Properties

Label 6-4560e3-1.1-c1e3-0-3
Degree $6$
Conductor $94818816000$
Sign $1$
Analytic cond. $48275.3$
Root an. cond. $6.03421$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s + 3·5-s + 6·9-s − 6·11-s + 4·13-s − 9·15-s + 12·17-s − 3·19-s + 4·23-s + 6·25-s − 10·27-s + 6·31-s + 18·33-s − 4·37-s − 12·39-s + 2·41-s − 2·43-s + 18·45-s − 4·47-s − 7·49-s − 36·51-s − 10·53-s − 18·55-s + 9·57-s − 2·59-s + 14·61-s + 12·65-s + ⋯
L(s)  = 1  − 1.73·3-s + 1.34·5-s + 2·9-s − 1.80·11-s + 1.10·13-s − 2.32·15-s + 2.91·17-s − 0.688·19-s + 0.834·23-s + 6/5·25-s − 1.92·27-s + 1.07·31-s + 3.13·33-s − 0.657·37-s − 1.92·39-s + 0.312·41-s − 0.304·43-s + 2.68·45-s − 0.583·47-s − 49-s − 5.04·51-s − 1.37·53-s − 2.42·55-s + 1.19·57-s − 0.260·59-s + 1.79·61-s + 1.48·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 5^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 5^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{12} \cdot 3^{3} \cdot 5^{3} \cdot 19^{3}\)
Sign: $1$
Analytic conductor: \(48275.3\)
Root analytic conductor: \(6.03421\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{4560} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 2^{12} \cdot 3^{3} \cdot 5^{3} \cdot 19^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(3.586262925\)
\(L(\frac12)\) \(\approx\) \(3.586262925\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{3} \)
5$C_1$ \( ( 1 - T )^{3} \)
19$C_1$ \( ( 1 + T )^{3} \)
good7$S_4\times C_2$ \( 1 + p T^{2} - 12 T^{3} + p^{2} T^{4} + p^{3} T^{6} \)
11$S_4\times C_2$ \( 1 + 6 T + 31 T^{2} + 100 T^{3} + 31 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
13$S_4\times C_2$ \( 1 - 4 T + T^{2} + 60 T^{3} + p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
17$C_2$ \( ( 1 - 4 T + p T^{2} )^{3} \)
23$S_4\times C_2$ \( 1 - 4 T + 25 T^{2} - 152 T^{3} + 25 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 + 73 T^{2} - 12 T^{3} + 73 p T^{4} + p^{3} T^{6} \)
31$C_2$ \( ( 1 - 2 T + p T^{2} )^{3} \)
37$S_4\times C_2$ \( 1 + 4 T + 33 T^{2} + 476 T^{3} + 33 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 - 2 T + p T^{2} + 180 T^{3} + p^{2} T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 + 2 T + 87 T^{2} + 252 T^{3} + 87 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 + 4 T + 97 T^{2} + 344 T^{3} + 97 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 + 10 T + 143 T^{2} + 964 T^{3} + 143 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 + 2 T + 33 T^{2} - 452 T^{3} + 33 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 - 14 T + 3 p T^{2} - 1564 T^{3} + 3 p^{2} T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{3} \)
71$S_4\times C_2$ \( 1 - 4 T + 45 T^{2} - 1208 T^{3} + 45 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
73$C_2$ \( ( 1 - 6 T + p T^{2} )^{3} \)
79$S_4\times C_2$ \( 1 - 2 T + 45 T^{2} + 708 T^{3} + 45 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 - 6 T + 205 T^{2} - 796 T^{3} + 205 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 - 10 T + 217 T^{2} - 1828 T^{3} + 217 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 - 32 T + 549 T^{2} - 6244 T^{3} + 549 p T^{4} - 32 p^{2} T^{5} + p^{3} T^{6} \)
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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

−7.48603282466043085512740391034, −6.80629329401620300053390826437, −6.75751791862380277356311734273, −6.70805336271148444250210561791, −6.17428763166814308738176458374, −6.09980711429328337916898057058, −6.01883073793229236287095203401, −5.49561365059393983937230213453, −5.37354205278993409376935154125, −5.32183268793334977547399771920, −4.93729325026524221985319285821, −4.87250610194109816541097932883, −4.60362157323689074996289912652, −4.03662244302245061224126887879, −3.83358719649212264276649746141, −3.41701869388460455000460819819, −3.21210915509482836846883328993, −2.84643309068377165297398054818, −2.73392550128240915909749650632, −2.04416368777916527683001091977, −1.84703216240304355420527762186, −1.54321547353715918194906029431, −1.12402033041330793032536183521, −0.63260094896920489915215132055, −0.57158648188473683718577188158, 0.57158648188473683718577188158, 0.63260094896920489915215132055, 1.12402033041330793032536183521, 1.54321547353715918194906029431, 1.84703216240304355420527762186, 2.04416368777916527683001091977, 2.73392550128240915909749650632, 2.84643309068377165297398054818, 3.21210915509482836846883328993, 3.41701869388460455000460819819, 3.83358719649212264276649746141, 4.03662244302245061224126887879, 4.60362157323689074996289912652, 4.87250610194109816541097932883, 4.93729325026524221985319285821, 5.32183268793334977547399771920, 5.37354205278993409376935154125, 5.49561365059393983937230213453, 6.01883073793229236287095203401, 6.09980711429328337916898057058, 6.17428763166814308738176458374, 6.70805336271148444250210561791, 6.75751791862380277356311734273, 6.80629329401620300053390826437, 7.48603282466043085512740391034

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