Properties

Label 6-4830e3-1.1-c1e3-0-5
Degree $6$
Conductor $112678587000$
Sign $-1$
Analytic cond. $57368.3$
Root an. cond. $6.21029$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $3$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·2-s − 3·3-s + 6·4-s + 3·5-s − 9·6-s − 3·7-s + 10·8-s + 6·9-s + 9·10-s − 18·12-s − 2·13-s − 9·14-s − 9·15-s + 15·16-s − 10·17-s + 18·18-s − 4·19-s + 18·20-s + 9·21-s − 3·23-s − 30·24-s + 6·25-s − 6·26-s − 10·27-s − 18·28-s − 10·29-s − 27·30-s + ⋯
L(s)  = 1  + 2.12·2-s − 1.73·3-s + 3·4-s + 1.34·5-s − 3.67·6-s − 1.13·7-s + 3.53·8-s + 2·9-s + 2.84·10-s − 5.19·12-s − 0.554·13-s − 2.40·14-s − 2.32·15-s + 15/4·16-s − 2.42·17-s + 4.24·18-s − 0.917·19-s + 4.02·20-s + 1.96·21-s − 0.625·23-s − 6.12·24-s + 6/5·25-s − 1.17·26-s − 1.92·27-s − 3.40·28-s − 1.85·29-s − 4.92·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 5^{3} \cdot 7^{3} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 5^{3} \cdot 7^{3} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{3} \cdot 3^{3} \cdot 5^{3} \cdot 7^{3} \cdot 23^{3}\)
Sign: $-1$
Analytic conductor: \(57368.3\)
Root analytic conductor: \(6.21029\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 2^{3} \cdot 3^{3} \cdot 5^{3} \cdot 7^{3} \cdot 23^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ -1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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)$
bad2$C_1$ \( ( 1 - T )^{3} \)
3$C_1$ \( ( 1 + T )^{3} \)
5$C_1$ \( ( 1 - T )^{3} \)
7$C_1$ \( ( 1 + T )^{3} \)
23$C_1$ \( ( 1 + T )^{3} \)
good11$S_4\times C_2$ \( 1 + 5 T^{2} - 16 T^{3} + 5 p T^{4} + p^{3} T^{6} \)
13$S_4\times C_2$ \( 1 + 2 T + 11 T^{2} + 60 T^{3} + 11 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 + 10 T + 67 T^{2} + 324 T^{3} + 67 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
19$S_4\times C_2$ \( 1 + 4 T + 45 T^{2} + 136 T^{3} + 45 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 + 10 T + 55 T^{2} + 228 T^{3} + 55 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 + 16 T + 161 T^{2} + 1056 T^{3} + 161 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{3} \)
41$D_{6}$ \( 1 - 6 T + 23 T^{2} - 148 T^{3} + 23 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 + 101 T^{2} - 16 T^{3} + 101 p T^{4} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 + 16 T + 209 T^{2} + 1568 T^{3} + 209 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \)
53$C_2$ \( ( 1 + 2 T + p T^{2} )^{3} \)
59$S_4\times C_2$ \( 1 - 4 T + 65 T^{2} - 536 T^{3} + 65 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 + 22 T + 275 T^{2} + 2532 T^{3} + 275 p T^{4} + 22 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 + 24 T + 365 T^{2} + 3488 T^{3} + 365 p T^{4} + 24 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 - 4 T + 89 T^{2} - 920 T^{3} + 89 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 - 6 T + 119 T^{2} - 788 T^{3} + 119 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 - 8 T + 13 T^{2} + 784 T^{3} + 13 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 + 12 T + 125 T^{2} + 1016 T^{3} + 125 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 + 14 T + 271 T^{2} + 2484 T^{3} + 271 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 - 2 T + 231 T^{2} - 524 T^{3} + 231 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
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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

−7.55826022596277079237808524213, −7.25064564511646999484719368352, −6.90653998730145253116072152906, −6.70669960394848853777563262090, −6.51158306345302832101760092888, −6.44986088391800520271767178123, −6.11913848364259396892779726878, −5.84497124316030675576925467976, −5.73992435055738851160200441897, −5.64940958832398280654317385196, −5.09727880524874182749835968721, −5.03494696822095687431215451465, −4.91390044075319578795666397992, −4.32326660621029644589673164273, −4.28249427116960562158335093770, −4.20503516396632186275172572632, −3.54729911653278432888407697855, −3.48505270480025521225016092856, −3.28725086743968923343685784456, −2.56956886015848851961545998029, −2.39066675045315860429966336180, −2.33973895189327634017971602452, −1.61115875675658131705344343854, −1.58230154230768104140161974414, −1.50670693193645715410800391439, 0, 0, 0, 1.50670693193645715410800391439, 1.58230154230768104140161974414, 1.61115875675658131705344343854, 2.33973895189327634017971602452, 2.39066675045315860429966336180, 2.56956886015848851961545998029, 3.28725086743968923343685784456, 3.48505270480025521225016092856, 3.54729911653278432888407697855, 4.20503516396632186275172572632, 4.28249427116960562158335093770, 4.32326660621029644589673164273, 4.91390044075319578795666397992, 5.03494696822095687431215451465, 5.09727880524874182749835968721, 5.64940958832398280654317385196, 5.73992435055738851160200441897, 5.84497124316030675576925467976, 6.11913848364259396892779726878, 6.44986088391800520271767178123, 6.51158306345302832101760092888, 6.70669960394848853777563262090, 6.90653998730145253116072152906, 7.25064564511646999484719368352, 7.55826022596277079237808524213

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