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Properties

Label 6-45e3-1.1-c17e3-0-1
Degree $6$
Conductor $91125$
Sign $-1$
Analytic cond. $560494.$
Root an. cond. $9.08019$
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 − 1.68e5·4-s + 1.17e6·5-s − 4.33e6·7-s − 5.35e7·8-s + 2.96e8·10-s − 9.43e8·11-s + 4.25e9·13-s − 1.09e9·14-s + 7.11e9·16-s − 3.06e9·17-s − 7.81e10·19-s − 1.97e11·20-s − 2.38e11·22-s − 2.34e11·23-s + 9.15e11·25-s + 1.07e12·26-s + 7.28e11·28-s − 5.77e12·29-s + 5.56e12·31-s + 2.87e12·32-s − 7.74e11·34-s − 5.07e12·35-s + 3.17e13·37-s − 1.97e13·38-s − 6.28e13·40-s − 1.67e14·41-s + ⋯
L(s)  = 1  + 0.698·2-s − 1.28·4-s + 1.34·5-s − 0.284·7-s − 1.12·8-s + 0.937·10-s − 1.32·11-s + 1.44·13-s − 0.198·14-s + 0.413·16-s − 0.106·17-s − 1.05·19-s − 1.72·20-s − 0.927·22-s − 0.623·23-s + 6/5·25-s + 1.01·26-s + 0.364·28-s − 2.14·29-s + 1.17·31-s + 0.461·32-s − 0.0744·34-s − 0.381·35-s + 1.48·37-s − 0.737·38-s − 1.51·40-s − 3.27·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91125 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91125 ^{s/2} \, \Gamma_{\C}(s+17/2)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(91125\)    =    \(3^{6} \cdot 5^{3}\)
Sign: $-1$
Analytic conductor: \(560494.\)
Root analytic conductor: \(9.08019\)
Motivic weight: \(17\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 91125,\ (\ :17/2, 17/2, 17/2),\ -1)\)

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 \( 1 \)
5$C_1$ \( ( 1 - p^{8} T )^{3} \)
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

−11.09457326488875425903134169903, −10.79974113491880739172605522028, −10.54057657345617261853634319713, −10.15676504008202141076081923242, −9.450553104859155196056290996956, −9.427521008066590571667480347610, −9.111652947437378626490363471638, −8.366344869445227507930701842675, −8.317963772419130301038256632226, −7.83526654679336641580376081499, −7.19349972721325030591577170358, −6.41431970857116435352113833570, −6.40126368851457271001763672453, −5.73624799059088650467166197601, −5.52401213444246262034670541430, −5.22358436313140554860731542348, −4.49541944877359502726347362713, −4.33173029252975611397362777166, −3.95416611337942967607098449762, −3.24926609113799537714308398938, −2.96699917377600389050225979290, −2.39888557727368092282079160358, −1.95291889765795639343791989593, −1.27514058164638536391875828638, −1.24312582195358554110135494461, 0, 0, 0, 1.24312582195358554110135494461, 1.27514058164638536391875828638, 1.95291889765795639343791989593, 2.39888557727368092282079160358, 2.96699917377600389050225979290, 3.24926609113799537714308398938, 3.95416611337942967607098449762, 4.33173029252975611397362777166, 4.49541944877359502726347362713, 5.22358436313140554860731542348, 5.52401213444246262034670541430, 5.73624799059088650467166197601, 6.40126368851457271001763672453, 6.41431970857116435352113833570, 7.19349972721325030591577170358, 7.83526654679336641580376081499, 8.317963772419130301038256632226, 8.366344869445227507930701842675, 9.111652947437378626490363471638, 9.427521008066590571667480347610, 9.450553104859155196056290996956, 10.15676504008202141076081923242, 10.54057657345617261853634319713, 10.79974113491880739172605522028, 11.09457326488875425903134169903

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