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

Origins of factors

Downloads

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Normalization:  

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

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