Properties

Degree 6
Conductor $ 2^{3} \cdot 3^{3} \cdot 7^{6} \cdot 11^{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  + 3·2-s − 3·3-s + 6·4-s − 9·6-s + 10·8-s + 6·9-s + 3·11-s − 18·12-s + 15·16-s − 3·17-s + 18·18-s + 3·19-s + 9·22-s + 9·23-s − 30·24-s − 6·25-s − 10·27-s + 9·29-s − 6·31-s + 21·32-s − 9·33-s − 9·34-s + 36·36-s + 9·37-s + 9·38-s − 12·41-s + 9·43-s + ⋯
L(s)  = 1  + 2.12·2-s − 1.73·3-s + 3·4-s − 3.67·6-s + 3.53·8-s + 2·9-s + 0.904·11-s − 5.19·12-s + 15/4·16-s − 0.727·17-s + 4.24·18-s + 0.688·19-s + 1.91·22-s + 1.87·23-s − 6.12·24-s − 6/5·25-s − 1.92·27-s + 1.67·29-s − 1.07·31-s + 3.71·32-s − 1.56·33-s − 1.54·34-s + 6·36-s + 1.47·37-s + 1.45·38-s − 1.87·41-s + 1.37·43-s + ⋯

Functional equation

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

Invariants

\( d \)  =  \(6\)
\( N \)  =  \(2^{3} \cdot 3^{3} \cdot 7^{6} \cdot 11^{3}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{3234} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((6,\ 2^{3} \cdot 3^{3} \cdot 7^{6} \cdot 11^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )\)
\(L(1)\)  \(\approx\)  \(13.51962232\)
\(L(\frac12)\)  \(\approx\)  \(13.51962232\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7,\;11\}$,\(F_p(T)\) is a polynomial of degree 6. If $p \in \{2,\;3,\;7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 5.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_1$ \( ( 1 - T )^{3} \)
3$C_1$ \( ( 1 + T )^{3} \)
7 \( 1 \)
11$C_1$ \( ( 1 - T )^{3} \)
good5$S_4\times C_2$ \( 1 + 6 T^{2} - 4 T^{3} + 6 p T^{4} + p^{3} T^{6} \)
13$S_4\times C_2$ \( 1 + 3 T^{2} - 32 T^{3} + 3 p T^{4} + p^{3} T^{6} \)
17$C_2$ \( ( 1 + T + p T^{2} )^{3} \)
19$S_4\times C_2$ \( 1 - 3 T + 21 T^{2} - 150 T^{3} + 21 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 - 9 T + 72 T^{2} - 393 T^{3} + 72 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 - 9 T + 3 p T^{2} - 430 T^{3} + 3 p^{2} T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 + 6 T - 3 T^{2} - 140 T^{3} - 3 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 - 9 T + 99 T^{2} - 502 T^{3} + 99 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 + 12 T + 144 T^{2} + 22 p T^{3} + 144 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 - 9 T + 117 T^{2} - 610 T^{3} + 117 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 - 15 T + 192 T^{2} - 1391 T^{3} + 192 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 + 123 T^{2} + 32 T^{3} + 123 p T^{4} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 + 9 T + 81 T^{2} + 294 T^{3} + 81 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 - 6 T + 54 T^{2} - 506 T^{3} + 54 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 - 6 T + 186 T^{2} - 720 T^{3} + 186 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 - 3 T + 189 T^{2} - 362 T^{3} + 189 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 + 123 T^{2} + 192 T^{3} + 123 p T^{4} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 - 12 T + 204 T^{2} - 1528 T^{3} + 204 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 - 6 T + 138 T^{2} - 1184 T^{3} + 138 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 - 36 T + 663 T^{2} - 7736 T^{3} + 663 p T^{4} - 36 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 + 9 T + 210 T^{2} + 1753 T^{3} + 210 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \)
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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

−7.49451183911909946579603274035, −7.28695198134806312197846061524, −7.01634175439641223572151640372, −6.67464780534178965950668735202, −6.44484662389975323630698182426, −6.26982016991356758216391574279, −6.26441365391880491461536260210, −5.72642615451475320633362944333, −5.67755008610389425682331262641, −5.25760474525030452728667552672, −4.98800200704514162585494332539, −4.97493418352912517494582393066, −4.63549863711891001640384890407, −4.26449030872215427077904287456, −4.06029465601077444653575257761, −3.96138197381220719775762658231, −3.35531447810459730823915449348, −3.13906998293170211770781339160, −3.12538746655731341828080346638, −2.19817288396331089543519404403, −2.18105242605040375493198853465, −1.96624841020152280249330656048, −1.19143127551374545663880035965, −0.854915983329421438109737707473, −0.68777103118958090452718532573, 0.68777103118958090452718532573, 0.854915983329421438109737707473, 1.19143127551374545663880035965, 1.96624841020152280249330656048, 2.18105242605040375493198853465, 2.19817288396331089543519404403, 3.12538746655731341828080346638, 3.13906998293170211770781339160, 3.35531447810459730823915449348, 3.96138197381220719775762658231, 4.06029465601077444653575257761, 4.26449030872215427077904287456, 4.63549863711891001640384890407, 4.97493418352912517494582393066, 4.98800200704514162585494332539, 5.25760474525030452728667552672, 5.67755008610389425682331262641, 5.72642615451475320633362944333, 6.26441365391880491461536260210, 6.26982016991356758216391574279, 6.44484662389975323630698182426, 6.67464780534178965950668735202, 7.01634175439641223572151640372, 7.28695198134806312197846061524, 7.49451183911909946579603274035

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