Properties

Degree $6$
Conductor $217298593728$
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  − 4·11-s + 2·13-s + 6·17-s − 4·19-s − 4·23-s − 11·25-s + 4·31-s + 10·37-s − 14·41-s − 20·43-s + 16·47-s − 5·49-s − 6·53-s − 24·59-s − 2·61-s + 10·67-s − 12·71-s − 10·73-s − 2·79-s − 4·83-s − 12·89-s + 18·97-s + 20·101-s + 6·103-s − 8·107-s − 6·109-s − 20·113-s + ⋯
L(s)  = 1  − 1.20·11-s + 0.554·13-s + 1.45·17-s − 0.917·19-s − 0.834·23-s − 2.19·25-s + 0.718·31-s + 1.64·37-s − 2.18·41-s − 3.04·43-s + 2.33·47-s − 5/7·49-s − 0.824·53-s − 3.12·59-s − 0.256·61-s + 1.22·67-s − 1.42·71-s − 1.17·73-s − 0.225·79-s − 0.439·83-s − 1.27·89-s + 1.82·97-s + 1.99·101-s + 0.591·103-s − 0.773·107-s − 0.574·109-s − 1.88·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 167^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 167^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{6} \cdot 3^{6} \cdot 167^{3}\)
Sign: $-1$
Motivic weight: \(1\)
Character: induced by $\chi_{6012} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 2^{6} \cdot 3^{6} \cdot 167^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ -1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3 \( 1 \)
167$C_1$ \( ( 1 - T )^{3} \)
good5$S_4\times C_2$ \( 1 + 11 T^{2} - 2 T^{3} + 11 p T^{4} + p^{3} T^{6} \)
7$S_4\times C_2$ \( 1 + 5 T^{2} + 16 T^{3} + 5 p T^{4} + p^{3} T^{6} \)
11$S_4\times C_2$ \( 1 + 4 T + 3 p T^{2} + 84 T^{3} + 3 p^{2} T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
13$D_{6}$ \( 1 - 2 T + 27 T^{2} - 44 T^{3} + 27 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 - 6 T + 59 T^{2} - 206 T^{3} + 59 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
19$S_4\times C_2$ \( 1 + 4 T + 41 T^{2} + 120 T^{3} + 41 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 + 4 T + 21 T^{2} + 120 T^{3} + 21 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 + 23 T^{2} - 128 T^{3} + 23 p T^{4} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 - 4 T + 61 T^{2} - 280 T^{3} + 61 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 - 10 T + 91 T^{2} - 748 T^{3} + 91 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 + 14 T + 151 T^{2} + 1026 T^{3} + 151 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 + 20 T + 241 T^{2} + 1838 T^{3} + 241 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 - 16 T + 221 T^{2} - 1628 T^{3} + 221 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 + 6 T + 59 T^{2} + 170 T^{3} + 59 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 + 24 T + 353 T^{2} + 3232 T^{3} + 353 p T^{4} + 24 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 + 2 T + 179 T^{2} + 240 T^{3} + 179 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 - 10 T + 221 T^{2} - 1314 T^{3} + 221 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
71$C_2$ \( ( 1 + 4 T + p T^{2} )^{3} \)
73$S_4\times C_2$ \( 1 + 10 T + 231 T^{2} + 1420 T^{3} + 231 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 + 2 T + 37 T^{2} + 650 T^{3} + 37 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 + 4 T + 161 T^{2} + 680 T^{3} + 161 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 + 12 T + 251 T^{2} + 1816 T^{3} + 251 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 - 18 T + 255 T^{2} - 2412 T^{3} + 255 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \)
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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

−7.69351671241809048579920522787, −7.25861687254358280458469294381, −7.19069949958222107642056482055, −6.71674303040775610606304936155, −6.37672818122130384300543132686, −6.29930270981859626091961289981, −6.07535659185797601654896217454, −5.86112989580193061797586144262, −5.71545693289591363208641375503, −5.33424519521832652976416456619, −4.90763486570166228243254261752, −4.84708064954987266911537640589, −4.83368052695664527226375766411, −4.19899988928164585437810440648, −3.97544583953630139217965415944, −3.90221585893798248416976063916, −3.33035234859659137801959153193, −3.22789536614514541721656648704, −3.17654083489179157909077119095, −2.41515282276789002075468673689, −2.30727559459510490378305363951, −2.28197009010112592803274883800, −1.36162988100844992625867441643, −1.36158697416898819048187976333, −1.33725707974921100276106148428, 0, 0, 0, 1.33725707974921100276106148428, 1.36158697416898819048187976333, 1.36162988100844992625867441643, 2.28197009010112592803274883800, 2.30727559459510490378305363951, 2.41515282276789002075468673689, 3.17654083489179157909077119095, 3.22789536614514541721656648704, 3.33035234859659137801959153193, 3.90221585893798248416976063916, 3.97544583953630139217965415944, 4.19899988928164585437810440648, 4.83368052695664527226375766411, 4.84708064954987266911537640589, 4.90763486570166228243254261752, 5.33424519521832652976416456619, 5.71545693289591363208641375503, 5.86112989580193061797586144262, 6.07535659185797601654896217454, 6.29930270981859626091961289981, 6.37672818122130384300543132686, 6.71674303040775610606304936155, 7.19069949958222107642056482055, 7.25861687254358280458469294381, 7.69351671241809048579920522787

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