Properties

Label 6-3549e3-1.1-c1e3-0-8
Degree $6$
Conductor $44701078149$
Sign $-1$
Analytic cond. $22758.7$
Root an. cond. $5.32343$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $3$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 3·3-s − 4-s − 6·6-s − 3·7-s − 5·8-s + 6·9-s + 11-s + 3·12-s − 6·14-s − 16-s + 2·17-s + 12·18-s − 6·19-s + 9·21-s + 2·22-s + 23-s + 15·24-s − 15·25-s − 10·27-s + 3·28-s − 9·29-s − 10·31-s + 4·32-s − 3·33-s + 4·34-s − 6·36-s + ⋯
L(s)  = 1  + 1.41·2-s − 1.73·3-s − 1/2·4-s − 2.44·6-s − 1.13·7-s − 1.76·8-s + 2·9-s + 0.301·11-s + 0.866·12-s − 1.60·14-s − 1/4·16-s + 0.485·17-s + 2.82·18-s − 1.37·19-s + 1.96·21-s + 0.426·22-s + 0.208·23-s + 3.06·24-s − 3·25-s − 1.92·27-s + 0.566·28-s − 1.67·29-s − 1.79·31-s + 0.707·32-s − 0.522·33-s + 0.685·34-s − 36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 7^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 7^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(3^{3} \cdot 7^{3} \cdot 13^{6}\)
Sign: $-1$
Analytic conductor: \(22758.7\)
Root analytic conductor: \(5.32343\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 3^{3} \cdot 7^{3} \cdot 13^{6} ,\ ( \ : 1/2, 1/2, 1/2 ),\ -1 )\)

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)$
bad3$C_1$ \( ( 1 + T )^{3} \)
7$C_1$ \( ( 1 + T )^{3} \)
13 \( 1 \)
good2$A_4\times C_2$ \( 1 - p T + 5 T^{2} - 7 T^{3} + 5 p T^{4} - p^{3} T^{5} + p^{3} T^{6} \)
5$C_2$ \( ( 1 + p T^{2} )^{3} \)
11$A_4\times C_2$ \( 1 - T + 31 T^{2} - 21 T^{3} + 31 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
17$A_4\times C_2$ \( 1 - 2 T + 43 T^{2} - 60 T^{3} + 43 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
19$A_4\times C_2$ \( 1 + 6 T + 41 T^{2} + 124 T^{3} + 41 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
23$A_4\times C_2$ \( 1 - T + 11 T^{2} - 59 T^{3} + 11 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
29$A_4\times C_2$ \( 1 + 9 T + 93 T^{2} + 479 T^{3} + 93 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \)
31$A_4\times C_2$ \( 1 + 10 T + 117 T^{2} + 628 T^{3} + 117 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
37$A_4\times C_2$ \( 1 - 7 T + 97 T^{2} - 427 T^{3} + 97 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \)
41$A_4\times C_2$ \( 1 - 20 T + 247 T^{2} - 1872 T^{3} + 247 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \)
43$A_4\times C_2$ \( 1 - 13 T + 169 T^{2} - 1147 T^{3} + 169 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \)
47$A_4\times C_2$ \( 1 + 57 T^{2} + 56 T^{3} + 57 p T^{4} + p^{3} T^{6} \)
53$A_4\times C_2$ \( 1 + 5 T + 67 T^{2} + 153 T^{3} + 67 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \)
59$A_4\times C_2$ \( 1 - 4 T + 145 T^{2} - 408 T^{3} + 145 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
61$A_4\times C_2$ \( 1 - 4 T + 67 T^{2} - 592 T^{3} + 67 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
67$A_4\times C_2$ \( 1 + 19 T + 305 T^{2} + 2673 T^{3} + 305 p T^{4} + 19 p^{2} T^{5} + p^{3} T^{6} \)
71$A_4\times C_2$ \( 1 - 7 T + 227 T^{2} - 1001 T^{3} + 227 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \)
73$A_4\times C_2$ \( 1 - 14 T + 163 T^{2} - 1316 T^{3} + 163 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \)
79$A_4\times C_2$ \( 1 + 9 T + 173 T^{2} + 1379 T^{3} + 173 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \)
83$A_4\times C_2$ \( 1 - 2 T + 185 T^{2} - 100 T^{3} + 185 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
89$A_4\times C_2$ \( 1 + 183 T^{2} - 56 T^{3} + 183 p T^{4} + p^{3} T^{6} \)
97$A_4\times C_2$ \( 1 + 8 T + 247 T^{2} + 1208 T^{3} + 247 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.88210362234912192971354117378, −7.61123823981702451769172069956, −7.47476446093461702887038254835, −7.15204311864564227770618277844, −6.76645023634159351738190654416, −6.44251749339125590125516042468, −6.36182873153641458930453345500, −5.99663150385625078290473303968, −5.79062283797436752517516650231, −5.73001515188790220164916583500, −5.35535735579066103139230824585, −5.23783858570004571355693055513, −5.15283230582480510889087146447, −4.31556509356929526147197632600, −4.27830701469773898361853559325, −4.20765150873937466656233016506, −3.79592130236799325064256972063, −3.79440082859112124987149664669, −3.72949027479360698551151896790, −2.76847349774158037462300991223, −2.70780906034079410160858045141, −2.30765345400777804571184392578, −1.84492851978741433876148691632, −1.23005394268309851941272204682, −1.11330538909057658040393773774, 0, 0, 0, 1.11330538909057658040393773774, 1.23005394268309851941272204682, 1.84492851978741433876148691632, 2.30765345400777804571184392578, 2.70780906034079410160858045141, 2.76847349774158037462300991223, 3.72949027479360698551151896790, 3.79440082859112124987149664669, 3.79592130236799325064256972063, 4.20765150873937466656233016506, 4.27830701469773898361853559325, 4.31556509356929526147197632600, 5.15283230582480510889087146447, 5.23783858570004571355693055513, 5.35535735579066103139230824585, 5.73001515188790220164916583500, 5.79062283797436752517516650231, 5.99663150385625078290473303968, 6.36182873153641458930453345500, 6.44251749339125590125516042468, 6.76645023634159351738190654416, 7.15204311864564227770618277844, 7.47476446093461702887038254835, 7.61123823981702451769172069956, 7.88210362234912192971354117378

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