Properties

Label 6-4332e3-1.1-c1e3-0-1
Degree $6$
Conductor $81295282368$
Sign $-1$
Analytic cond. $41390.1$
Root an. cond. $5.88142$
Motivic weight $1$
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  + 3·3-s + 3·5-s − 3·7-s + 6·9-s − 3·11-s − 9·13-s + 9·15-s − 3·17-s − 9·21-s − 12·23-s − 6·25-s + 10·27-s + 3·29-s + 6·31-s − 9·33-s − 9·35-s − 6·37-s − 27·39-s − 18·41-s − 21·43-s + 18·45-s − 15·47-s − 6·49-s − 9·51-s + 6·53-s − 9·55-s + 9·59-s + ⋯
L(s)  = 1  + 1.73·3-s + 1.34·5-s − 1.13·7-s + 2·9-s − 0.904·11-s − 2.49·13-s + 2.32·15-s − 0.727·17-s − 1.96·21-s − 2.50·23-s − 6/5·25-s + 1.92·27-s + 0.557·29-s + 1.07·31-s − 1.56·33-s − 1.52·35-s − 0.986·37-s − 4.32·39-s − 2.81·41-s − 3.20·43-s + 2.68·45-s − 2.18·47-s − 6/7·49-s − 1.26·51-s + 0.824·53-s − 1.21·55-s + 1.17·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{3} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{3} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{6} \cdot 3^{3} \cdot 19^{6}\)
Sign: $-1$
Analytic conductor: \(41390.1\)
Root analytic conductor: \(5.88142\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{4332} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 2^{6} \cdot 3^{3} \cdot 19^{6} ,\ ( \ : 1/2, 1/2, 1/2 ),\ -1 )\)

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 \( 1 \)
3$C_1$ \( ( 1 - T )^{3} \)
19 \( 1 \)
good5$A_4\times C_2$ \( 1 - 3 T + 3 p T^{2} - 27 T^{3} + 3 p^{2} T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \)
7$A_4\times C_2$ \( 1 + 3 T + 15 T^{2} + 25 T^{3} + 15 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
11$A_4\times C_2$ \( 1 + 3 T + 15 T^{2} + 63 T^{3} + 15 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
13$A_4\times C_2$ \( 1 + 9 T + 54 T^{2} + 217 T^{3} + 54 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \)
17$A_4\times C_2$ \( 1 + 3 T + 33 T^{2} + 99 T^{3} + 33 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
23$A_4\times C_2$ \( 1 + 12 T + 96 T^{2} + 495 T^{3} + 96 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \)
29$A_4\times C_2$ \( 1 - 3 T + 42 T^{2} - 63 T^{3} + 42 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \)
31$A_4\times C_2$ \( 1 - 6 T + 78 T^{2} - 353 T^{3} + 78 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
37$A_4\times C_2$ \( 1 + 6 T + 102 T^{2} + 427 T^{3} + 102 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
41$A_4\times C_2$ \( 1 + 18 T + 204 T^{2} + 1503 T^{3} + 204 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \)
43$A_4\times C_2$ \( 1 + 21 T + 240 T^{2} + 1825 T^{3} + 240 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} \)
47$A_4\times C_2$ \( 1 + 15 T + 213 T^{2} + 1521 T^{3} + 213 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \)
53$A_4\times C_2$ \( 1 - 6 T + 114 T^{2} - 423 T^{3} + 114 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
59$A_4\times C_2$ \( 1 - 9 T + 141 T^{2} - 1071 T^{3} + 141 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \)
61$A_4\times C_2$ \( 1 - 15 T + 249 T^{2} - 1901 T^{3} + 249 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \)
67$A_4\times C_2$ \( 1 + 6 T + 3 p T^{2} + 796 T^{3} + 3 p^{2} T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
71$A_4\times C_2$ \( 1 - 9 T + 132 T^{2} - 765 T^{3} + 132 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \)
73$A_4\times C_2$ \( 1 + 6 T + 192 T^{2} + 895 T^{3} + 192 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
79$A_4\times C_2$ \( 1 + 27 T + 468 T^{2} + 4879 T^{3} + 468 p T^{4} + 27 p^{2} T^{5} + p^{3} T^{6} \)
83$A_4\times C_2$ \( 1 + 15 T + 213 T^{2} + 2493 T^{3} + 213 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \)
89$A_4\times C_2$ \( 1 - 12 T + 186 T^{2} - 1755 T^{3} + 186 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \)
97$A_4\times C_2$ \( 1 - 6 T + 186 T^{2} - 785 T^{3} + 186 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
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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

−8.077248767350792655722011349524, −7.46653754515956496378078409801, −7.30884313479398226268590885442, −7.04704844059734516184952481446, −6.70411091627965346639914496498, −6.68378800929815819077939688638, −6.52094437352877995551163102861, −6.11626791364696536639153149592, −5.81424451788120511546486821620, −5.48834176287466956490867364927, −5.18657189877457230624214479430, −5.04972966881000745092309407006, −4.90141789711959435266160939196, −4.31555085808263500305659614353, −4.27381946715023235967370966879, −3.88004394070892481923350646010, −3.50374919251587703703258511009, −3.20549949383699262227413033705, −3.14327371152579108624660100075, −2.58070106847330282080429155948, −2.46979851169930612728100428583, −2.26232953584233414041983725922, −1.81984179502612656483197352897, −1.69232083052172181852257790814, −1.46256818258674725927074227038, 0, 0, 0, 1.46256818258674725927074227038, 1.69232083052172181852257790814, 1.81984179502612656483197352897, 2.26232953584233414041983725922, 2.46979851169930612728100428583, 2.58070106847330282080429155948, 3.14327371152579108624660100075, 3.20549949383699262227413033705, 3.50374919251587703703258511009, 3.88004394070892481923350646010, 4.27381946715023235967370966879, 4.31555085808263500305659614353, 4.90141789711959435266160939196, 5.04972966881000745092309407006, 5.18657189877457230624214479430, 5.48834176287466956490867364927, 5.81424451788120511546486821620, 6.11626791364696536639153149592, 6.52094437352877995551163102861, 6.68378800929815819077939688638, 6.70411091627965346639914496498, 7.04704844059734516184952481446, 7.30884313479398226268590885442, 7.46653754515956496378078409801, 8.077248767350792655722011349524

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