Properties

Degree 6
Conductor $ 3^{3} \cdot 5^{6} \cdot 7^{3} $
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  + 2-s − 3·3-s − 3·6-s − 3·7-s + 2·8-s + 6·9-s + 6·11-s − 6·13-s − 3·14-s + 3·16-s + 6·18-s + 6·19-s + 9·21-s + 6·22-s + 4·23-s − 6·24-s − 6·26-s − 10·27-s + 2·29-s + 2·31-s + 3·32-s − 18·33-s − 4·37-s + 6·38-s + 18·39-s + 2·41-s + 9·42-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.73·3-s − 1.22·6-s − 1.13·7-s + 0.707·8-s + 2·9-s + 1.80·11-s − 1.66·13-s − 0.801·14-s + 3/4·16-s + 1.41·18-s + 1.37·19-s + 1.96·21-s + 1.27·22-s + 0.834·23-s − 1.22·24-s − 1.17·26-s − 1.92·27-s + 0.371·29-s + 0.359·31-s + 0.530·32-s − 3.13·33-s − 0.657·37-s + 0.973·38-s + 2.88·39-s + 0.312·41-s + 1.38·42-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{6} \cdot 7^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{6} \cdot 7^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(6\)
\( N \)  =  \(3^{3} \cdot 5^{6} \cdot 7^{3}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{525} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((6,\ 3^{3} \cdot 5^{6} \cdot 7^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )\)
\(L(1)\)  \(\approx\)  \(1.85793\)
\(L(\frac12)\)  \(\approx\)  \(1.85793\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 6. If $p \in \{3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 5.
$p$$\Gal(F_p)$$F_p(T)$
bad3$C_1$ \( ( 1 + T )^{3} \)
5 \( 1 \)
7$C_1$ \( ( 1 + T )^{3} \)
good2$S_4\times C_2$ \( 1 - T + T^{2} - 3 T^{3} + p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
11$C_2$ \( ( 1 - 2 T + p T^{2} )^{3} \)
13$S_4\times C_2$ \( 1 + 6 T + 35 T^{2} + 148 T^{3} + 35 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 + 35 T^{2} - 16 T^{3} + 35 p T^{4} + p^{3} T^{6} \)
19$S_4\times C_2$ \( 1 - 6 T + 53 T^{2} - 188 T^{3} + 53 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 - 4 T + 61 T^{2} - 168 T^{3} + 61 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 - 2 T + 35 T^{2} - 76 T^{3} + 35 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 - 2 T + 41 T^{2} + 60 T^{3} + 41 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 + 4 T + 31 T^{2} + 232 T^{3} + 31 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 - 2 T + 63 T^{2} + 36 T^{3} + 63 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 - 4 T - 15 T^{2} + 488 T^{3} - 15 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 - 8 T + 109 T^{2} - 624 T^{3} + 109 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 - 14 T + 171 T^{2} - 1188 T^{3} + 171 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 - 16 T + 113 T^{2} - 608 T^{3} + 113 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 + 6 T + 131 T^{2} + 484 T^{3} + 131 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 - 8 T + 169 T^{2} - 944 T^{3} + 169 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
71$C_2$ \( ( 1 - 2 T + p T^{2} )^{3} \)
73$S_4\times C_2$ \( 1 + 18 T + 311 T^{2} + 2732 T^{3} + 311 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 - 12 T + 221 T^{2} - 1576 T^{3} + 221 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 - 8 T + 185 T^{2} - 1072 T^{3} + 185 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 - 14 T + 319 T^{2} - 2532 T^{3} + 319 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 + 22 T + 255 T^{2} + 2404 T^{3} + 255 p T^{4} + 22 p^{2} T^{5} + p^{3} T^{6} \)
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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

−9.914525071408972659806251594373, −9.316520234174628371140068914352, −9.284020288732556461940958544307, −9.156825295931349904739131572860, −8.362766972126603023432502689627, −8.230490685348470661746481354447, −7.54195956443254506082870524233, −7.33870709268040940690541670912, −7.06666112289600058325462250884, −6.87566073734968804379748228912, −6.64213211134235603508882279981, −6.18421278721861225303735663511, −5.92690994891109039255491340178, −5.34873866904477615142468144811, −5.27741902028662012725921251974, −5.14085153600714544753454493451, −4.30027321369541196988573082776, −4.22140353570702312656420354378, −4.17650345759519333851478902367, −3.35778747230251275316332301302, −3.14182764210480382492854285569, −2.43619215235081207014809590862, −1.89682845780111182442129770040, −0.956151394633825783233879657697, −0.799153297890762279963436329692, 0.799153297890762279963436329692, 0.956151394633825783233879657697, 1.89682845780111182442129770040, 2.43619215235081207014809590862, 3.14182764210480382492854285569, 3.35778747230251275316332301302, 4.17650345759519333851478902367, 4.22140353570702312656420354378, 4.30027321369541196988573082776, 5.14085153600714544753454493451, 5.27741902028662012725921251974, 5.34873866904477615142468144811, 5.92690994891109039255491340178, 6.18421278721861225303735663511, 6.64213211134235603508882279981, 6.87566073734968804379748228912, 7.06666112289600058325462250884, 7.33870709268040940690541670912, 7.54195956443254506082870524233, 8.230490685348470661746481354447, 8.362766972126603023432502689627, 9.156825295931349904739131572860, 9.284020288732556461940958544307, 9.316520234174628371140068914352, 9.914525071408972659806251594373

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