Properties

Label 6-8470e3-1.1-c1e3-0-1
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 + 2·3-s + 6·4-s + 3·5-s − 6·6-s − 3·7-s − 10·8-s + 9-s − 9·10-s + 12·12-s + 9·14-s + 6·15-s + 15·16-s − 4·17-s − 3·18-s − 6·19-s + 18·20-s − 6·21-s + 8·23-s − 20·24-s + 6·25-s − 18·28-s + 2·29-s − 18·30-s + 2·31-s − 21·32-s + 12·34-s + ⋯
L(s)  = 1  − 2.12·2-s + 1.15·3-s + 3·4-s + 1.34·5-s − 2.44·6-s − 1.13·7-s − 3.53·8-s + 1/3·9-s − 2.84·10-s + 3.46·12-s + 2.40·14-s + 1.54·15-s + 15/4·16-s − 0.970·17-s − 0.707·18-s − 1.37·19-s + 4.02·20-s − 1.30·21-s + 1.66·23-s − 4.08·24-s + 6/5·25-s − 3.40·28-s + 0.371·29-s − 3.28·30-s + 0.359·31-s − 3.71·32-s + 2.05·34-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

\(L(1)\) \(\approx\) \(3.062865145\)
\(L(\frac12)\) \(\approx\) \(3.062865145\)
\(L(\frac{3}{2})\) 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$S_4\times C_2$ \( 1 - 2 T + p T^{2} - 4 T^{3} + p^{2} T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
13$C_2$ \( ( 1 + p T^{2} )^{3} \)
17$S_4\times C_2$ \( 1 + 4 T + 39 T^{2} + 120 T^{3} + 39 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
19$S_4\times C_2$ \( 1 + 6 T + 35 T^{2} + 92 T^{3} + 35 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 - 8 T + 75 T^{2} - 324 T^{3} + 75 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 - 2 T + 73 T^{2} - 84 T^{3} + 73 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 - 2 T + 33 T^{2} - 260 T^{3} + 33 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 - 8 T + 117 T^{2} - 548 T^{3} + 117 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 + 57 T^{2} + 92 T^{3} + 57 p T^{4} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 + 101 T^{2} - 16 T^{3} + 101 p T^{4} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 - 8 T + 97 T^{2} - 768 T^{3} + 97 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 + 21 T^{2} - 596 T^{3} + 21 p T^{4} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 + 10 T + 193 T^{2} + 1164 T^{3} + 193 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 - 2 T + 119 T^{2} - 372 T^{3} + 119 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 + 14 T + 197 T^{2} + 1692 T^{3} + 197 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 - 4 T + 101 T^{2} - 632 T^{3} + 101 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 - 12 T + 239 T^{2} - 1688 T^{3} + 239 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 - 2 T + 103 T^{2} - 500 T^{3} + 103 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 + 4 T + 185 T^{2} + 536 T^{3} + 185 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 - 14 T + 215 T^{2} - 1828 T^{3} + 215 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 - 44 T + 921 T^{2} - 11444 T^{3} + 921 p T^{4} - 44 p^{2} T^{5} + p^{3} T^{6} \)
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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

−6.91488933067814726053653728994, −6.70964770507679914891098966921, −6.44046765344276995638926741637, −6.37427263985763106421805804295, −5.98603825888204882559521096742, −5.95558643533625662167774421323, −5.94041434271820886963181240772, −5.13330555486621668149687958241, −5.10345893632829975520543937120, −4.82830616182995868583518684837, −4.53899548492450040581406166516, −4.05011782991061299022536111286, −4.03490312361028819506401755127, −3.37496728909433603178155820525, −3.27937032450332379375251778688, −3.12756591285512362338790717020, −2.63743298800753939922094079438, −2.50084636401619643182605497391, −2.46066635391127671907511046327, −1.91879505144826918516533917913, −1.89654747882353093101561332155, −1.47264789099290257727590035121, −0.941826935418002484495265503402, −0.63169999725042214737349061505, −0.46099944266943986910857999232, 0.46099944266943986910857999232, 0.63169999725042214737349061505, 0.941826935418002484495265503402, 1.47264789099290257727590035121, 1.89654747882353093101561332155, 1.91879505144826918516533917913, 2.46066635391127671907511046327, 2.50084636401619643182605497391, 2.63743298800753939922094079438, 3.12756591285512362338790717020, 3.27937032450332379375251778688, 3.37496728909433603178155820525, 4.03490312361028819506401755127, 4.05011782991061299022536111286, 4.53899548492450040581406166516, 4.82830616182995868583518684837, 5.10345893632829975520543937120, 5.13330555486621668149687958241, 5.94041434271820886963181240772, 5.95558643533625662167774421323, 5.98603825888204882559521096742, 6.37427263985763106421805804295, 6.44046765344276995638926741637, 6.70964770507679914891098966921, 6.91488933067814726053653728994

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