# Properties

 Degree 4 Conductor 169 Sign $1$ Self-dual yes Motivic weight 1

# Related objects

## Dirichlet series

 L(A,s)  = 1 − 3·2-s − 2·3-s + 4·4-s + 6·6-s − 3·8-s + 3·9-s − 8·12-s − 5·13-s + 3·16-s + 3·17-s − 9·18-s − 6·19-s + 6·23-s + 6·24-s + 7·25-s + 15·26-s − 10·27-s − 3·29-s − 6·32-s − 9·34-s + 12·36-s + 15·37-s + 18·38-s + 10·39-s − 9·41-s − 8·43-s − 18·46-s + ⋯
 L(s,A)  = 1 − 2.121·2-s − 1.154·3-s + 2·4-s + 2.449·6-s − 1.060·8-s + 9-s − 2.309·12-s − 1.386·13-s + 0.75·16-s + 0.727·17-s − 2.121·18-s − 1.376·19-s + 1.251·23-s + 1.224·24-s + 1.4·25-s + 2.941·26-s − 1.924·27-s − 0.557·29-s − 1.060·32-s − 1.543·34-s + 2·36-s + 2.465·37-s + 2.919·38-s + 1.601·39-s − 1.405·41-s − 1.219·43-s − 2.653·46-s + ⋯

## Functional equation

\begin{align} \Lambda(A,s)=\mathstrut & 169 ^{s/2} \Gamma_{\C}(s) ^{2} \cdot L(A,s)\cr =\mathstrut & \Lambda(A, 2-s) \end{align}
\begin{align} \Lambda(s,A)=\mathstrut & 169 ^{s/2} \Gamma_{\C}(s+0.5) ^{2} \cdot L(s,A)\cr =\mathstrut & \Lambda(1-s,A) \end{align}

## Invariants

 $d$ = $4$ $N$ = $169$    =    $13^{2}$ $\varepsilon$ = $1$ weight = 1 character : $\chi_{169} (1, \cdot )$ Sato-Tate : $E_6$ primitive : no self-dual : yes analytic rank = 0 Selberg data = $(4,\ 169,\ (\ :1/2, 1/2),\ 1)$ $L(A,1)$ $\approx$ $0.0904903908$ $L(\frac12,A)$ $\approx$ $0.0904903908$ $L(A,\frac{3}{2})$ not available $L(1,A)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \neq 13$, $F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4$with $b_p = a_p^2 - a_{p^2}$. If $p = 13$, then $F_p$ is a polynomial of degree at most 3.
$p$$F_p$$\Gal(F_p)$
bad13$1+5T+13T^{2}$$C_2 good21+3T+5T^{2}+6T^{3}+4T^{4}$$V_4$
3$1+2T+T^{2}+6T^{3}+9T^{4}$$V_4 51-7T^{2}+25T^{4}$$V_4$
7$1+7T^{2}+49T^{4}$$V_4 111+11T^{2}+121T^{4}$$V_4$
17$1-3T-8T^{2}-51T^{3}+289T^{4}$$V_4 19(1-T+19T^{2})(1+7T+19T^{2})$$C_2$
23$1-6T+13T^{2}-138T^{3}+529T^{4}$$V_4 291+3T-20T^{2}+87T^{3}+841T^{4}$$V_4$
31$1-50T^{2}+961T^{4}$$V_4 371-15T+112T^{2}-555T^{3}+1369T^{4}$$V_4$
41$1+9T+68T^{2}+369T^{3}+1681T^{4}$$V_4 43(1-5T+43T^{2})(1+13T+43T^{2})$$C_2$
47$1-82T^{2}+2209T^{4}$$V_4 53(1+3T+53T^{2})^{2}$$C_2$
59$1-12T+107T^{2}-708T^{3}+3481T^{4}$$V_4 61(1-13T+61T^{2})(1+14T+61T^{2})$$C_2$
67$(1-11T+67T^{2})(1+5T+67T^{2})$$C_2 711-6T+83T^{2}-426T^{3}+5041T^{4}$$V_4$
73$(1-17T+73T^{2})(1+17T+73T^{2})$$C_2 79(1-4T+79T^{2})^{2}$$C_2$
83$1+26T^{2}+6889T^{4}$$V_4 891+12T+137T^{2}+1068T^{3}+7921T^{4}$$V_4$
97$1-12T+145T^{2}-1164T^{3}+9409T^{4}$$V_4$
$$$L(s,A) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$$