Properties

Label 6-75e3-1.1-c17e3-0-1
Degree $6$
Conductor $421875$
Sign $-1$
Analytic cond. $2.59487\times 10^{6}$
Root an. cond. $11.7224$
Motivic weight $17$
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  + 253·2-s − 1.96e4·3-s − 1.68e5·4-s − 4.97e6·6-s + 4.33e6·7-s − 5.35e7·8-s + 2.58e8·9-s + 9.43e8·11-s + 3.30e9·12-s − 4.25e9·13-s + 1.09e9·14-s + 7.11e9·16-s − 3.06e9·17-s + 6.53e10·18-s − 7.81e10·19-s − 8.52e10·21-s + 2.38e11·22-s − 2.34e11·23-s + 1.05e12·24-s − 1.07e12·26-s − 2.82e12·27-s − 7.28e11·28-s + 5.77e12·29-s + 5.56e12·31-s + 2.87e12·32-s − 1.85e13·33-s − 7.74e11·34-s + ⋯
L(s)  = 1  + 0.698·2-s − 1.73·3-s − 1.28·4-s − 1.21·6-s + 0.284·7-s − 1.12·8-s + 2·9-s + 1.32·11-s + 2.22·12-s − 1.44·13-s + 0.198·14-s + 0.413·16-s − 0.106·17-s + 1.39·18-s − 1.05·19-s − 0.491·21-s + 0.927·22-s − 0.623·23-s + 1.95·24-s − 1.01·26-s − 1.92·27-s − 0.364·28-s + 2.14·29-s + 1.17·31-s + 0.461·32-s − 2.29·33-s − 0.0744·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 421875 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 421875 ^{s/2} \, \Gamma_{\C}(s+17/2)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(421875\)    =    \(3^{3} \cdot 5^{6}\)
Sign: $-1$
Analytic conductor: \(2.59487\times 10^{6}\)
Root analytic conductor: \(11.7224\)
Motivic weight: \(17\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 421875,\ (\ :17/2, 17/2, 17/2),\ -1)\)

