61923043099, 8.9693018007, 9.60053715731, 10.0445417434, 10.2233909024, 10.778879429, 11.179977311, 11.7625322305, 12.1030651912, 12.5423592213, 13.2077106818, 13.5431774477, 13.9370107649, 14.7706469544

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