Properties

Degree 6
Conductor $ 2^{3} $
Sign $1$
Motivic weight 83
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 6.59e12·2-s − 1.01e20·3-s + 2.90e25·4-s − 3.01e27·5-s + 6.70e32·6-s − 5.68e34·7-s − 1.06e38·8-s − 1.46e39·9-s + 1.99e40·10-s − 1.49e43·11-s − 2.95e45·12-s − 1.24e45·13-s + 3.75e47·14-s + 3.06e47·15-s + 3.50e50·16-s + 3.08e50·17-s + 9.65e51·18-s − 1.15e53·19-s − 8.75e52·20-s + 5.78e54·21-s + 9.86e55·22-s + 1.94e56·23-s + 1.08e58·24-s − 9.05e57·25-s + 8.23e57·26-s + 5.37e59·27-s − 1.64e60·28-s + ⋯
L(s)  = 1  − 2.12·2-s − 1.60·3-s + 3·4-s − 0.0296·5-s + 3.41·6-s − 0.482·7-s − 3.53·8-s − 0.366·9-s + 0.0629·10-s − 0.905·11-s − 4.82·12-s − 0.0737·13-s + 1.02·14-s + 0.0477·15-s + 15/4·16-s + 0.266·17-s + 0.777·18-s − 0.987·19-s − 0.0890·20-s + 0.776·21-s + 1.92·22-s + 0.599·23-s + 5.69·24-s − 0.875·25-s + 0.156·26-s + 2.13·27-s − 1.44·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(84-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+83/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(6\)
\( N \)  =  \(8\)    =    \(2^{3}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(83\)
character  :  induced by $\chi_{2} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(6,\ 8,\ (\ :83/2, 83/2, 83/2),\ 1)$
$L(42)$  $\approx$  $0.01857966202$
$L(\frac12)$  $\approx$  $0.01857966202$
$L(\frac{85}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 2$, \(F_p(T)\) is a polynomial of degree 6. If $p = 2$, then $F_p(T)$ is a polynomial of degree at most 5.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_1$ \( ( 1 + p^{41} T )^{3} \)
good3$S_4\times C_2$ \( 1 + 3766768380958173188 p^{3} T + \)\(66\!\cdots\!59\)\( p^{11} T^{2} + \)\(95\!\cdots\!44\)\( p^{25} T^{3} + \)\(66\!\cdots\!59\)\( p^{94} T^{4} + 3766768380958173188 p^{169} T^{5} + p^{249} T^{6} \)
5$S_4\times C_2$ \( 1 + \)\(48\!\cdots\!06\)\( p^{4} T + \)\(18\!\cdots\!43\)\( p^{11} T^{2} - \)\(33\!\cdots\!28\)\( p^{22} T^{3} + \)\(18\!\cdots\!43\)\( p^{94} T^{4} + \)\(48\!\cdots\!06\)\( p^{170} T^{5} + p^{249} T^{6} \)
7$S_4\times C_2$ \( 1 + \)\(11\!\cdots\!12\)\( p^{2} T + \)\(59\!\cdots\!77\)\( p^{8} T^{2} + \)\(45\!\cdots\!04\)\( p^{16} T^{3} + \)\(59\!\cdots\!77\)\( p^{91} T^{4} + \)\(11\!\cdots\!12\)\( p^{168} T^{5} + p^{249} T^{6} \)
11$S_4\times C_2$ \( 1 + \)\(13\!\cdots\!24\)\( p T + \)\(46\!\cdots\!25\)\( p^{2} T^{2} + \)\(34\!\cdots\!40\)\( p^{7} T^{3} + \)\(46\!\cdots\!25\)\( p^{85} T^{4} + \)\(13\!\cdots\!24\)\( p^{167} T^{5} + p^{249} T^{6} \)
13$S_4\times C_2$ \( 1 + \)\(73\!\cdots\!54\)\( p^{2} T + \)\(79\!\cdots\!87\)\( p^{6} T^{2} + \)\(12\!\cdots\!36\)\( p^{11} T^{3} + \)\(79\!\cdots\!87\)\( p^{89} T^{4} + \)\(73\!\cdots\!54\)\( p^{168} T^{5} + p^{249} T^{6} \)
17$S_4\times C_2$ \( 1 - \)\(18\!\cdots\!06\)\( p T + \)\(13\!\cdots\!63\)\( p^{2} T^{2} + \)\(12\!\cdots\!92\)\( p^{5} T^{3} + \)\(13\!\cdots\!63\)\( p^{85} T^{4} - \)\(18\!\cdots\!06\)\( p^{167} T^{5} + p^{249} T^{6} \)
