Properties

Label 6-4788e3-1.1-c1e3-0-1
Degree $6$
Conductor $109764631872$
Sign $-1$
Analytic cond. $55884.8$
Root an. cond. $6.18323$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $3$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s − 3·7-s + 3·11-s + 8·13-s − 9·17-s − 3·19-s − 4·23-s − 2·25-s − 11·29-s − 5·31-s + 6·35-s + 4·37-s − 11·41-s − 4·43-s + 2·47-s + 6·49-s − 9·53-s − 6·55-s + 12·59-s − 2·61-s − 16·65-s − 9·67-s − 21·73-s − 9·77-s + 6·79-s + 7·83-s + 18·85-s + ⋯
L(s)  = 1  − 0.894·5-s − 1.13·7-s + 0.904·11-s + 2.21·13-s − 2.18·17-s − 0.688·19-s − 0.834·23-s − 2/5·25-s − 2.04·29-s − 0.898·31-s + 1.01·35-s + 0.657·37-s − 1.71·41-s − 0.609·43-s + 0.291·47-s + 6/7·49-s − 1.23·53-s − 0.809·55-s + 1.56·59-s − 0.256·61-s − 1.98·65-s − 1.09·67-s − 2.45·73-s − 1.02·77-s + 0.675·79-s + 0.768·83-s + 1.95·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 7^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 7^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{6} \cdot 3^{6} \cdot 7^{3} \cdot 19^{3}\)
Sign: $-1$
Analytic conductor: \(55884.8\)
Root analytic conductor: \(6.18323\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 2^{6} \cdot 3^{6} \cdot 7^{3} \cdot 19^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ -1 )\)

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 \( 1 \)
7$C_1$ \( ( 1 + T )^{3} \)
19$C_1$ \( ( 1 + T )^{3} \)
good5$S_4\times C_2$ \( 1 + 2 T + 6 T^{2} + 6 T^{3} + 6 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
11$S_4\times C_2$ \( 1 - 3 T + 26 T^{2} - 46 T^{3} + 26 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \)
13$S_4\times C_2$ \( 1 - 8 T + 50 T^{2} - 192 T^{3} + 50 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 + 9 T + 4 p T^{2} + 292 T^{3} + 4 p^{2} T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 + 4 T + 64 T^{2} + 180 T^{3} + 64 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 + 11 T + 102 T^{2} + 636 T^{3} + 102 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 + 5 T + 76 T^{2} + 314 T^{3} + 76 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 - 4 T + 92 T^{2} - 246 T^{3} + 92 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 + 11 T + 110 T^{2} + 622 T^{3} + 110 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 + 4 T + 37 T^{2} + 376 T^{3} + 37 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 - 2 T + 66 T^{2} - 372 T^{3} + 66 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 + 9 T + 120 T^{2} + 632 T^{3} + 120 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 - 12 T + 132 T^{2} - 1308 T^{3} + 132 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 + 2 T + 26 T^{2} + 42 T^{3} + 26 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 + 9 T + 218 T^{2} + 1192 T^{3} + 218 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 + 158 T^{2} + 58 T^{3} + 158 p T^{4} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 + 21 T + 302 T^{2} + 2764 T^{3} + 302 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 - 6 T + 89 T^{2} + 68 T^{3} + 89 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 - 7 T + 204 T^{2} - 1062 T^{3} + 204 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 - 18 T + 335 T^{2} - 3268 T^{3} + 335 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 - 28 T + 330 T^{2} - 2896 T^{3} + 330 p T^{4} - 28 p^{2} T^{5} + p^{3} T^{6} \)
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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

−7.80233357009246873797216254831, −7.45709637976400741588544457565, −7.05525600287532077528601368695, −7.01460571540713490223087234466, −6.58981154981396501174413316534, −6.48949630681350100625244847123, −6.35892855877094674435060863559, −5.89121501983364901012169717262, −5.87936516085587305840767800565, −5.81180814346290804813736336848, −5.08894741655729006122253346542, −4.84329458615620407145141925152, −4.83133918485081641807896272980, −4.14156748431979660425516907669, −4.01055706408249825884400235926, −3.95715600855877940973954997520, −3.61417168403294593911068923119, −3.48509581821501164855341846140, −3.33193055776557401191519542690, −2.63091160861477335470633531621, −2.40920811542236548713162993931, −2.12635607869483977801507754761, −1.71142352880627929744876132350, −1.32804388299010437186613276852, −1.14207886106608566229236838372, 0, 0, 0, 1.14207886106608566229236838372, 1.32804388299010437186613276852, 1.71142352880627929744876132350, 2.12635607869483977801507754761, 2.40920811542236548713162993931, 2.63091160861477335470633531621, 3.33193055776557401191519542690, 3.48509581821501164855341846140, 3.61417168403294593911068923119, 3.95715600855877940973954997520, 4.01055706408249825884400235926, 4.14156748431979660425516907669, 4.83133918485081641807896272980, 4.84329458615620407145141925152, 5.08894741655729006122253346542, 5.81180814346290804813736336848, 5.87936516085587305840767800565, 5.89121501983364901012169717262, 6.35892855877094674435060863559, 6.48949630681350100625244847123, 6.58981154981396501174413316534, 7.01460571540713490223087234466, 7.05525600287532077528601368695, 7.45709637976400741588544457565, 7.80233357009246873797216254831

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