# Properties

 Degree 4 Conductor $2^{2} \cdot 7^{2} \cdot 67$ Sign $1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 0

# Origins

## Dirichlet series

 L(s)  = 1 − 2·2-s + 3·4-s − 7-s − 4·8-s + 9-s + 2·14-s + 5·16-s − 2·18-s + 9·23-s + 8·25-s − 3·28-s − 6·32-s + 3·36-s + 4·37-s + 7·43-s − 18·46-s − 6·49-s − 16·50-s − 18·53-s + 4·56-s − 63-s + 7·64-s + 15·67-s + 3·71-s − 4·72-s − 8·74-s − 2·79-s + ⋯
 L(s)  = 1 − 1.41·2-s + 3/2·4-s − 0.377·7-s − 1.41·8-s + 1/3·9-s + 0.534·14-s + 5/4·16-s − 0.471·18-s + 1.87·23-s + 8/5·25-s − 0.566·28-s − 1.06·32-s + 1/2·36-s + 0.657·37-s + 1.06·43-s − 2.65·46-s − 6/7·49-s − 2.26·50-s − 2.47·53-s + 0.534·56-s − 0.125·63-s + 7/8·64-s + 1.83·67-s + 0.356·71-s − 0.471·72-s − 0.929·74-s − 0.225·79-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 13132 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 13132 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$13132$$    =    $$2^{2} \cdot 7^{2} \cdot 67$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{13132} (1, \cdot )$ Sato-Tate : $\mathrm{SU}(2)$ primitive : yes self-dual : yes analytic rank = 0 Selberg data = $(4,\ 13132,\ (\ :1/2, 1/2),\ 1)$ $L(1)$ $\approx$ $0.5711183854$ $L(\frac12)$ $\approx$ $0.5711183854$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;7,\;67\}$, $F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4$with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;7,\;67\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_1$ $$( 1 + T )^{2}$$
7$C_2$ $$1 + T + p T^{2}$$
67$C_1$$\times$$C_2$ $$( 1 - T )( 1 - 14 T + p T^{2} )$$
good3$V_4$ $$1 - T^{2} + p^{2} T^{4}$$
5$V_4$ $$1 - 8 T^{2} + p^{2} T^{4}$$
11$C_2$ $$( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} )$$
13$V_4$ $$1 - 8 T^{2} + p^{2} T^{4}$$
17$C_2$ $$( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} )$$
19$V_4$ $$1 - 20 T^{2} + p^{2} T^{4}$$
23$C_2$$\times$$C_2$ $$( 1 - 6 T + p T^{2} )( 1 - 3 T + p T^{2} )$$
29$C_2$ $$( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} )$$
31$V_4$ $$1 + 19 T^{2} + p^{2} T^{4}$$
37$C_2$ $$( 1 - 2 T + p T^{2} )^{2}$$
41$V_4$ $$1 - 8 T^{2} + p^{2} T^{4}$$
43$C_2$$\times$$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} )$$
47$V_4$ $$1 + 40 T^{2} + p^{2} T^{4}$$
53$C_2$$\times$$C_2$ $$( 1 + 6 T + p T^{2} )( 1 + 12 T + p T^{2} )$$
59$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
61$V_4$ $$1 - 86 T^{2} + p^{2} T^{4}$$
71$C_2$$\times$$C_2$ $$( 1 - 9 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
73$V_4$ $$1 - 29 T^{2} + p^{2} T^{4}$$
79$C_2$$\times$$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
83$V_4$ $$1 - 32 T^{2} + p^{2} T^{4}$$
89$V_4$ $$1 - 83 T^{2} + p^{2} T^{4}$$
97$V_4$ $$1 + 148 T^{2} + p^{2} T^{4}$$
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\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

−11.06242556734797633316617654325, −10.75142325686499068465966747941, −10.03580089431797300060009499074, −9.496854124180573508165716157921, −9.145978517862082072816079858900, −8.570611933157460167149885055115, −7.946093092912831285360621515242, −7.34391638071145319827988985542, −6.73529519082847451308985885003, −6.38183167189614169029600475754, −5.35998631133052630855741203130, −4.60746508936177210269135914867, −3.34534227552155671660153330050, −2.62780247878860436110860336078, −1.19064202541676050214281881703, 1.19064202541676050214281881703, 2.62780247878860436110860336078, 3.34534227552155671660153330050, 4.60746508936177210269135914867, 5.35998631133052630855741203130, 6.38183167189614169029600475754, 6.73529519082847451308985885003, 7.34391638071145319827988985542, 7.946093092912831285360621515242, 8.570611933157460167149885055115, 9.145978517862082072816079858900, 9.496854124180573508165716157921, 10.03580089431797300060009499074, 10.75142325686499068465966747941, 11.06242556734797633316617654325