Particular Values

\(L(9)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{19}{2})\) 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$C_1$ \( ( 1 + p^{8} T )^{3} \)
5 \( 1 \)
good2$S_4\times C_2$ \( 1 - 253 T + 58039 p^{2} T^{2} - 744997 p^{6} T^{3} + 58039 p^{19} T^{4} - 253 p^{34} T^{5} + p^{51} T^{6} \)
7$S_4\times C_2$ \( 1 - 4332484 T + 944156890755 p^{3} T^{2} - 1789629686794678136 p^{4} T^{3} + 944156890755 p^{20} T^{4} - 4332484 p^{34} T^{5} + p^{51} T^{6} \)
11$S_4\times C_2$ \( 1 - 943563680 T + 111526088429338691 p T^{2} - \)\(59\!\cdots\!44\)\( p^{2} T^{3} + 111526088429338691 p^{18} T^{4} - 943563680 p^{34} T^{5} + p^{51} T^{6} \)
13$S_4\times C_2$ \( 1 + 327462550 p T + 61448977723329179 p^{2} T^{2} + \)\(47\!\cdots\!76\)\( p^{3} T^{3} + 61448977723329179 p^{19} T^{4} + 327462550 p^{35} T^{5} + p^{51} T^{6} \)
17$S_4\times C_2$ \( 1 + 180186442 p T + \)\(33\!\cdots\!63\)\( T^{2} + \)\(63\!\cdots\!28\)\( T^{3} + \)\(33\!\cdots\!63\)\( p^{17} T^{4} + 180186442 p^{35} T^{5} + p^{51} T^{6} \)
19$S_4\times C_2$ \( 1 + 78122492996 T + \)\(17\!\cdots\!77\)\( T^{2} + \)\(84\!\cdots\!88\)\( T^{3} + \)\(17\!\cdots\!77\)\( p^{17} T^{4} + 78122492996 p^{34} T^{5} + p^{51} T^{6} \)
23$S_4\times C_2$ \( 1 + 234308204088 T + \)\(29\!\cdots\!85\)\( T^{2} + \)\(55\!\cdots\!88\)\( T^{3} + \)\(29\!\cdots\!85\)\( p^{17} T^{4} + 234308204088 p^{34} T^{5} + p^{51} T^{6} \)
29$S_4\times C_2$ \( 1 - 5775268588078 T + \)\(31\!\cdots\!87\)\( T^{2} - \)\(88\!\cdots\!04\)\( T^{3} + \)\(31\!\cdots\!87\)\( p^{17} T^{4} - 5775268588078 p^{34} T^{5} + p^{51} T^{6} \)
31$S_4\times C_2$ \( 1 - 5565463149104 T + \)\(58\!\cdots\!37\)\( T^{2} - \)\(19\!\cdots\!88\)\( T^{3} + \)\(58\!\cdots\!37\)\( p^{17} T^{4} - 5565463149104 p^{34} T^{5} + p^{51} T^{6} \)
37$S_4\times C_2$ \( 1 + 31751809399326 T + \)\(11\!\cdots\!75\)\( T^{2} + \)\(20\!\cdots\!24\)\( T^{3} + \)\(11\!\cdots\!75\)\( p^{17} T^{4} + 31751809399326 p^{34} T^{5} + p^{51} T^{6} \)
41$S_4\times C_2$ \( 1 - 167461457288254 T + \)\(15\!\cdots\!27\)\( T^{2} - \)\(93\!\cdots\!08\)\( T^{3} + \)\(15\!\cdots\!27\)\( p^{17} T^{4} - 167461457288254 p^{34} T^{5} + p^{51} T^{6} \)
43$S_4\times C_2$ \( 1 + 3504169349788 p T + \)\(22\!\cdots\!33\)\( T^{2} - \)\(57\!\cdots\!40\)\( T^{3} + \)\(22\!\cdots\!33\)\( p^{17} T^{4} + 3504169349788 p^{35} T^{5} + p^{51} T^{6} \)
47$S_4\times C_2$ \( 1 - 72221896199032 T + \)\(20\!\cdots\!17\)\( T^{2} + \)\(35\!\cdots\!40\)\( T^{3} + \)\(20\!\cdots\!17\)\( p^{17} T^{4} - 72221896199032 p^{34} T^{5} + p^{51} T^{6} \)
53$S_4\times C_2$ \( 1 + 76290818594558 T - \)\(16\!\cdots\!65\)\( T^{2} - \)\(11\!\cdots\!72\)\( T^{3} - \)\(16\!\cdots\!65\)\( p^{17} T^{4} + 76290818594558 p^{34} T^{5} + p^{51} T^{6} \)
59$S_4\times C_2$ \( 1 - 465601947196256 T + \)\(32\!\cdots\!37\)\( T^{2} - \)\(11\!\cdots\!28\)\( T^{3} + \)\(32\!\cdots\!37\)\( p^{17} T^{4} - 465601947196256 p^{34} T^{5} + p^{51} T^{6} \)
61$S_4\times C_2$ \( 1 + 2317809676510478 T + \)\(76\!\cdots\!19\)\( T^{2} + \)\(10\!\cdots\!44\)\( T^{3} + \)\(76\!\cdots\!19\)\( p^{17} T^{4} + 2317809676510478 p^{34} T^{5} + p^{51} T^{6} \)
67$S_4\times C_2$ \( 1 - 6392459657973196 T + \)\(46\!\cdots\!53\)\( T^{2} - \)\(14\!\cdots\!52\)\( T^{3} + \)\(46\!\cdots\!53\)\( p^{17} T^{4} - 6392459657973196 p^{34} T^{5} + p^{51} T^{6} \)
71$S_4\times C_2$ \( 1 - 6465608483990656 T + \)\(10\!\cdots\!85\)\( T^{2} - \)\(38\!\cdots\!00\)\( T^{3} + \)\(10\!\cdots\!85\)\( p^{17} T^{4} - 6465608483990656 p^{34} T^{5} + p^{51} T^{6} \)
73$S_4\times C_2$ \( 1 + 5113761577485238 T + \)\(61\!\cdots\!75\)\( T^{2} + \)\(41\!\cdots\!48\)\( T^{3} + \)\(61\!\cdots\!75\)\( p^{17} T^{4} + 5113761577485238 p^{34} T^{5} + p^{51} T^{6} \)
79$S_4\times C_2$ \( 1 - 8740508940658880 T + \)\(43\!\cdots\!77\)\( T^{2} - \)\(31\!\cdots\!40\)\( T^{3} + \)\(43\!\cdots\!77\)\( p^{17} T^{4} - 8740508940658880 p^{34} T^{5} + p^{51} T^{6} \)
83$S_4\times C_2$ \( 1 + 38179195227158436 T + \)\(12\!\cdots\!09\)\( T^{2} + \)\(25\!\cdots\!04\)\( T^{3} + \)\(12\!\cdots\!09\)\( p^{17} T^{4} + 38179195227158436 p^{34} T^{5} + p^{51} T^{6} \)
89$S_4\times C_2$ \( 1 + 17217755358726426 T + \)\(12\!\cdots\!87\)\( T^{2} - \)\(30\!\cdots\!92\)\( T^{3} + \)\(12\!\cdots\!87\)\( p^{17} T^{4} + 17217755358726426 p^{34} T^{5} + p^{51} T^{6} \)
97$S_4\times C_2$ \( 1 - 94917192725726586 T + \)\(13\!\cdots\!43\)\( T^{2} - \)\(71\!\cdots\!92\)\( T^{3} + \)\(13\!\cdots\!43\)\( p^{17} T^{4} - 94917192725726586 p^{34} T^{5} + p^{51} T^{6} \)
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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

−10.47223710517683609081144269440, −9.768712747601371220505388565120, −9.765901108973634635455191667009, −9.431131941371853998247364746720, −8.855128161346867013976142612340, −8.421789788179125957729656082104, −8.260798910378193121821223102763, −7.57017895313920992307143239658, −7.24934990968072451796307912546, −6.55328219690790516150394447425, −6.49701797192131751526520332084, −6.25034148228164231389302647366, −5.71380656174635369304865743533, −4.97702005459455884461826643239, −4.96212999564976014538957491104, −4.90245127087309667361621992254, −4.23528713247734365556383451858, −3.94510052231758049065401026602, −3.92276642971808452743018285612, −3.00961914406732640514621493233, −2.51692356904465120434084502970, −2.15870462980458677366354752332, −1.42100658935723234815386505130, −1.11708888442867196441015978010, −0.888640997163160059385205579590, 0, 0, 0, 0.888640997163160059385205579590, 1.11708888442867196441015978010, 1.42100658935723234815386505130, 2.15870462980458677366354752332, 2.51692356904465120434084502970, 3.00961914406732640514621493233, 3.92276642971808452743018285612, 3.94510052231758049065401026602, 4.23528713247734365556383451858, 4.90245127087309667361621992254, 4.96212999564976014538957491104, 4.97702005459455884461826643239, 5.71380656174635369304865743533, 6.25034148228164231389302647366, 6.49701797192131751526520332084, 6.55328219690790516150394447425, 7.24934990968072451796307912546, 7.57017895313920992307143239658, 8.260798910378193121821223102763, 8.421789788179125957729656082104, 8.855128161346867013976142612340, 9.431131941371853998247364746720, 9.765901108973634635455191667009, 9.768712747601371220505388565120, 10.47223710517683609081144269440

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