Properties

Label 6-2160e3-1.1-c3e3-0-0
Degree $6$
Conductor $10077696000$
Sign $1$
Analytic cond. $2.06994\times 10^{6}$
Root an. cond. $11.2891$
Motivic weight $3$
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  − 15·5-s − 9·7-s + 18·11-s − 21·13-s − 84·17-s − 21·19-s + 48·23-s + 150·25-s + 36·29-s − 324·31-s + 135·35-s + 33·37-s − 114·41-s − 282·43-s + 282·47-s − 360·49-s − 222·53-s − 270·55-s − 276·59-s + 303·61-s + 315·65-s − 1.03e3·67-s + 510·71-s + 447·73-s − 162·77-s − 777·79-s + 78·83-s + ⋯
L(s)  = 1  − 1.34·5-s − 0.485·7-s + 0.493·11-s − 0.448·13-s − 1.19·17-s − 0.253·19-s + 0.435·23-s + 6/5·25-s + 0.230·29-s − 1.87·31-s + 0.651·35-s + 0.146·37-s − 0.434·41-s − 1.00·43-s + 0.875·47-s − 1.04·49-s − 0.575·53-s − 0.661·55-s − 0.609·59-s + 0.635·61-s + 0.601·65-s − 1.88·67-s + 0.852·71-s + 0.716·73-s − 0.239·77-s − 1.10·79-s + 0.103·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{9} \cdot 5^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{9} \cdot 5^{3}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{12} \cdot 3^{9} \cdot 5^{3}\)
Sign: $1$
Analytic conductor: \(2.06994\times 10^{6}\)
Root analytic conductor: \(11.2891\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2160} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 2^{12} \cdot 3^{9} \cdot 5^{3} ,\ ( \ : 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(0.3111797289\)
\(L(\frac12)\) \(\approx\) \(0.3111797289\)
\(L(\frac{5}{2})\) 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 \( 1 \)
3 \( 1 \)
5$C_1$ \( ( 1 + p T )^{3} \)
good7$S_4\times C_2$ \( 1 + 9 T + 9 p^{2} T^{2} + 650 T^{3} + 9 p^{5} T^{4} + 9 p^{6} T^{5} + p^{9} T^{6} \)
11$S_4\times C_2$ \( 1 - 18 T + 912 T^{2} + 28510 T^{3} + 912 p^{3} T^{4} - 18 p^{6} T^{5} + p^{9} T^{6} \)
13$S_4\times C_2$ \( 1 + 21 T + 2910 T^{2} + 4721 p T^{3} + 2910 p^{3} T^{4} + 21 p^{6} T^{5} + p^{9} T^{6} \)
17$S_4\times C_2$ \( 1 + 84 T + 14706 T^{2} + 738652 T^{3} + 14706 p^{3} T^{4} + 84 p^{6} T^{5} + p^{9} T^{6} \)
19$S_4\times C_2$ \( 1 + 21 T + 15189 T^{2} + 149614 T^{3} + 15189 p^{3} T^{4} + 21 p^{6} T^{5} + p^{9} T^{6} \)
23$S_4\times C_2$ \( 1 - 48 T + 36654 T^{2} - 1165994 T^{3} + 36654 p^{3} T^{4} - 48 p^{6} T^{5} + p^{9} T^{6} \)
29$S_4\times C_2$ \( 1 - 36 T + 31512 T^{2} + 1278578 T^{3} + 31512 p^{3} T^{4} - 36 p^{6} T^{5} + p^{9} T^{6} \)
31$S_4\times C_2$ \( 1 + 324 T + 70560 T^{2} + 10091392 T^{3} + 70560 p^{3} T^{4} + 324 p^{6} T^{5} + p^{9} T^{6} \)
37$S_4\times C_2$ \( 1 - 33 T + 107211 T^{2} - 491142 T^{3} + 107211 p^{3} T^{4} - 33 p^{6} T^{5} + p^{9} T^{6} \)
41$S_4\times C_2$ \( 1 + 114 T + 125619 T^{2} + 13866756 T^{3} + 125619 p^{3} T^{4} + 114 p^{6} T^{5} + p^{9} T^{6} \)
43$S_4\times C_2$ \( 1 + 282 T + 220578 T^{2} + 43728712 T^{3} + 220578 p^{3} T^{4} + 282 p^{6} T^{5} + p^{9} T^{6} \)
47$S_4\times C_2$ \( 1 - 6 p T + 152826 T^{2} - 69794468 T^{3} + 152826 p^{3} T^{4} - 6 p^{7} T^{5} + p^{9} T^{6} \)
53$S_4\times C_2$ \( 1 + 222 T + 282615 T^{2} + 72010284 T^{3} + 282615 p^{3} T^{4} + 222 p^{6} T^{5} + p^{9} T^{6} \)
59$C_2$ \( ( 1 + 92 T + p^{3} T^{2} )^{3} \)
61$S_4\times C_2$ \( 1 - 303 T + 620067 T^{2} - 124220266 T^{3} + 620067 p^{3} T^{4} - 303 p^{6} T^{5} + p^{9} T^{6} \)
67$S_4\times C_2$ \( 1 + 1035 T + 63873 T^{2} - 227096926 T^{3} + 63873 p^{3} T^{4} + 1035 p^{6} T^{5} + p^{9} T^{6} \)
71$S_4\times C_2$ \( 1 - 510 T + 1098285 T^{2} - 353629860 T^{3} + 1098285 p^{3} T^{4} - 510 p^{6} T^{5} + p^{9} T^{6} \)
73$S_4\times C_2$ \( 1 - 447 T + 657111 T^{2} - 269206770 T^{3} + 657111 p^{3} T^{4} - 447 p^{6} T^{5} + p^{9} T^{6} \)
79$S_4\times C_2$ \( 1 + 777 T + 14268 p T^{2} + 780182781 T^{3} + 14268 p^{4} T^{4} + 777 p^{6} T^{5} + p^{9} T^{6} \)
83$S_4\times C_2$ \( 1 - 78 T + 882717 T^{2} - 176745340 T^{3} + 882717 p^{3} T^{4} - 78 p^{6} T^{5} + p^{9} T^{6} \)
89$S_4\times C_2$ \( 1 - 324 T + 1606299 T^{2} - 549274824 T^{3} + 1606299 p^{3} T^{4} - 324 p^{6} T^{5} + p^{9} T^{6} \)
97$S_4\times C_2$ \( 1 - 1191 T + 2555007 T^{2} - 2142250738 T^{3} + 2555007 p^{3} T^{4} - 1191 p^{6} T^{5} + p^{9} T^{6} \)
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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

