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Properties

Label 6-45e3-1.1-c21e3-0-0
Degree $6$
Conductor $91125$
Sign $1$
Analytic cond. $1.98919\times 10^{6}$
Root an. cond. $11.2144$
Motivic weight $21$
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  + 2.30e3·2-s + 1.31e5·4-s − 2.92e7·5-s + 4.65e8·7-s − 4.08e9·8-s − 6.73e10·10-s + 1.67e11·11-s − 5.45e11·13-s + 1.07e12·14-s − 3.67e12·16-s + 8.10e12·17-s + 3.93e12·19-s − 3.85e12·20-s + 3.84e14·22-s + 1.56e14·23-s + 5.72e14·25-s − 1.25e15·26-s + 6.12e13·28-s + 9.30e14·29-s − 6.25e15·31-s − 1.18e15·32-s + 1.86e16·34-s − 1.36e16·35-s − 2.22e16·37-s + 9.05e15·38-s + 1.19e17·40-s + 1.86e17·41-s + ⋯
L(s)  = 1  + 1.58·2-s + 0.0626·4-s − 1.34·5-s + 0.623·7-s − 1.34·8-s − 2.13·10-s + 1.94·11-s − 1.09·13-s + 0.989·14-s − 0.835·16-s + 0.975·17-s + 0.147·19-s − 0.0841·20-s + 3.08·22-s + 0.786·23-s + 6/5·25-s − 1.74·26-s + 0.0390·28-s + 0.410·29-s − 1.37·31-s − 0.186·32-s + 1.54·34-s − 0.835·35-s − 0.760·37-s + 0.234·38-s + 1.80·40-s + 2.16·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91125 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91125 ^{s/2} \, \Gamma_{\C}(s+21/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(91125\)    =    \(3^{6} \cdot 5^{3}\)
Sign: $1$
Analytic conductor: \(1.98919\times 10^{6}\)
Root analytic conductor: \(11.2144\)
Motivic weight: \(21\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 91125,\ (\ :21/2, 21/2, 21/2),\ 1)\)

