# Properties

 Label 6-7600e3-1.1-c1e3-0-7 Degree $6$ Conductor $438976000000$ Sign $1$ Analytic cond. $223497.$ Root an. cond. $7.79014$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 2·3-s + 2·7-s − 2·9-s + 11-s − 13-s − 2·17-s + 3·19-s + 4·21-s + 8·23-s − 5·27-s − 7·29-s − 11·31-s + 2·33-s + 5·37-s − 2·39-s + 13·41-s + 11·43-s + 19·47-s − 14·49-s − 4·51-s − 11·53-s + 6·57-s + 6·59-s + 7·61-s − 4·63-s + 3·67-s + 16·69-s + ⋯
 L(s)  = 1 + 1.15·3-s + 0.755·7-s − 2/3·9-s + 0.301·11-s − 0.277·13-s − 0.485·17-s + 0.688·19-s + 0.872·21-s + 1.66·23-s − 0.962·27-s − 1.29·29-s − 1.97·31-s + 0.348·33-s + 0.821·37-s − 0.320·39-s + 2.03·41-s + 1.67·43-s + 2.77·47-s − 2·49-s − 0.560·51-s − 1.51·53-s + 0.794·57-s + 0.781·59-s + 0.896·61-s − 0.503·63-s + 0.366·67-s + 1.92·69-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$6$$ Conductor: $$2^{12} \cdot 5^{6} \cdot 19^{3}$$ Sign: $1$ Analytic conductor: $$223497.$$ Root analytic conductor: $$7.79014$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(6,\ 2^{12} \cdot 5^{6} \cdot 19^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$8.607193680$$ $$L(\frac12)$$ $$\approx$$ $$8.607193680$$ $$L(\frac{3}{2})$$ 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)$
bad2 $$1$$
5 $$1$$
19$C_1$ $$( 1 - T )^{3}$$
good3$S_4\times C_2$ $$1 - 2 T + 2 p T^{2} - 11 T^{3} + 2 p^{2} T^{4} - 2 p^{2} T^{5} + p^{3} T^{6}$$
7$S_4\times C_2$ $$1 - 2 T + 18 T^{2} - 27 T^{3} + 18 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6}$$
11$S_4\times C_2$ $$1 - T + 7 T^{2} - 37 T^{3} + 7 p T^{4} - p^{2} T^{5} + p^{3} T^{6}$$
13$S_4\times C_2$ $$1 + T + 31 T^{2} + 23 T^{3} + 31 p T^{4} + p^{2} T^{5} + p^{3} T^{6}$$
17$S_4\times C_2$ $$1 + 2 T + 48 T^{2} + 67 T^{3} + 48 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6}$$
23$S_4\times C_2$ $$1 - 8 T + 66 T^{2} - 359 T^{3} + 66 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6}$$
29$S_4\times C_2$ $$1 + 7 T + 77 T^{2} + 381 T^{3} + 77 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6}$$
31$S_4\times C_2$ $$1 + 11 T + 87 T^{2} + 441 T^{3} + 87 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6}$$
37$S_4\times C_2$ $$1 - 5 T + 39 T^{2} + 35 T^{3} + 39 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6}$$
41$S_4\times C_2$ $$1 - 13 T + 159 T^{2} - 1091 T^{3} + 159 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6}$$
43$S_4\times C_2$ $$1 - 11 T + 152 T^{2} - 919 T^{3} + 152 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6}$$
47$S_4\times C_2$ $$1 - 19 T + 209 T^{2} - 1723 T^{3} + 209 p T^{4} - 19 p^{2} T^{5} + p^{3} T^{6}$$
53$S_4\times C_2$ $$1 + 11 T + 182 T^{2} + 1139 T^{3} + 182 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6}$$
59$S_4\times C_2$ $$1 - 6 T + T^{2} + 148 T^{3} + p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6}$$
61$S_4\times C_2$ $$1 - 7 T + 153 T^{2} - 879 T^{3} + 153 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6}$$
67$S_4\times C_2$ $$1 - 3 T + 73 T^{2} + 197 T^{3} + 73 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6}$$
71$S_4\times C_2$ $$1 - 5 T + 140 T^{2} - 833 T^{3} + 140 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6}$$
73$S_4\times C_2$ $$1 + 9 T + 201 T^{2} + 1287 T^{3} + 201 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6}$$
79$S_4\times C_2$ $$1 - 18 T + 238 T^{2} - 2237 T^{3} + 238 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6}$$
83$S_4\times C_2$ $$1 - 3 T + T^{2} + 251 T^{3} + p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6}$$
89$S_4\times C_2$ $$1 + 84 T^{2} - 857 T^{3} + 84 p T^{4} + p^{3} T^{6}$$
97$S_4\times C_2$ $$1 - 3 T + 219 T^{2} - 501 T^{3} + 219 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6}$$
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$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$

## Imaginary part of the first few zeros on the critical line

−7.20279465082218790374115710318, −6.70942305793420864334865046293, −6.54696321966464319842208253689, −6.26219957888206299547120104688, −5.98152244628634054109554951483, −5.73772137014100447043382119776, −5.59343338583778743086150691117, −5.17622249340051583061973477565, −5.16857182462666947910263707181, −4.89320604935356090279562578078, −4.44888481856705684894240139854, −4.32798573214418665451654452009, −4.03790681969940499561913894166, −3.63199833478364263223087851656, −3.55424844461036963261181383574, −3.29950452511557309929500664282, −2.82656049234235038194951431632, −2.73482891957050133018774811866, −2.50025379862495986320978025376, −2.06675794172931910919606615043, −1.99412980880902145376971346087, −1.56790185805339350452333605946, −1.08237037827346774527484834707, −0.66051616729244648808885548679, −0.51484683691178072834808714237, 0.51484683691178072834808714237, 0.66051616729244648808885548679, 1.08237037827346774527484834707, 1.56790185805339350452333605946, 1.99412980880902145376971346087, 2.06675794172931910919606615043, 2.50025379862495986320978025376, 2.73482891957050133018774811866, 2.82656049234235038194951431632, 3.29950452511557309929500664282, 3.55424844461036963261181383574, 3.63199833478364263223087851656, 4.03790681969940499561913894166, 4.32798573214418665451654452009, 4.44888481856705684894240139854, 4.89320604935356090279562578078, 5.16857182462666947910263707181, 5.17622249340051583061973477565, 5.59343338583778743086150691117, 5.73772137014100447043382119776, 5.98152244628634054109554951483, 6.26219957888206299547120104688, 6.54696321966464319842208253689, 6.70942305793420864334865046293, 7.20279465082218790374115710318