Properties

Label 6-7728e3-1.1-c1e3-0-4
Degree $6$
Conductor $461531492352$
Sign $1$
Analytic cond. $234980.$
Root an. cond. $7.85546$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s + 5-s − 3·7-s + 6·9-s + 2·11-s − 3·13-s + 3·15-s + 2·17-s + 8·19-s − 9·21-s − 3·23-s − 7·25-s + 10·27-s − 2·29-s − 10·31-s + 6·33-s − 3·35-s + 12·37-s − 9·39-s + 16·41-s + 9·43-s + 6·45-s − 14·47-s + 6·49-s + 6·51-s + 15·53-s + 2·55-s + ⋯
L(s)  = 1  + 1.73·3-s + 0.447·5-s − 1.13·7-s + 2·9-s + 0.603·11-s − 0.832·13-s + 0.774·15-s + 0.485·17-s + 1.83·19-s − 1.96·21-s − 0.625·23-s − 7/5·25-s + 1.92·27-s − 0.371·29-s − 1.79·31-s + 1.04·33-s − 0.507·35-s + 1.97·37-s − 1.44·39-s + 2.49·41-s + 1.37·43-s + 0.894·45-s − 2.04·47-s + 6/7·49-s + 0.840·51-s + 2.06·53-s + 0.269·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 7^{3} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 7^{3} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{12} \cdot 3^{3} \cdot 7^{3} \cdot 23^{3}\)
Sign: $1$
Analytic conductor: \(234980.\)
Root analytic conductor: \(7.85546\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{7728} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 2^{12} \cdot 3^{3} \cdot 7^{3} \cdot 23^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(15.40505883\)
\(L(\frac12)\) \(\approx\) \(15.40505883\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{3} \)
7$C_1$ \( ( 1 + T )^{3} \)
23$C_1$ \( ( 1 + T )^{3} \)
good5$S_4\times C_2$ \( 1 - T + 8 T^{2} - 4 T^{3} + 8 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
11$S_4\times C_2$ \( 1 - 2 T + 14 T^{2} - 8 T^{3} + 14 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
13$S_4\times C_2$ \( 1 + 3 T + 28 T^{2} + 84 T^{3} + 28 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 - 2 T + 30 T^{2} - 50 T^{3} + 30 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
19$S_4\times C_2$ \( 1 - 8 T + 56 T^{2} - 260 T^{3} + 56 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 + 2 T + 66 T^{2} + 98 T^{3} + 66 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 + 10 T + 106 T^{2} + 612 T^{3} + 106 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 - 12 T + 76 T^{2} - 334 T^{3} + 76 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 - 16 T + 188 T^{2} - 1378 T^{3} + 188 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 - 9 T + 142 T^{2} - 778 T^{3} + 142 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 + 14 T + 41 T^{2} - 156 T^{3} + 41 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 - 15 T + 118 T^{2} - 668 T^{3} + 118 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 - 11 T + 152 T^{2} - 950 T^{3} + 152 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 - 19 T + 168 T^{2} - 1112 T^{3} + 168 p T^{4} - 19 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 - 13 T + 192 T^{2} - 1698 T^{3} + 192 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 - 13 T + 224 T^{2} - 1822 T^{3} + 224 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 + 10 T + 122 T^{2} + 782 T^{3} + 122 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 - 20 T + 348 T^{2} - 3304 T^{3} + 348 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 + 2 T + 230 T^{2} + 296 T^{3} + 230 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 - 17 T + 318 T^{2} - 2860 T^{3} + 318 p T^{4} - 17 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 - 20 T + 404 T^{2} - 4018 T^{3} + 404 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \)
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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

−7.11769474891368807835500575392, −6.68932833347238910136892598409, −6.52727030024421432156569661391, −6.35364537974451138304594311103, −5.97785695123838158487111004150, −5.71632095607667363932008038772, −5.69453574052191907269160742152, −5.29724511215228100678591657205, −5.06299242621250682555869722905, −4.87528907941776354008510640344, −4.26820836182968314401506966161, −4.24421396934927834986293557814, −3.93511147958218462646876969022, −3.59073246894269986626381581739, −3.57708541260758834980409212335, −3.37160319587115260599304289965, −2.99975756083796142834340276070, −2.55050074414854664675864156041, −2.45500633881841111008354546450, −2.08815254016717030275962245558, −2.03652969213474175099983400934, −1.71720138899966393676174568787, −0.846769875939663071435630144965, −0.74734194918063996957800009168, −0.71025124504326412618283983753, 0.71025124504326412618283983753, 0.74734194918063996957800009168, 0.846769875939663071435630144965, 1.71720138899966393676174568787, 2.03652969213474175099983400934, 2.08815254016717030275962245558, 2.45500633881841111008354546450, 2.55050074414854664675864156041, 2.99975756083796142834340276070, 3.37160319587115260599304289965, 3.57708541260758834980409212335, 3.59073246894269986626381581739, 3.93511147958218462646876969022, 4.24421396934927834986293557814, 4.26820836182968314401506966161, 4.87528907941776354008510640344, 5.06299242621250682555869722905, 5.29724511215228100678591657205, 5.69453574052191907269160742152, 5.71632095607667363932008038772, 5.97785695123838158487111004150, 6.35364537974451138304594311103, 6.52727030024421432156569661391, 6.68932833347238910136892598409, 7.11769474891368807835500575392

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