Properties

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

Related objects

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Normalization:  

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 + ⋯
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 + ⋯

Functional equation

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

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(169\)    =    \(13^{2}\)
\( \varepsilon \)  =  $1$
weight  =  1
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

\[L(A,s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] 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}$$C_2$
good2$1+3T+5T^{2}+6T^{3}+4T^{4}$$V_4$
3$1+2T+T^{2}+6T^{3}+9T^{4}$$V_4$
5$1-7T^{2}+25T^{4}$$V_4$
7$1+7T^{2}+49T^{4}$$V_4$
11$1+11T^{2}+121T^{4}$$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}$$V_4$
29$1+3T-20T^{2}+87T^{3}+841T^{4}$$V_4$
31$1-50T^{2}+961T^{4}$$V_4$
37$1-15T+112T^{2}-555T^{3}+1369T^{4}$$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})$$C_2$
71$1-6T+83T^{2}-426T^{3}+5041T^{4}$$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}$$V_4$
89$1+12T+137T^{2}+1068T^{3}+7921T^{4}$$V_4$
97$1-12T+145T^{2}-1164T^{3}+9409T^{4}$$V_4$
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\[\begin{equation} L(s,A) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{equation}\]

Imaginary part of the first few zeros on the critical line

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