# Properties

 Label 6-7e3-1.1-c9e3-0-0 Degree $6$ Conductor $343$ Sign $1$ Analytic cond. $46.8604$ Root an. cond. $1.89874$ Motivic weight $9$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 21·2-s + 84·3-s + 231·4-s + 1.55e3·5-s + 1.76e3·6-s + 7.20e3·7-s + 2.86e3·8-s − 3.89e4·9-s + 3.26e4·10-s − 3.44e3·11-s + 1.94e4·12-s − 1.97e4·13-s + 1.51e5·14-s + 1.30e5·15-s − 4.49e4·16-s + 1.01e6·17-s − 8.18e5·18-s + 2.22e5·19-s + 3.58e5·20-s + 6.05e5·21-s − 7.23e4·22-s + 1.88e6·23-s + 2.40e5·24-s − 1.85e5·25-s − 4.15e5·26-s − 3.84e6·27-s + 1.66e6·28-s + ⋯
 L(s)  = 1 + 0.928·2-s + 0.598·3-s + 0.451·4-s + 1.11·5-s + 0.555·6-s + 1.13·7-s + 0.247·8-s − 1.98·9-s + 1.03·10-s − 0.0709·11-s + 0.270·12-s − 0.192·13-s + 1.05·14-s + 0.665·15-s − 0.171·16-s + 2.95·17-s − 1.83·18-s + 0.392·19-s + 0.501·20-s + 0.678·21-s − 0.0658·22-s + 1.40·23-s + 0.148·24-s − 0.0950·25-s − 0.178·26-s − 1.39·27-s + 0.511·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 343 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 343 ^{s/2} \, \Gamma_{\C}(s+9/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$6$$ Conductor: $$343$$    =    $$7^{3}$$ Sign: $1$ Analytic conductor: $$46.8604$$ Root analytic conductor: $$1.89874$$ Motivic weight: $$9$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{7} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(6,\ 343,\ (\ :9/2, 9/2, 9/2),\ 1)$$

