Properties

Degree 4
Conductor $ 2^{2} \cdot 7 \cdot 293 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 2

Origins

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 4·3-s + 2·4-s − 3·5-s + 8·6-s − 5·7-s + 7·9-s + 6·10-s − 5·11-s − 8·12-s − 2·13-s + 10·14-s + 12·15-s − 4·16-s − 14·18-s − 6·19-s − 6·20-s + 20·21-s + 10·22-s + 23-s + 2·25-s + 4·26-s − 4·27-s − 10·28-s − 2·29-s − 24·30-s + 31-s + ⋯
L(s)  = 1  − 1.41·2-s − 2.30·3-s + 4-s − 1.34·5-s + 3.26·6-s − 1.88·7-s + 7/3·9-s + 1.89·10-s − 1.50·11-s − 2.30·12-s − 0.554·13-s + 2.67·14-s + 3.09·15-s − 16-s − 3.29·18-s − 1.37·19-s − 1.34·20-s + 4.36·21-s + 2.13·22-s + 0.208·23-s + 2/5·25-s + 0.784·26-s − 0.769·27-s − 1.88·28-s − 0.371·29-s − 4.38·30-s + 0.179·31-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 8204 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 8204 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(8204\)    =    \(2^{2} \cdot 7 \cdot 293\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{8204} (1, \cdot )$
Sato-Tate  :  $\mathrm{USp}(4)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  2
Selberg data  =  $(4,\ 8204,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;7,\;293\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;7,\;293\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_2$ \( 1 + p T + p T^{2} \)
7$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 4 T + p T^{2} ) \)
293$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 19 T + p T^{2} ) \)
good3$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + p T + p T^{2} ) \)
5$D_{4}$ \( 1 + 3 T + 7 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + 5 T + 19 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 2 T + 12 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 6 T + 40 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - T - 35 T^{2} - p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 2 T + 11 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - T - 9 T^{2} - p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 6 T + 42 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 10 T + 95 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 2 T + 14 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 2 T + 63 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 14 T + 107 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 3 T - 25 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 17 T + 148 T^{2} + 17 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 4 T + 63 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 4 T + 9 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 2 T + 31 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 3 T - 3 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 9 T + 97 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 14 T + 168 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
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

−17.444663256, −16.7712615124, −16.5074212328, −16.1172246247, −15.650503663, −15.5458946076, −14.6289150761, −13.4646093399, −12.9851591694, −12.5069422593, −12.1106180484, −11.57065993, −10.9441038362, −10.557355852, −10.3800228538, −9.56501471858, −9.00041523929, −8.09165463658, −7.58864841972, −6.94855742021, −6.38266258649, −5.82513734068, −4.95978083087, −4.18609383229, −2.85476568525, 0, 0, 2.85476568525, 4.18609383229, 4.95978083087, 5.82513734068, 6.38266258649, 6.94855742021, 7.58864841972, 8.09165463658, 9.00041523929, 9.56501471858, 10.3800228538, 10.557355852, 10.9441038362, 11.57065993, 12.1106180484, 12.5069422593, 12.9851591694, 13.4646093399, 14.6289150761, 15.5458946076, 15.650503663, 16.1172246247, 16.5074212328, 16.7712615124, 17.444663256

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