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Properties

Label 6-75e3-1.1-c21e3-0-2
Degree $6$
Conductor $421875$
Sign $-1$
Analytic cond. $9.20923\times 10^{6}$
Root an. cond. $14.4778$
Motivic weight $21$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $3$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.30e3·2-s − 1.77e5·3-s + 1.31e5·4-s − 4.07e8·6-s − 4.65e8·7-s − 4.08e9·8-s + 2.09e10·9-s − 1.67e11·11-s − 2.32e10·12-s + 5.45e11·13-s − 1.07e12·14-s − 3.67e12·16-s + 8.10e12·17-s + 4.81e13·18-s + 3.93e12·19-s + 8.24e13·21-s − 3.84e14·22-s + 1.56e14·23-s + 7.22e14·24-s + 1.25e15·26-s − 2.05e15·27-s − 6.12e13·28-s − 9.30e14·29-s − 6.25e15·31-s − 1.18e15·32-s + 2.96e16·33-s + 1.86e16·34-s + ⋯
L(s)  = 1  + 1.58·2-s − 1.73·3-s + 0.0626·4-s − 2.75·6-s − 0.623·7-s − 1.34·8-s + 2·9-s − 1.94·11-s − 0.108·12-s + 1.09·13-s − 0.989·14-s − 0.835·16-s + 0.975·17-s + 3.17·18-s + 0.147·19-s + 1.07·21-s − 3.08·22-s + 0.786·23-s + 2.32·24-s + 1.74·26-s − 1.92·27-s − 0.0390·28-s − 0.410·29-s − 1.37·31-s − 0.186·32-s + 3.36·33-s + 1.54·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 421875 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 421875 ^{s/2} \, \Gamma_{\C}(s+21/2)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(421875\)    =    \(3^{3} \cdot 5^{6}\)
Sign: $-1$
Analytic conductor: \(9.20923\times 10^{6}\)
Root analytic conductor: \(14.4778\)
Motivic weight: \(21\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 421875,\ (\ :21/2, 21/2, 21/2),\ -1)\)

Particular Values

\(L(11)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{23}{2})\) 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$C_1$ \( ( 1 + p^{10} T )^{3} \)
5 \( 1 \)
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

−10.08386317434043130255547046666, −9.226434097597783526434955504308, −9.186056095221464982794758674743, −8.969371611845011947780187610136, −8.071241210127974596493881750820, −7.87206648912958637161171157265, −7.41723481989945095467099955879, −7.25352453154079232782140459795, −6.48043573026159608975663586783, −6.37838523850441532043543525995, −5.69800179184089843233709104568, −5.67151350806435644400970446257, −5.52267417409517508723218044831, −4.93856447043331759867500412093, −4.79557492972913619577258477114, −4.49401085686233049692723495980, −4.00818323985150359425661694648, −3.53726703008427095111706237110, −3.41845255834709053588244521575, −3.03308872062672741686854547053, −2.40163921182545775483721216692, −2.08323946446096562469763829785, −1.37127465923005077770130264313, −1.07794294927113487338004297358, −0.802520746609033305877807490256, 0, 0, 0, 0.802520746609033305877807490256, 1.07794294927113487338004297358, 1.37127465923005077770130264313, 2.08323946446096562469763829785, 2.40163921182545775483721216692, 3.03308872062672741686854547053, 3.41845255834709053588244521575, 3.53726703008427095111706237110, 4.00818323985150359425661694648, 4.49401085686233049692723495980, 4.79557492972913619577258477114, 4.93856447043331759867500412093, 5.52267417409517508723218044831, 5.67151350806435644400970446257, 5.69800179184089843233709104568, 6.37838523850441532043543525995, 6.48043573026159608975663586783, 7.25352453154079232782140459795, 7.41723481989945095467099955879, 7.87206648912958637161171157265, 8.071241210127974596493881750820, 8.969371611845011947780187610136, 9.186056095221464982794758674743, 9.226434097597783526434955504308, 10.08386317434043130255547046666

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