## Particular Values

 $$L(5)$$ $$\approx$$ $$5.623412405$$ $$L(\frac12)$$ $$\approx$$ $$5.623412405$$ $$L(\frac{11}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$\Gal(F_p)$$F_p(T)$
bad7$C_1$ $$( 1 - p^{4} T )^{3}$$
good2$S_4\times C_2$ $$1 - 21 T + 105 p T^{2} - 303 p^{3} T^{3} + 105 p^{10} T^{4} - 21 p^{18} T^{5} + p^{27} T^{6}$$
3$S_4\times C_2$ $$1 - 28 p T + 5117 p^{2} T^{2} - 122360 p^{3} T^{3} + 5117 p^{11} T^{4} - 28 p^{19} T^{5} + p^{27} T^{6}$$
5$S_4\times C_2$ $$1 - 1554 T + 520107 p T^{2} - 46663764 p^{2} T^{3} + 520107 p^{10} T^{4} - 1554 p^{18} T^{5} + p^{27} T^{6}$$
11$S_4\times C_2$ $$1 + 3444 T + 455343105 T^{2} + 125101303155960 T^{3} + 455343105 p^{9} T^{4} + 3444 p^{18} T^{5} + p^{27} T^{6}$$
13$S_4\times C_2$ $$1 + 19782 T + 18882215055 T^{2} + 378008000651932 T^{3} + 18882215055 p^{9} T^{4} + 19782 p^{18} T^{5} + p^{27} T^{6}$$
17$S_4\times C_2$ $$1 - 1016694 T + 649258140783 T^{2} - 263109059935862868 T^{3} + 649258140783 p^{9} T^{4} - 1016694 p^{18} T^{5} + p^{27} T^{6}$$
19$S_4\times C_2$ $$1 - 222852 T + 614081373717 T^{2} - 186835068238407176 T^{3} + 614081373717 p^{9} T^{4} - 222852 p^{18} T^{5} + p^{27} T^{6}$$
23$S_4\times C_2$ $$1 - 81984 p T + 5417652680517 T^{2} - 5817973976209389696 T^{3} + 5417652680517 p^{9} T^{4} - 81984 p^{19} T^{5} + p^{27} T^{6}$$
29$S_4\times C_2$ $$1 - 4081818 T + 38739015783987 T^{2} -$$$$11\!\cdots\!84$$$$T^{3} + 38739015783987 p^{9} T^{4} - 4081818 p^{18} T^{5} + p^{27} T^{6}$$
31$S_4\times C_2$ $$1 - 2869440 T + 21142500166221 T^{2} -$$$$22\!\cdots\!64$$$$T^{3} + 21142500166221 p^{9} T^{4} - 2869440 p^{18} T^{5} + p^{27} T^{6}$$
37$S_4\times C_2$ $$1 - 1395618 T + 262675972194027 T^{2} -$$$$70\!\cdots\!00$$$$T^{3} + 262675972194027 p^{9} T^{4} - 1395618 p^{18} T^{5} + p^{27} T^{6}$$
41$S_4\times C_2$ $$1 + 14420658 T + 764979654799959 T^{2} +$$$$74\!\cdots\!64$$$$T^{3} + 764979654799959 p^{9} T^{4} + 14420658 p^{18} T^{5} + p^{27} T^{6}$$
43$S_4\times C_2$ $$1 + 61631172 T + 2687603003165025 T^{2} +$$$$16\!\cdots\!04$$$$p T^{3} + 2687603003165025 p^{9} T^{4} + 61631172 p^{18} T^{5} + p^{27} T^{6}$$
47$S_4\times C_2$ $$1 + 10368960 T + 2946826961339709 T^{2} +$$$$18\!\cdots\!24$$$$T^{3} + 2946826961339709 p^{9} T^{4} + 10368960 p^{18} T^{5} + p^{27} T^{6}$$
53$S_4\times C_2$ $$1 - 67502610 T + 6295046710287531 T^{2} -$$$$20\!\cdots\!32$$$$T^{3} + 6295046710287531 p^{9} T^{4} - 67502610 p^{18} T^{5} + p^{27} T^{6}$$
59$S_4\times C_2$ $$1 + 42590100 T + 19076976504365997 T^{2} +$$$$78\!\cdots\!00$$$$T^{3} + 19076976504365997 p^{9} T^{4} + 42590100 p^{18} T^{5} + p^{27} T^{6}$$
61$S_4\times C_2$ $$1 - 191746842 T + 38596678668907359 T^{2} -$$$$44\!\cdots\!36$$$$T^{3} + 38596678668907359 p^{9} T^{4} - 191746842 p^{18} T^{5} + p^{27} T^{6}$$
67$S_4\times C_2$ $$1 + 255175788 T + 80172518705654361 T^{2} +$$$$11\!\cdots\!08$$$$T^{3} + 80172518705654361 p^{9} T^{4} + 255175788 p^{18} T^{5} + p^{27} T^{6}$$
71$S_4\times C_2$ $$1 - 296514504 T + 147895725194380437 T^{2} -$$$$25\!\cdots\!68$$$$T^{3} + 147895725194380437 p^{9} T^{4} - 296514504 p^{18} T^{5} + p^{27} T^{6}$$
73$S_4\times C_2$ $$1 - 344213310 T + 83298340302082311 T^{2} -$$$$20\!\cdots\!12$$$$T^{3} + 83298340302082311 p^{9} T^{4} - 344213310 p^{18} T^{5} + p^{27} T^{6}$$
79$S_4\times C_2$ $$1 + 960412656 T + 566786434394061357 T^{2} +$$$$21\!\cdots\!28$$$$T^{3} + 566786434394061357 p^{9} T^{4} + 960412656 p^{18} T^{5} + p^{27} T^{6}$$
83$S_4\times C_2$ $$1 + 1100517180 T + 873896700882341301 T^{2} +$$$$43\!\cdots\!28$$$$T^{3} + 873896700882341301 p^{9} T^{4} + 1100517180 p^{18} T^{5} + p^{27} T^{6}$$
89$S_4\times C_2$ $$1 - 506816478 T + 956194525794688887 T^{2} -$$$$35\!\cdots\!64$$$$T^{3} + 956194525794688887 p^{9} T^{4} - 506816478 p^{18} T^{5} + p^{27} T^{6}$$
97$S_4\times C_2$ $$1 + 647498250 T + 819696244591424799 T^{2} +$$$$48\!\cdots\!84$$$$T^{3} + 819696244591424799 p^{9} T^{4} + 647498250 p^{18} T^{5} + p^{27} T^{6}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$