−7.72482161689766603668784861581, −7.42281997367145703824912514491, −7.11628665538161078952480085531, −7.03314455088647665289434335523, −6.55699466174677142709572415380, −6.49536378972613917695361239799, −6.32682588842465689425818093719, −5.79449024619700863114740012463, −5.50374486558184156591473013830, −5.30517279458625158544587232190, −4.76160230446448533195050346214, −4.73117351206125087779913195293, −4.46595466996278286007341673799, −3.99223094162855468619207340301, −3.81194870470042003794609257339, −3.65140508153700652794686853834, −3.06807169338575925706661682475, −2.95361400622317892568025384428, −2.76176448811212950260594728768, −1.96972612622166197729846886069, −1.80085170543626496335521538349, −1.65872956166561162131142991731, −0.69715916994554157241350064901, −0.69680486274299395389776843223, −0.10528843580233186873871118968, 0.10528843580233186873871118968, 0.69680486274299395389776843223, 0.69715916994554157241350064901, 1.65872956166561162131142991731, 1.80085170543626496335521538349, 1.96972612622166197729846886069, 2.76176448811212950260594728768, 2.95361400622317892568025384428, 3.06807169338575925706661682475, 3.65140508153700652794686853834, 3.81194870470042003794609257339, 3.99223094162855468619207340301, 4.46595466996278286007341673799, 4.73117351206125087779913195293, 4.76160230446448533195050346214, 5.30517279458625158544587232190, 5.50374486558184156591473013830, 5.79449024619700863114740012463, 6.32682588842465689425818093719, 6.49536378972613917695361239799, 6.55699466174677142709572415380, 7.03314455088647665289434335523, 7.11628665538161078952480085531, 7.42281997367145703824912514491, 7.72482161689766603668784861581

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