Properties

Label 6-9800e3-1.1-c1e3-0-1
Degree $6$
Conductor $941192000000$
Sign $1$
Analytic cond. $479191.$
Root an. cond. $8.84609$
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  + 3·11-s − 3·13-s − 6·17-s + 3·19-s − 3·23-s + 6·27-s + 12·29-s + 12·31-s − 9·37-s + 9·41-s − 12·43-s + 15·47-s − 9·53-s + 24·59-s − 6·61-s − 6·67-s − 18·79-s + 30·83-s + 6·97-s − 18·101-s + 6·103-s − 18·107-s + 6·109-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  + 0.904·11-s − 0.832·13-s − 1.45·17-s + 0.688·19-s − 0.625·23-s + 1.15·27-s + 2.22·29-s + 2.15·31-s − 1.47·37-s + 1.40·41-s − 1.82·43-s + 2.18·47-s − 1.23·53-s + 3.12·59-s − 0.768·61-s − 0.733·67-s − 2.02·79-s + 3.29·83-s + 0.609·97-s − 1.79·101-s + 0.591·103-s − 1.74·107-s + 0.574·109-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 5^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 5^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{9} \cdot 5^{6} \cdot 7^{6}\)
Sign: $1$
Analytic conductor: \(479191.\)
Root analytic conductor: \(8.84609\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{9800} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 2^{9} \cdot 5^{6} \cdot 7^{6} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.275626224\)
\(L(\frac12)\) \(\approx\) \(2.275626224\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 \)
good3$D_{6}$ \( 1 - 2 p T^{3} + p^{3} T^{6} \)
11$S_4\times C_2$ \( 1 - 3 T + 9 T^{2} - 2 p T^{3} + 9 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \)
13$S_4\times C_2$ \( 1 + 3 T + 15 T^{2} + 10 T^{3} + 15 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{3} \)
19$S_4\times C_2$ \( 1 - 3 T + 33 T^{2} - 130 T^{3} + 33 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 + 3 T + 54 T^{2} + 145 T^{3} + 54 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 - 12 T + 108 T^{2} - 670 T^{3} + 108 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 - 12 T + 105 T^{2} - 616 T^{3} + 105 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 + 9 T + 3 p T^{2} + 570 T^{3} + 3 p^{2} T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 - 9 T + 78 T^{2} - 357 T^{3} + 78 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 + 12 T + 168 T^{2} + 1054 T^{3} + 168 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 - 15 T + 45 T^{2} + 178 T^{3} + 45 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 + 9 T + 87 T^{2} + 330 T^{3} + 87 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \)
59$C_2$ \( ( 1 - 8 T + p T^{2} )^{3} \)
61$S_4\times C_2$ \( 1 + 6 T + 96 T^{2} + 188 T^{3} + 96 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 + 6 T + 132 T^{2} + 812 T^{3} + 132 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
71$C_2$ \( ( 1 + p T^{2} )^{3} \)
73$S_4\times C_2$ \( 1 + 111 T^{2} - 336 T^{3} + 111 p T^{4} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 + 18 T + 3 p T^{2} + 2076 T^{3} + 3 p^{2} T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 - 30 T + 540 T^{2} - 5884 T^{3} + 540 p T^{4} - 30 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 + 240 T^{2} + 42 T^{3} + 240 p T^{4} + p^{3} T^{6} \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{3} \)
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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

−6.68122409044435126426704402157, −6.64420061917747104628153655563, −6.40828839799576500647650957395, −6.36586205418435968618229585060, −5.67872403750231755162011090531, −5.63908813239856305697544764078, −5.53214096904552982464867162563, −5.03586168565196019868266340216, −4.89317413861182099315269531892, −4.52306765862731935660300389227, −4.48781379162705647760970014303, −4.35695122352475839599642793456, −4.04607450602788549480953838947, −3.58064795246866162811108579113, −3.45440135599920505531046224731, −3.18739309050141279171177749702, −2.65143889871719567822435566408, −2.61339436289299654333652692531, −2.53947019993699431631820961406, −1.96905965839362431693077911145, −1.85787885630984362399323552514, −1.19846087975663751007791416174, −1.05057541153073013845727373776, −0.816572692592439060531618931317, −0.23214942092026475231784630549, 0.23214942092026475231784630549, 0.816572692592439060531618931317, 1.05057541153073013845727373776, 1.19846087975663751007791416174, 1.85787885630984362399323552514, 1.96905965839362431693077911145, 2.53947019993699431631820961406, 2.61339436289299654333652692531, 2.65143889871719567822435566408, 3.18739309050141279171177749702, 3.45440135599920505531046224731, 3.58064795246866162811108579113, 4.04607450602788549480953838947, 4.35695122352475839599642793456, 4.48781379162705647760970014303, 4.52306765862731935660300389227, 4.89317413861182099315269531892, 5.03586168565196019868266340216, 5.53214096904552982464867162563, 5.63908813239856305697544764078, 5.67872403750231755162011090531, 6.36586205418435968618229585060, 6.40828839799576500647650957395, 6.64420061917747104628153655563, 6.68122409044435126426704402157

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