Properties

Degree 6
Conductor $ 2^{3} \cdot 7^{3} \cdot 13^{6} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·2-s + 2·3-s + 6·4-s + 6·6-s − 3·7-s + 10·8-s − 4·9-s + 11-s + 12·12-s − 9·14-s + 15·16-s + 2·17-s − 12·18-s + 15·19-s − 6·21-s + 3·22-s + 23-s + 20·24-s − 8·25-s − 13·27-s − 18·28-s + 5·29-s + 4·31-s + 21·32-s + 2·33-s + 6·34-s − 24·36-s + ⋯
L(s)  = 1  + 2.12·2-s + 1.15·3-s + 3·4-s + 2.44·6-s − 1.13·7-s + 3.53·8-s − 4/3·9-s + 0.301·11-s + 3.46·12-s − 2.40·14-s + 15/4·16-s + 0.485·17-s − 2.82·18-s + 3.44·19-s − 1.30·21-s + 0.639·22-s + 0.208·23-s + 4.08·24-s − 8/5·25-s − 2.50·27-s − 3.40·28-s + 0.928·29-s + 0.718·31-s + 3.71·32-s + 0.348·33-s + 1.02·34-s − 4·36-s + ⋯

Functional equation

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

Invariants

\( d \)  =  \(6\)
\( N \)  =  \(2^{3} \cdot 7^{3} \cdot 13^{6}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{2366} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((6,\ 2^{3} \cdot 7^{3} \cdot 13^{6} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )\)
\(L(1)\)  \(\approx\)  \(22.89990524\)
\(L(\frac12)\)  \(\approx\)  \(22.89990524\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;7,\;13\}$,\(F_p(T)\) is a polynomial of degree 6. If $p \in \{2,\;7,\;13\}$, then $F_p(T)$ is a polynomial of degree at most 5.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_1$ \( ( 1 - T )^{3} \)
7$C_1$ \( ( 1 + T )^{3} \)
13 \( 1 \)
good3$A_4\times C_2$ \( 1 - 2 T + 8 T^{2} - 11 T^{3} + 8 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
5$A_4\times C_2$ \( 1 + 8 T^{2} + 7 T^{3} + 8 p T^{4} + p^{3} T^{6} \)
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$A_4\times C_2$ \( 1 - 2 T + 36 T^{2} - 81 T^{3} + 36 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
19$A_4\times C_2$ \( 1 - 15 T + 125 T^{2} - 653 T^{3} + 125 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \)
23$A_4\times C_2$ \( 1 - T + 53 T^{2} - 59 T^{3} + 53 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
29$A_4\times C_2$ \( 1 - 5 T + 65 T^{2} - 193 T^{3} + 65 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \)
31$A_4\times C_2$ \( 1 - 4 T + 54 T^{2} - 289 T^{3} + 54 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
37$A_4\times C_2$ \( 1 - 28 T + 10 p T^{2} - 2863 T^{3} + 10 p^{2} T^{4} - 28 p^{2} T^{5} + p^{3} T^{6} \)
41$A_4\times C_2$ \( 1 - 13 T + 149 T^{2} - 1067 T^{3} + 149 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \)
43$A_4\times C_2$ \( 1 + T + 85 T^{2} + 169 T^{3} + 85 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
47$A_4\times C_2$ \( 1 - 7 T + 43 T^{2} - 21 T^{3} + 43 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \)
53$A_4\times C_2$ \( 1 + 12 T + 186 T^{2} + 1259 T^{3} + 186 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \)
59$A_4\times C_2$ \( 1 + 3 T + 117 T^{2} + 481 T^{3} + 117 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
61$A_4\times C_2$ \( 1 + 3 T + 95 T^{2} + 479 T^{3} + 95 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
67$A_4\times C_2$ \( 1 + 5 T + 151 T^{2} + 545 T^{3} + 151 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \)
71$A_4\times C_2$ \( 1 + 164 T^{2} - 91 T^{3} + 164 p T^{4} + p^{3} T^{6} \)
73$A_4\times C_2$ \( 1 - 21 T + 149 T^{2} - 707 T^{3} + 149 p T^{4} - 21 p^{2} T^{5} + p^{3} T^{6} \)
79$A_4\times C_2$ \( 1 + 2 T + 12 T^{2} - 931 T^{3} + 12 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
83$A_4\times C_2$ \( 1 + 5 T + 241 T^{2} + 789 T^{3} + 241 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \)
89$A_4\times C_2$ \( 1 - 14 T + 176 T^{2} - 2191 T^{3} + 176 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \)
97$A_4\times C_2$ \( 1 + 15 T + 359 T^{2} + 31 p T^{3} + 359 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−7.83683571391805489191070037880, −7.58677532408271765610362148939, −7.57366572163028341905736455192, −7.39434499374166393153848589420, −6.53500235615994553079807682967, −6.48511654029689579116965473402, −6.43191263404335210442896837874, −5.95982217573503243218932671641, −5.79185349235949292139111204141, −5.73522778857034344744748916912, −5.18429092421811213887281254807, −5.04785349472335484044784897120, −4.88145457634050711964668802649, −4.07704227736615626234234253296, −4.02012316849193116834220697501, −4.01947128822591181734032450504, −3.40020875293993581659968882287, −3.16101185412149583870684179121, −2.99465980874160581162293047811, −2.58060668054311888099501511624, −2.50498974873250465675792636204, −2.48481634720681423580776757605, −1.40850576997079563377082833895, −1.14421363680449441349199471809, −0.65161611358665726622005258345, 0.65161611358665726622005258345, 1.14421363680449441349199471809, 1.40850576997079563377082833895, 2.48481634720681423580776757605, 2.50498974873250465675792636204, 2.58060668054311888099501511624, 2.99465980874160581162293047811, 3.16101185412149583870684179121, 3.40020875293993581659968882287, 4.01947128822591181734032450504, 4.02012316849193116834220697501, 4.07704227736615626234234253296, 4.88145457634050711964668802649, 5.04785349472335484044784897120, 5.18429092421811213887281254807, 5.73522778857034344744748916912, 5.79185349235949292139111204141, 5.95982217573503243218932671641, 6.43191263404335210442896837874, 6.48511654029689579116965473402, 6.53500235615994553079807682967, 7.39434499374166393153848589420, 7.57366572163028341905736455192, 7.58677532408271765610362148939, 7.83683571391805489191070037880

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