Particular Values

\(L(11)\) \(\approx\) \(3.947493893\)
\(L(\frac12)\) \(\approx\) \(3.947493893\)
\(L(\frac{23}{2})\) 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)$
bad3 \( 1 \)
5$C_1$ \( ( 1 + p^{10} T )^{3} \)
good2$S_4\times C_2$ \( 1 - 575 p^{2} T + 40301 p^{7} T^{2} - 913219 p^{13} T^{3} + 40301 p^{28} T^{4} - 575 p^{44} T^{5} + p^{63} T^{6} \)
7$S_4\times C_2$ \( 1 - 465666872 T + 177059815791280803 p T^{2} - \)\(96\!\cdots\!92\)\( p^{2} T^{3} + 177059815791280803 p^{22} T^{4} - 465666872 p^{42} T^{5} + p^{63} T^{6} \)
11$S_4\times C_2$ \( 1 - 167336332556 T + \)\(26\!\cdots\!33\)\( T^{2} - \)\(22\!\cdots\!00\)\( T^{3} + \)\(26\!\cdots\!33\)\( p^{21} T^{4} - 167336332556 p^{42} T^{5} + p^{63} T^{6} \)
13$S_4\times C_2$ \( 1 + 41967002606 p T + \)\(27\!\cdots\!39\)\( T^{2} + \)\(83\!\cdots\!48\)\( p T^{3} + \)\(27\!\cdots\!39\)\( p^{21} T^{4} + 41967002606 p^{43} T^{5} + p^{63} T^{6} \)
17$S_4\times C_2$ \( 1 - 8104424487194 T + \)\(72\!\cdots\!39\)\( p T^{2} - \)\(40\!\cdots\!92\)\( p^{2} T^{3} + \)\(72\!\cdots\!39\)\( p^{22} T^{4} - 8104424487194 p^{42} T^{5} + p^{63} T^{6} \)
19$S_4\times C_2$ \( 1 - 207247407412 p T + \)\(21\!\cdots\!53\)\( p^{2} T^{2} - \)\(23\!\cdots\!76\)\( p^{3} T^{3} + \)\(21\!\cdots\!53\)\( p^{23} T^{4} - 207247407412 p^{43} T^{5} + p^{63} T^{6} \)
23$S_4\times C_2$ \( 1 - 156235274730744 T + \)\(40\!\cdots\!69\)\( T^{2} - \)\(26\!\cdots\!24\)\( T^{3} + \)\(40\!\cdots\!69\)\( p^{21} T^{4} - 156235274730744 p^{42} T^{5} + p^{63} T^{6} \)
29$S_4\times C_2$ \( 1 - 930273612785494 T + \)\(10\!\cdots\!51\)\( T^{2} - \)\(38\!\cdots\!12\)\( T^{3} + \)\(10\!\cdots\!51\)\( p^{21} T^{4} - 930273612785494 p^{42} T^{5} + p^{63} T^{6} \)
31$S_4\times C_2$ \( 1 + 6257709152718928 T + \)\(67\!\cdots\!93\)\( T^{2} + \)\(26\!\cdots\!36\)\( T^{3} + \)\(67\!\cdots\!93\)\( p^{21} T^{4} + 6257709152718928 p^{42} T^{5} + p^{63} T^{6} \)
37$S_4\times C_2$ \( 1 + 22246337613227118 T + \)\(14\!\cdots\!71\)\( T^{2} + \)\(42\!\cdots\!32\)\( T^{3} + \)\(14\!\cdots\!71\)\( p^{21} T^{4} + 22246337613227118 p^{42} T^{5} + p^{63} T^{6} \)
41$S_4\times C_2$ \( 1 - 186265908060974338 T + \)\(78\!\cdots\!43\)\( p T^{2} - \)\(29\!\cdots\!16\)\( T^{3} + \)\(78\!\cdots\!43\)\( p^{22} T^{4} - 186265908060974338 p^{42} T^{5} + p^{63} T^{6} \)
43$S_4\times C_2$ \( 1 + 268609288174096316 T + \)\(78\!\cdots\!53\)\( T^{2} + \)\(11\!\cdots\!60\)\( T^{3} + \)\(78\!\cdots\!53\)\( p^{21} T^{4} + 268609288174096316 p^{42} T^{5} + p^{63} T^{6} \)
47$S_4\times C_2$ \( 1 - 900034127817222032 T + \)\(59\!\cdots\!37\)\( T^{2} - \)\(23\!\cdots\!80\)\( T^{3} + \)\(59\!\cdots\!37\)\( p^{21} T^{4} - 900034127817222032 p^{42} T^{5} + p^{63} T^{6} \)
53$S_4\times C_2$ \( 1 - 1269623243180583374 T + \)\(14\!\cdots\!39\)\( T^{2} + \)\(47\!\cdots\!56\)\( T^{3} + \)\(14\!\cdots\!39\)\( p^{21} T^{4} - 1269623243180583374 p^{42} T^{5} + p^{63} T^{6} \)
59$S_4\times C_2$ \( 1 - 8551487099411338268 T + \)\(56\!\cdots\!73\)\( T^{2} - \)\(26\!\cdots\!84\)\( T^{3} + \)\(56\!\cdots\!73\)\( p^{21} T^{4} - 8551487099411338268 p^{42} T^{5} + p^{63} T^{6} \)
61$S_4\times C_2$ \( 1 + 7181148471323735222 T + \)\(10\!\cdots\!59\)\( T^{2} + \)\(44\!\cdots\!56\)\( T^{3} + \)\(10\!\cdots\!59\)\( p^{21} T^{4} + 7181148471323735222 p^{42} T^{5} + p^{63} T^{6} \)
67$S_4\times C_2$ \( 1 - 2946635148405656396 T + \)\(36\!\cdots\!73\)\( T^{2} + \)\(10\!\cdots\!68\)\( T^{3} + \)\(36\!\cdots\!73\)\( p^{21} T^{4} - 2946635148405656396 p^{42} T^{5} + p^{63} T^{6} \)
71$S_4\times C_2$ \( 1 - 37849731561832987624 T + \)\(84\!\cdots\!05\)\( T^{2} - \)\(58\!\cdots\!20\)\( T^{3} + \)\(84\!\cdots\!05\)\( p^{21} T^{4} - 37849731561832987624 p^{42} T^{5} + p^{63} T^{6} \)
73$S_4\times C_2$ \( 1 + 7149058835824826594 T + \)\(83\!\cdots\!39\)\( T^{2} + \)\(81\!\cdots\!24\)\( T^{3} + \)\(83\!\cdots\!39\)\( p^{21} T^{4} + 7149058835824826594 p^{42} T^{5} + p^{63} T^{6} \)
79$S_4\times C_2$ \( 1 - \)\(30\!\cdots\!00\)\( T + \)\(50\!\cdots\!37\)\( T^{2} - \)\(52\!\cdots\!00\)\( T^{3} + \)\(50\!\cdots\!37\)\( p^{21} T^{4} - \)\(30\!\cdots\!00\)\( p^{42} T^{5} + p^{63} T^{6} \)
83$S_4\times C_2$ \( 1 + \)\(19\!\cdots\!80\)\( T + \)\(67\!\cdots\!37\)\( T^{2} + \)\(77\!\cdots\!44\)\( T^{3} + \)\(67\!\cdots\!37\)\( p^{21} T^{4} + \)\(19\!\cdots\!80\)\( p^{42} T^{5} + p^{63} T^{6} \)
89$S_4\times C_2$ \( 1 + \)\(16\!\cdots\!38\)\( T + \)\(12\!\cdots\!03\)\( T^{2} + \)\(41\!\cdots\!64\)\( T^{3} + \)\(12\!\cdots\!03\)\( p^{21} T^{4} + \)\(16\!\cdots\!38\)\( p^{42} T^{5} + p^{63} T^{6} \)
97$S_4\times C_2$ \( 1 + \)\(12\!\cdots\!54\)\( T + \)\(13\!\cdots\!63\)\( T^{2} + \)\(10\!\cdots\!08\)\( T^{3} + \)\(13\!\cdots\!63\)\( p^{21} T^{4} + \)\(12\!\cdots\!54\)\( p^{42} T^{5} + p^{63} T^{6} \)
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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