19$S_4\times C_2$ \( 1 + \)\(60\!\cdots\!00\)\( p T + \)\(20\!\cdots\!03\)\( p^{3} T^{2} - \)\(23\!\cdots\!00\)\( p^{6} T^{3} + \)\(20\!\cdots\!03\)\( p^{86} T^{4} + \)\(60\!\cdots\!00\)\( p^{167} T^{5} + p^{249} T^{6} \)
23$S_4\times C_2$ \( 1 - \)\(84\!\cdots\!48\)\( p T + \)\(41\!\cdots\!37\)\( p^{2} T^{2} - \)\(76\!\cdots\!48\)\( p^{4} T^{3} + \)\(41\!\cdots\!37\)\( p^{85} T^{4} - \)\(84\!\cdots\!48\)\( p^{167} T^{5} + p^{249} T^{6} \)
29$S_4\times C_2$ \( 1 + \)\(13\!\cdots\!70\)\( T + \)\(41\!\cdots\!23\)\( p T^{2} + \)\(27\!\cdots\!40\)\( p^{3} T^{3} + \)\(41\!\cdots\!23\)\( p^{84} T^{4} + \)\(13\!\cdots\!70\)\( p^{166} T^{5} + p^{249} T^{6} \)
31$S_4\times C_2$ \( 1 + \)\(15\!\cdots\!44\)\( T + \)\(49\!\cdots\!35\)\( p T^{2} - \)\(32\!\cdots\!00\)\( p^{3} T^{3} + \)\(49\!\cdots\!35\)\( p^{84} T^{4} + \)\(15\!\cdots\!44\)\( p^{166} T^{5} + p^{249} T^{6} \)
37$S_4\times C_2$ \( 1 + \)\(39\!\cdots\!54\)\( p T + \)\(35\!\cdots\!59\)\( p^{3} T^{2} - \)\(18\!\cdots\!88\)\( p^{5} T^{3} + \)\(35\!\cdots\!59\)\( p^{86} T^{4} + \)\(39\!\cdots\!54\)\( p^{167} T^{5} + p^{249} T^{6} \)
41$S_4\times C_2$ \( 1 + \)\(36\!\cdots\!54\)\( p T + \)\(14\!\cdots\!95\)\( p^{2} T^{2} + \)\(27\!\cdots\!60\)\( p^{3} T^{3} + \)\(14\!\cdots\!95\)\( p^{85} T^{4} + \)\(36\!\cdots\!54\)\( p^{167} T^{5} + p^{249} T^{6} \)
43$S_4\times C_2$ \( 1 + \)\(18\!\cdots\!96\)\( T + \)\(21\!\cdots\!93\)\( T^{2} + \)\(36\!\cdots\!84\)\( p T^{3} + \)\(21\!\cdots\!93\)\( p^{83} T^{4} + \)\(18\!\cdots\!96\)\( p^{166} T^{5} + p^{249} T^{6} \)
47$S_4\times C_2$ \( 1 + \)\(34\!\cdots\!88\)\( T + \)\(15\!\cdots\!17\)\( T^{2} + \)\(74\!\cdots\!72\)\( p T^{3} + \)\(15\!\cdots\!17\)\( p^{83} T^{4} + \)\(34\!\cdots\!88\)\( p^{166} T^{5} + p^{249} T^{6} \)
53$S_4\times C_2$ \( 1 + \)\(38\!\cdots\!66\)\( T + \)\(37\!\cdots\!11\)\( p T^{2} + \)\(80\!\cdots\!68\)\( p^{2} T^{3} + \)\(37\!\cdots\!11\)\( p^{84} T^{4} + \)\(38\!\cdots\!66\)\( p^{166} T^{5} + p^{249} T^{6} \)
59$S_4\times C_2$ \( 1 - \)\(31\!\cdots\!60\)\( T + \)\(25\!\cdots\!37\)\( T^{2} - \)\(83\!\cdots\!20\)\( p T^{3} + \)\(25\!\cdots\!37\)\( p^{83} T^{4} - \)\(31\!\cdots\!60\)\( p^{166} T^{5} + p^{249} T^{6} \)
61$S_4\times C_2$ \( 1 + \)\(49\!\cdots\!54\)\( p T + \)\(16\!\cdots\!55\)\( p^{2} T^{2} + \)\(41\!\cdots\!20\)\( p^{3} T^{3} + \)\(16\!\cdots\!55\)\( p^{85} T^{4} + \)\(49\!\cdots\!54\)\( p^{167} T^{5} + p^{249} T^{6} \)
67$S_4\times C_2$ \( 1 + \)\(98\!\cdots\!28\)\( T + \)\(49\!\cdots\!51\)\( p T^{2} - \)\(47\!\cdots\!64\)\( p^{2} T^{3} + \)\(49\!\cdots\!51\)\( p^{84} T^{4} + \)\(98\!\cdots\!28\)\( p^{166} T^{5} + p^{249} T^{6} \)
71$S_4\times C_2$ \( 1 - \)\(81\!\cdots\!96\)\( T + \)\(18\!\cdots\!55\)\( p T^{2} - \)\(13\!\cdots\!80\)\( p^{2} T^{3} + \)\(18\!\cdots\!55\)\( p^{84} T^{4} - \)\(81\!\cdots\!96\)\( p^{166} T^{5} + p^{249} T^{6} \)
