Properties

Label 6-8550e3-1.1-c1e3-0-14
Degree $6$
Conductor $625026375000$
Sign $-1$
Analytic cond. $318221.$
Root an. cond. $8.26269$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $3$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3·2-s + 6·4-s − 2·7-s + 10·8-s − 5·11-s − 6·14-s + 15·16-s − 6·17-s + 3·19-s − 15·22-s + 23-s − 12·28-s − 15·29-s − 31-s + 21·32-s − 18·34-s + 4·37-s + 9·38-s − 6·41-s − 10·43-s − 30·44-s + 3·46-s − 7·49-s + 5·53-s − 20·56-s − 45·58-s − 12·59-s + ⋯
L(s)  = 1  + 2.12·2-s + 3·4-s − 0.755·7-s + 3.53·8-s − 1.50·11-s − 1.60·14-s + 15/4·16-s − 1.45·17-s + 0.688·19-s − 3.19·22-s + 0.208·23-s − 2.26·28-s − 2.78·29-s − 0.179·31-s + 3.71·32-s − 3.08·34-s + 0.657·37-s + 1.45·38-s − 0.937·41-s − 1.52·43-s − 4.52·44-s + 0.442·46-s − 49-s + 0.686·53-s − 2.67·56-s − 5.90·58-s − 1.56·59-s + ⋯

Functional equation

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

Invariants

Degree: \(6\)
Conductor: \(2^{3} \cdot 3^{6} \cdot 5^{6} \cdot 19^{3}\)
Sign: $-1$
Analytic conductor: \(318221.\)
Root analytic conductor: \(8.26269\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{8550} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 2^{3} \cdot 3^{6} \cdot 5^{6} \cdot 19^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ -1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{3} \)
3 \( 1 \)
5 \( 1 \)
19$C_1$ \( ( 1 - T )^{3} \)
good7$S_4\times C_2$ \( 1 + 2 T + 11 T^{2} + 26 T^{3} + 11 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
11$S_4\times C_2$ \( 1 + 5 T + 36 T^{2} + 107 T^{3} + 36 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \)
13$S_4\times C_2$ \( 1 + 3 T^{2} - 82 T^{3} + 3 p T^{4} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 + 6 T + 45 T^{2} + 150 T^{3} + 45 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 - T + 60 T^{2} - 49 T^{3} + 60 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 + 15 T + 144 T^{2} + 879 T^{3} + 144 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 + T + 8 T^{2} - 47 T^{3} + 8 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 - 4 T + 95 T^{2} - 280 T^{3} + 95 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 + 6 T + 51 T^{2} + 168 T^{3} + 51 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 + 10 T + 157 T^{2} + 878 T^{3} + 157 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 + 117 T^{2} - 36 T^{3} + 117 p T^{4} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 - 5 T + 102 T^{2} - 347 T^{3} + 102 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 + 12 T - 27 T^{2} - 1050 T^{3} - 27 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 - T + 46 T^{2} + 517 T^{3} + 46 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 + T + 154 T^{2} - 7 T^{3} + 154 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 + 24 T + 387 T^{2} + 3750 T^{3} + 387 p T^{4} + 24 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 + 9 T + 138 T^{2} + 773 T^{3} + 138 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 - 5 T + 224 T^{2} - 749 T^{3} + 224 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 - 11 T + 222 T^{2} - 1787 T^{3} + 222 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 + 23 T + 234 T^{2} + 1787 T^{3} + 234 p T^{4} + 23 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 + 14 T + 197 T^{2} + 1742 T^{3} + 197 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.10055176916182136889016000132, −6.78982424834440923513471138298, −6.71069816325063153335758091446, −6.67639189454609935946124162125, −6.02721452491203185840066350506, −5.90896633249390032780526893519, −5.84758208841967961099351965069, −5.65165485098506055893059094066, −5.25434344823429865720489882691, −5.14406886455350205191750309987, −4.86666834561171968793557830264, −4.64972975425285027265591795153, −4.46032401552084108776135170454, −4.14587323602029661179714942754, −3.79774528044929844846937749985, −3.69716321751102060155838167055, −3.33185719055502394665719115404, −3.22522014947094594637093432025, −2.87981253027303691496926626566, −2.49355133254412103781942907692, −2.45078492778120993851874409195, −2.26457949940225757702892949909, −1.50099055647686889364296435066, −1.49719587439711256755670440635, −1.33658403437998747671368320557, 0, 0, 0, 1.33658403437998747671368320557, 1.49719587439711256755670440635, 1.50099055647686889364296435066, 2.26457949940225757702892949909, 2.45078492778120993851874409195, 2.49355133254412103781942907692, 2.87981253027303691496926626566, 3.22522014947094594637093432025, 3.33185719055502394665719115404, 3.69716321751102060155838167055, 3.79774528044929844846937749985, 4.14587323602029661179714942754, 4.46032401552084108776135170454, 4.64972975425285027265591795153, 4.86666834561171968793557830264, 5.14406886455350205191750309987, 5.25434344823429865720489882691, 5.65165485098506055893059094066, 5.84758208841967961099351965069, 5.90896633249390032780526893519, 6.02721452491203185840066350506, 6.67639189454609935946124162125, 6.71069816325063153335758091446, 6.78982424834440923513471138298, 7.10055176916182136889016000132

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