Properties

Label 6-7098e3-1.1-c1e3-0-4
Degree $6$
Conductor $357608625192$
Sign $1$
Analytic cond. $182070.$
Root an. cond. $7.52846$
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  − 3·2-s + 3·3-s + 6·4-s + 8·5-s − 9·6-s − 3·7-s − 10·8-s + 6·9-s − 24·10-s + 5·11-s + 18·12-s + 9·14-s + 24·15-s + 15·16-s − 3·17-s − 18·18-s + 5·19-s + 48·20-s − 9·21-s − 15·22-s − 7·23-s − 30·24-s + 30·25-s + 10·27-s − 18·28-s − 4·29-s − 72·30-s + ⋯
L(s)  = 1  − 2.12·2-s + 1.73·3-s + 3·4-s + 3.57·5-s − 3.67·6-s − 1.13·7-s − 3.53·8-s + 2·9-s − 7.58·10-s + 1.50·11-s + 5.19·12-s + 2.40·14-s + 6.19·15-s + 15/4·16-s − 0.727·17-s − 4.24·18-s + 1.14·19-s + 10.7·20-s − 1.96·21-s − 3.19·22-s − 1.45·23-s − 6.12·24-s + 6·25-s + 1.92·27-s − 3.40·28-s − 0.742·29-s − 13.1·30-s + ⋯

Functional equation

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

Invariants

Degree: \(6\)
Conductor: \(2^{3} \cdot 3^{3} \cdot 7^{3} \cdot 13^{6}\)
Sign: $1$
Analytic conductor: \(182070.\)
Root analytic conductor: \(7.52846\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{7098} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 2^{3} \cdot 3^{3} \cdot 7^{3} \cdot 13^{6} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(12.45798882\)
\(L(\frac12)\) \(\approx\) \(12.45798882\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{3} \)
7$C_1$ \( ( 1 + T )^{3} \)
13 \( 1 \)
good5$A_4\times C_2$ \( 1 - 8 T + 34 T^{2} - 93 T^{3} + 34 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
11$A_4\times C_2$ \( 1 - 5 T + 39 T^{2} - 111 T^{3} + 39 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \)
17$A_4\times C_2$ \( 1 + 3 T + 33 T^{2} + 89 T^{3} + 33 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
19$A_4\times C_2$ \( 1 - 5 T + 49 T^{2} - 149 T^{3} + 49 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \)
23$A_4\times C_2$ \( 1 + 7 T + 55 T^{2} + 315 T^{3} + 55 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \)
29$A_4\times C_2$ \( 1 + 4 T + 62 T^{2} + 161 T^{3} + 62 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
31$A_4\times C_2$ \( 1 + 4 T + 26 T^{2} + 219 T^{3} + 26 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
37$A_4\times C_2$ \( 1 + 2 T + 82 T^{2} + 77 T^{3} + 82 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
41$A_4\times C_2$ \( 1 - 18 T + 168 T^{2} - 1125 T^{3} + 168 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \)
43$A_4\times C_2$ \( 1 - 5 T + 121 T^{2} - 389 T^{3} + 121 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \)
47$A_4\times C_2$ \( 1 - 23 T + 308 T^{2} - 2539 T^{3} + 308 p T^{4} - 23 p^{2} T^{5} + p^{3} T^{6} \)
53$A_4\times C_2$ \( 1 + 12 T + 116 T^{2} + 1175 T^{3} + 116 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \)
59$A_4\times C_2$ \( 1 - 16 T + 162 T^{2} - 1425 T^{3} + 162 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \)
61$A_4\times C_2$ \( 1 - 17 T + 214 T^{2} - 2033 T^{3} + 214 p T^{4} - 17 p^{2} T^{5} + p^{3} T^{6} \)
67$A_4\times C_2$ \( 1 - 10 T + 162 T^{2} - 991 T^{3} + 162 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
71$A_4\times C_2$ \( 1 + 14 T + 241 T^{2} + 1932 T^{3} + 241 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \)
73$A_4\times C_2$ \( 1 - 23 T + 379 T^{2} - 3707 T^{3} + 379 p T^{4} - 23 p^{2} T^{5} + p^{3} T^{6} \)
79$A_4\times C_2$ \( 1 - 4 T + 170 T^{2} - 603 T^{3} + 170 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
83$A_4\times C_2$ \( 1 - 37 T + 689 T^{2} - 7835 T^{3} + 689 p T^{4} - 37 p^{2} T^{5} + p^{3} T^{6} \)
89$A_4\times C_2$ \( 1 + 10 T + 186 T^{2} + 1893 T^{3} + 186 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
97$A_4\times C_2$ \( 1 - 20 T + 338 T^{2} - 3909 T^{3} + 338 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \)
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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

−7.21473090899583183576305195750, −6.76551576867480791327665307365, −6.54430901868271242832400255156, −6.51120791627527345169796495883, −6.04120353154658636596807537365, −6.03656092603715557601653120040, −5.92639000381105323835630765296, −5.37706745576314254711604458785, −5.30499126794722552981238092420, −5.28409232415865219573659405202, −4.41818335538045572226665808833, −4.09484832310546995637359119872, −4.05931665015565197269955221776, −3.55398520768614027396546789652, −3.52444660066486916469985331939, −3.04417419752888305151159145831, −2.68201468070754340615045202452, −2.40785716535563627563862590728, −2.29603819832773762097162237368, −1.97264816010419433091291413053, −1.93907660289185312049490548665, −1.74653466238650491327028478612, −0.997337556971012053703626726953, −0.847390897144414027727064016826, −0.73398630358007951865083919730, 0.73398630358007951865083919730, 0.847390897144414027727064016826, 0.997337556971012053703626726953, 1.74653466238650491327028478612, 1.93907660289185312049490548665, 1.97264816010419433091291413053, 2.29603819832773762097162237368, 2.40785716535563627563862590728, 2.68201468070754340615045202452, 3.04417419752888305151159145831, 3.52444660066486916469985331939, 3.55398520768614027396546789652, 4.05931665015565197269955221776, 4.09484832310546995637359119872, 4.41818335538045572226665808833, 5.28409232415865219573659405202, 5.30499126794722552981238092420, 5.37706745576314254711604458785, 5.92639000381105323835630765296, 6.03656092603715557601653120040, 6.04120353154658636596807537365, 6.51120791627527345169796495883, 6.54430901868271242832400255156, 6.76551576867480791327665307365, 7.21473090899583183576305195750

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