Properties

Label 6-8470e3-1.1-c1e3-0-5
Degree $6$
Conductor $607645423000$
Sign $1$
Analytic cond. $309372.$
Root an. cond. $8.22394$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·2-s + 6·4-s − 3·5-s + 3·7-s + 10·8-s − 2·9-s − 9·10-s + 9·14-s + 15·16-s + 11·17-s − 6·18-s + 5·19-s − 18·20-s − 4·23-s + 6·25-s − 2·27-s + 18·28-s + 2·29-s + 21·32-s + 33·34-s − 9·35-s − 12·36-s − 2·37-s + 15·38-s − 30·40-s + 7·43-s + 6·45-s + ⋯
L(s)  = 1  + 2.12·2-s + 3·4-s − 1.34·5-s + 1.13·7-s + 3.53·8-s − 2/3·9-s − 2.84·10-s + 2.40·14-s + 15/4·16-s + 2.66·17-s − 1.41·18-s + 1.14·19-s − 4.02·20-s − 0.834·23-s + 6/5·25-s − 0.384·27-s + 3.40·28-s + 0.371·29-s + 3.71·32-s + 5.65·34-s − 1.52·35-s − 2·36-s − 0.328·37-s + 2.43·38-s − 4.74·40-s + 1.06·43-s + 0.894·45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 5^{3} \cdot 7^{3} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 5^{3} \cdot 7^{3} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{3} \cdot 5^{3} \cdot 7^{3} \cdot 11^{6}\)
Sign: $1$
Analytic conductor: \(309372.\)
Root analytic conductor: \(8.22394\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{8470} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 2^{3} \cdot 5^{3} \cdot 7^{3} \cdot 11^{6} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(30.86636874\)
\(L(\frac12)\) \(\approx\) \(30.86636874\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{3} \)
5$C_1$ \( ( 1 + T )^{3} \)
7$C_1$ \( ( 1 - T )^{3} \)
11 \( 1 \)
good3$S_4\times C_2$ \( 1 + 2 T^{2} + 2 T^{3} + 2 p T^{4} + p^{3} T^{6} \)
13$S_4\times C_2$ \( 1 + 32 T^{2} + 2 T^{3} + 32 p T^{4} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 - 11 T + 75 T^{2} - 342 T^{3} + 75 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \)
19$S_4\times C_2$ \( 1 - 5 T + 36 T^{2} - 173 T^{3} + 36 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 + 4 T + 42 T^{2} + 78 T^{3} + 42 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 - 2 T + 23 T^{2} - 244 T^{3} + 23 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
31$C_2$ \( ( 1 + p T^{2} )^{3} \)
37$S_4\times C_2$ \( 1 + 2 T + 43 T^{2} + 140 T^{3} + 43 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 + 95 T^{2} - 16 T^{3} + 95 p T^{4} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 - 7 T + 41 T^{2} - 54 T^{3} + 41 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 - 8 T + 93 T^{2} - 624 T^{3} + 93 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 - T + 143 T^{2} - 122 T^{3} + 143 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 - 7 T + 164 T^{2} - 743 T^{3} + 164 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 - 13 T + 3 p T^{2} - 1270 T^{3} + 3 p^{2} T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 + 9 T + 81 T^{2} + 214 T^{3} + 81 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 - 17 T + 277 T^{2} - 2478 T^{3} + 277 p T^{4} - 17 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 - 19 T + 307 T^{2} - 2758 T^{3} + 307 p T^{4} - 19 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 + 9 T + 236 T^{2} + 1381 T^{3} + 236 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 - 6 T + 254 T^{2} - 988 T^{3} + 254 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 - 12 T + 287 T^{2} - 2104 T^{3} + 287 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 + 7 T + 275 T^{2} + 1230 T^{3} + 275 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \)
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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

−6.97527323139160506707512202520, −6.52242380736624906563317439049, −6.38067569494574578714195185231, −5.99793704410532733246358230779, −5.71592395767365072962627520909, −5.61545770335549493674546235921, −5.54535513549560506451780030619, −5.11911930246173555734499484434, −5.01914665972444400290957349314, −4.90539225820075947015701057975, −4.48236543912557344994949318765, −4.14598666212608247994587603083, −4.01254080495435396836484820850, −3.70893360839727664214844433337, −3.66333959878010799718416476833, −3.40405684195895885562196598283, −2.97344882306496851739446658705, −2.82744891943637132716997706387, −2.64440171493433481428624307734, −2.10459948107619394671808123086, −1.97749595815253869048468488415, −1.53550203288305370453418088176, −1.10663253826053226605914316449, −0.70622398268489978735753856046, −0.67335676781879113890718087737, 0.67335676781879113890718087737, 0.70622398268489978735753856046, 1.10663253826053226605914316449, 1.53550203288305370453418088176, 1.97749595815253869048468488415, 2.10459948107619394671808123086, 2.64440171493433481428624307734, 2.82744891943637132716997706387, 2.97344882306496851739446658705, 3.40405684195895885562196598283, 3.66333959878010799718416476833, 3.70893360839727664214844433337, 4.01254080495435396836484820850, 4.14598666212608247994587603083, 4.48236543912557344994949318765, 4.90539225820075947015701057975, 5.01914665972444400290957349314, 5.11911930246173555734499484434, 5.54535513549560506451780030619, 5.61545770335549493674546235921, 5.71592395767365072962627520909, 5.99793704410532733246358230779, 6.38067569494574578714195185231, 6.52242380736624906563317439049, 6.97527323139160506707512202520

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