Properties

Label 6-9680e3-1.1-c1e3-0-4
Degree $6$
Conductor $907039232000$
Sign $1$
Analytic cond. $461803.$
Root an. cond. $8.79176$
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-s + 3·5-s + 7-s + 3·15-s + 7·17-s − 3·19-s + 21-s + 4·23-s + 6·25-s + 27-s + 5·29-s + 17·31-s + 3·35-s + 37-s − 2·41-s − 16·43-s + 6·47-s − 12·49-s + 7·51-s + 53-s − 3·57-s + 4·59-s + 11·61-s + 16·67-s + 4·69-s − 71-s − 22·73-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.34·5-s + 0.377·7-s + 0.774·15-s + 1.69·17-s − 0.688·19-s + 0.218·21-s + 0.834·23-s + 6/5·25-s + 0.192·27-s + 0.928·29-s + 3.05·31-s + 0.507·35-s + 0.164·37-s − 0.312·41-s − 2.43·43-s + 0.875·47-s − 1.71·49-s + 0.980·51-s + 0.137·53-s − 0.397·57-s + 0.520·59-s + 1.40·61-s + 1.95·67-s + 0.481·69-s − 0.118·71-s − 2.57·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{3} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{3} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{12} \cdot 5^{3} \cdot 11^{6}\)
Sign: $1$
Analytic conductor: \(461803.\)
Root analytic conductor: \(8.79176\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 2^{12} \cdot 5^{3} \cdot 11^{6} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(11.75985123\)
\(L(\frac12)\) \(\approx\) \(11.75985123\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{3} \)
11 \( 1 \)
good3$S_4\times C_2$ \( 1 - T + T^{2} - 2 T^{3} + p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
7$S_4\times C_2$ \( 1 - T + 13 T^{2} - 10 T^{3} + 13 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
13$S_4\times C_2$ \( 1 - T^{2} + 64 T^{3} - p T^{4} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 - 7 T + 59 T^{2} - 230 T^{3} + 59 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \)
19$S_4\times C_2$ \( 1 + 3 T + 17 T^{2} + 130 T^{3} + 17 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 - 4 T + 49 T^{2} - 120 T^{3} + 49 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 - 5 T + 71 T^{2} - 226 T^{3} + 71 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 - 17 T + 165 T^{2} - 1070 T^{3} + 165 p T^{4} - 17 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 - T + 87 T^{2} - 94 T^{3} + 87 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 + 2 T + 91 T^{2} + 132 T^{3} + 91 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 + 16 T + 189 T^{2} + 1360 T^{3} + 189 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 - 6 T + 113 T^{2} - 556 T^{3} + 113 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 - T + 135 T^{2} - 126 T^{3} + 135 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 - 4 T + 81 T^{2} - 600 T^{3} + 81 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 - 11 T + 95 T^{2} - 6 p T^{3} + 95 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 - 16 T + 165 T^{2} - 1168 T^{3} + 165 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 + T + 85 T^{2} + 654 T^{3} + 85 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 + 22 T + 355 T^{2} + 3388 T^{3} + 355 p T^{4} + 22 p^{2} T^{5} + p^{3} T^{6} \)
79$C_2$ \( ( 1 + 12 T + p T^{2} )^{3} \)
83$S_4\times C_2$ \( 1 + 21 T^{2} + 880 T^{3} + 21 p T^{4} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 + 3 T + 83 T^{2} + 1138 T^{3} + 83 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
97$C_2$ \( ( 1 - 6 T + p T^{2} )^{3} \)
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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

−6.63128699194487440843466359479, −6.62887424590716179400721677078, −6.24550499630155809537551097231, −6.14942655120269324759598888743, −5.95407448680879077986627440024, −5.68843655042114507259147379968, −5.31440924819345913393596772571, −5.14220123066984480179456506416, −4.84870048181756176888336655361, −4.84718785782607107581296774015, −4.55886187081697257717942202768, −4.20497667000711147105680773082, −3.97163335420475411829806373435, −3.45631293117335009361436578699, −3.45335110407644180131352012751, −3.08079211108406218832315421688, −2.70057732770798090186206355022, −2.63556625974732090561476218092, −2.60603100049963760950165201406, −1.89853826540730011602315577343, −1.73359930492885931557187196391, −1.46775641696807008546089622876, −1.14766842624033893569676927141, −0.796401126466010236973448118470, −0.46297039293501990590908250169, 0.46297039293501990590908250169, 0.796401126466010236973448118470, 1.14766842624033893569676927141, 1.46775641696807008546089622876, 1.73359930492885931557187196391, 1.89853826540730011602315577343, 2.60603100049963760950165201406, 2.63556625974732090561476218092, 2.70057732770798090186206355022, 3.08079211108406218832315421688, 3.45335110407644180131352012751, 3.45631293117335009361436578699, 3.97163335420475411829806373435, 4.20497667000711147105680773082, 4.55886187081697257717942202768, 4.84718785782607107581296774015, 4.84870048181756176888336655361, 5.14220123066984480179456506416, 5.31440924819345913393596772571, 5.68843655042114507259147379968, 5.95407448680879077986627440024, 6.14942655120269324759598888743, 6.24550499630155809537551097231, 6.62887424590716179400721677078, 6.63128699194487440843466359479

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