73$S_4\times C_2$ \( 1 - \)\(75\!\cdots\!18\)\( p T + \)\(43\!\cdots\!27\)\( p^{2} T^{2} - \)\(14\!\cdots\!44\)\( p^{3} T^{3} + \)\(43\!\cdots\!27\)\( p^{85} T^{4} - \)\(75\!\cdots\!18\)\( p^{167} T^{5} + p^{249} T^{6} \)
79$S_4\times C_2$ \( 1 + \)\(65\!\cdots\!80\)\( p^{2} T + \)\(10\!\cdots\!37\)\( p^{2} T^{2} + \)\(60\!\cdots\!60\)\( p^{3} T^{3} + \)\(10\!\cdots\!37\)\( p^{85} T^{4} + \)\(65\!\cdots\!80\)\( p^{168} T^{5} + p^{249} T^{6} \)
83$S_4\times C_2$ \( 1 - \)\(52\!\cdots\!84\)\( T + \)\(37\!\cdots\!13\)\( T^{2} - \)\(94\!\cdots\!68\)\( T^{3} + \)\(37\!\cdots\!13\)\( p^{83} T^{4} - \)\(52\!\cdots\!84\)\( p^{166} T^{5} + p^{249} T^{6} \)
89$S_4\times C_2$ \( 1 - \)\(73\!\cdots\!30\)\( T - \)\(73\!\cdots\!93\)\( T^{2} + \)\(70\!\cdots\!60\)\( T^{3} - \)\(73\!\cdots\!93\)\( p^{83} T^{4} - \)\(73\!\cdots\!30\)\( p^{166} T^{5} + p^{249} T^{6} \)
97$S_4\times C_2$ \( 1 - \)\(11\!\cdots\!82\)\( T + \)\(67\!\cdots\!27\)\( T^{2} - \)\(23\!\cdots\!56\)\( T^{3} + \)\(67\!\cdots\!27\)\( p^{83} T^{4} - \)\(11\!\cdots\!82\)\( p^{166} T^{5} + p^{249} T^{6} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−11.31778640454824613380333939006, −10.88167920773412221064390714451, −10.51144590641007809546441433925, −10.10692298276767702176028971077, −9.446593554235572662888117061235, −9.341850961182419423369906951027, −8.593948661947357553317514111734, −8.157539316703607536004586605043, −7.894911232942774342096213026871, −7.39198553362586102701435136574, −6.58905180427988984388776399401, −6.42883167997121748803007677234, −6.20535689093710138932044460470, −5.38990677386099909510799920598, −5.24326065145192439281204632879, −4.89231643953424577113814838626, −3.59039794397489261291743093431, −3.36477498500808209903509353717, −3.02793283270186350080828190482, −2.01471965848111635953369991381, −2.01410371192593249767918229774, −1.60325493786589936209995264031, −0.791014437359105890905061053263, −0.27143987742286290018096475355, −0.11179335629051929744315893201, 0.11179335629051929744315893201, 0.27143987742286290018096475355, 0.791014437359105890905061053263, 1.60325493786589936209995264031, 2.01410371192593249767918229774, 2.01471965848111635953369991381, 3.02793283270186350080828190482, 3.36477498500808209903509353717, 3.59039794397489261291743093431, 4.89231643953424577113814838626, 5.24326065145192439281204632879, 5.38990677386099909510799920598, 6.20535689093710138932044460470, 6.42883167997121748803007677234, 6.58905180427988984388776399401, 7.39198553362586102701435136574, 7.894911232942774342096213026871, 8.157539316703607536004586605043, 8.593948661947357553317514111734, 9.341850961182419423369906951027, 9.446593554235572662888117061235, 10.10692298276767702176028971077, 10.51144590641007809546441433925, 10.88167920773412221064390714451, 11.31778640454824613380333939006

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