Properties

Degree 6
Conductor $ 3^{6} \cdot 7^{3} \cdot 11^{6} $
Sign $-1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 3

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 3·4-s − 5-s + 3·7-s − 4·8-s + 2·10-s + 8·13-s − 6·14-s + 3·16-s − 8·17-s − 3·20-s − 7·23-s − 6·25-s − 16·26-s + 9·28-s − 13·31-s − 6·32-s + 16·34-s − 3·35-s − 17·37-s + 4·40-s − 16·41-s + 4·43-s + 14·46-s − 4·47-s + 6·49-s + 12·50-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s − 0.447·5-s + 1.13·7-s − 1.41·8-s + 0.632·10-s + 2.21·13-s − 1.60·14-s + 3/4·16-s − 1.94·17-s − 0.670·20-s − 1.45·23-s − 6/5·25-s − 3.13·26-s + 1.70·28-s − 2.33·31-s − 1.06·32-s + 2.74·34-s − 0.507·35-s − 2.79·37-s + 0.632·40-s − 2.49·41-s + 0.609·43-s + 2.06·46-s − 0.583·47-s + 6/7·49-s + 1.69·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 7^{3} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 7^{3} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(6\)
\( N \)  =  \(3^{6} \cdot 7^{3} \cdot 11^{6}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{7623} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  3
Selberg data  =  $(6,\ 3^{6} \cdot 7^{3} \cdot 11^{6} ,\ ( \ : 1/2, 1/2, 1/2 ),\ -1 )$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;7,\;11\}$,\(F_p(T)\) is a polynomial of degree 6. If $p \in \{3,\;7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 5.
$p$$\Gal(F_p)$$F_p(T)$
bad3 \( 1 \)
7$C_1$ \( ( 1 - T )^{3} \)
11 \( 1 \)
good2$D_{6}$ \( 1 + p T + T^{2} + p T^{4} + p^{3} T^{5} + p^{3} T^{6} \)
5$S_4\times C_2$ \( 1 + T + 7 T^{2} + 6 T^{3} + 7 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
13$S_4\times C_2$ \( 1 - 8 T + 41 T^{2} - 144 T^{3} + 41 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 + 8 T + 53 T^{2} + 208 T^{3} + 53 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
19$S_4\times C_2$ \( 1 + 17 T^{2} - 64 T^{3} + 17 p T^{4} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 + 7 T + 77 T^{2} + 314 T^{3} + 77 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 + 47 T^{2} - 64 T^{3} + 47 p T^{4} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 + 13 T + 143 T^{2} + 864 T^{3} + 143 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 + 17 T + 183 T^{2} + 1274 T^{3} + 183 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 + 16 T + 189 T^{2} + 1392 T^{3} + 189 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 - 4 T + 101 T^{2} - 312 T^{3} + 101 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 + 4 T + 127 T^{2} + 384 T^{3} + 127 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 - 10 T + 3 p T^{2} - 996 T^{3} + 3 p^{2} T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 + T + 171 T^{2} + 120 T^{3} + 171 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 + 16 T + 249 T^{2} + 2032 T^{3} + 249 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 + 3 T + 113 T^{2} - 22 T^{3} + 113 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 + 5 T + 5 T^{2} - 770 T^{3} + 5 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 - 16 T + 285 T^{2} - 2416 T^{3} + 285 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 - 28 T + 465 T^{2} - 4936 T^{3} + 465 p T^{4} - 28 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 + 8 T + 193 T^{2} + 1392 T^{3} + 193 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 - 21 T + 371 T^{2} - 3838 T^{3} + 371 p T^{4} - 21 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 + 11 T + 259 T^{2} + 1682 T^{3} + 259 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \)
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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

−7.29609222558563396266763543826, −7.12058810388824924021724994207, −7.06711674107894364964828474146, −6.51237889931001551178237694924, −6.41687336283557190778323553272, −6.39630004025078393758070455087, −6.03648302897845718053503583793, −5.71460385171020972356921982996, −5.41422456700486232422723491694, −5.27710038355053337136107573841, −5.05047250523704989371899821261, −4.61652567720975334695143655297, −4.52718885811727722323314499442, −3.93486308076744415584564911596, −3.86055606185165122179937970553, −3.57297981054057704622203547003, −3.53382557456664855930680214009, −3.33183148283014022041928849014, −2.56828221974674493143531529110, −2.34208739529183519494339866583, −1.92857691161388760442532291467, −1.90172089076921706656830620604, −1.74635018172508328530954454478, −1.28509090457716517419825559442, −0.991645112262633184237523442496, 0, 0, 0, 0.991645112262633184237523442496, 1.28509090457716517419825559442, 1.74635018172508328530954454478, 1.90172089076921706656830620604, 1.92857691161388760442532291467, 2.34208739529183519494339866583, 2.56828221974674493143531529110, 3.33183148283014022041928849014, 3.53382557456664855930680214009, 3.57297981054057704622203547003, 3.86055606185165122179937970553, 3.93486308076744415584564911596, 4.52718885811727722323314499442, 4.61652567720975334695143655297, 5.05047250523704989371899821261, 5.27710038355053337136107573841, 5.41422456700486232422723491694, 5.71460385171020972356921982996, 6.03648302897845718053503583793, 6.39630004025078393758070455087, 6.41687336283557190778323553272, 6.51237889931001551178237694924, 7.06711674107894364964828474146, 7.12058810388824924021724994207, 7.29609222558563396266763543826

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