−10.33345155017692909786317624407, −9.367521428896148039255941039408, −9.298404705597883102847862730836, −9.157216901054178425222219977391, −8.355381688447850844467796732603, −8.216298746334524550534206733170, −7.67713575976551129971450841988, −7.25804940060755980960109332552, −6.77580345857464314210887521041, −6.76376768866397102012494928622, −5.78012521229111048927917679296, −5.57474196846547401488620095613, −5.17745802409909081054781858951, −4.70747926316199739286147433175, −4.37755596931862454633534939895, −4.32942967644414584371168425737, −3.52582439359354811509256135224, −3.49878089601175179437164702551, −3.41340509201448685703420026433, −2.36458958221560632926128730066, −2.21983737381598459435588831124, −1.41172469132609629288555915333, −1.02713719181479015870592617451, −0.75906779354436926969320585849, −0.22887907292494402519623239194, 0.22887907292494402519623239194, 0.75906779354436926969320585849, 1.02713719181479015870592617451, 1.41172469132609629288555915333, 2.21983737381598459435588831124, 2.36458958221560632926128730066, 3.41340509201448685703420026433, 3.49878089601175179437164702551, 3.52582439359354811509256135224, 4.32942967644414584371168425737, 4.37755596931862454633534939895, 4.70747926316199739286147433175, 5.17745802409909081054781858951, 5.57474196846547401488620095613, 5.78012521229111048927917679296, 6.76376768866397102012494928622, 6.77580345857464314210887521041, 7.25804940060755980960109332552, 7.67713575976551129971450841988, 8.216298746334524550534206733170, 8.355381688447850844467796732603, 9.157216901054178425222219977391, 9.298404705597883102847862730836, 9.367521428896148039255941039408, 10.33345155017692909786317624407

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