Properties

Label 6-5290e3-1.1-c1e3-0-2
Degree $6$
Conductor $148035889000$
Sign $-1$
Analytic cond. $75369.9$
Root an. cond. $6.49929$
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  + 3·2-s − 3-s + 6·4-s − 3·5-s − 3·6-s − 3·7-s + 10·8-s − 9-s − 9·10-s − 5·11-s − 6·12-s − 3·13-s − 9·14-s + 3·15-s + 15·16-s − 13·17-s − 3·18-s + 5·19-s − 18·20-s + 3·21-s − 15·22-s − 10·24-s + 6·25-s − 9·26-s + 27-s − 18·28-s + 2·29-s + ⋯
L(s)  = 1  + 2.12·2-s − 0.577·3-s + 3·4-s − 1.34·5-s − 1.22·6-s − 1.13·7-s + 3.53·8-s − 1/3·9-s − 2.84·10-s − 1.50·11-s − 1.73·12-s − 0.832·13-s − 2.40·14-s + 0.774·15-s + 15/4·16-s − 3.15·17-s − 0.707·18-s + 1.14·19-s − 4.02·20-s + 0.654·21-s − 3.19·22-s − 2.04·24-s + 6/5·25-s − 1.76·26-s + 0.192·27-s − 3.40·28-s + 0.371·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 5^{3} \cdot 23^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 5^{3} \cdot 23^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{3} \cdot 5^{3} \cdot 23^{6}\)
Sign: $-1$
Analytic conductor: \(75369.9\)
Root analytic conductor: \(6.49929\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{5290} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 2^{3} \cdot 5^{3} \cdot 23^{6} ,\ ( \ : 1/2, 1/2, 1/2 ),\ -1 )\)

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$C_1$ \( ( 1 - T )^{3} \)
5$C_1$ \( ( 1 + T )^{3} \)
23 \( 1 \)
good3$S_4\times C_2$ \( 1 + T + 2 T^{2} + 2 T^{3} + 2 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
7$S_4\times C_2$ \( 1 + 3 T + 6 T^{2} + 6 T^{3} + 6 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
11$S_4\times C_2$ \( 1 + 5 T + 34 T^{2} + 104 T^{3} + 34 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \)
13$S_4\times C_2$ \( 1 + 3 T + 24 T^{2} + 80 T^{3} + 24 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 + 13 T + 100 T^{2} + 490 T^{3} + 100 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \)
19$S_4\times C_2$ \( 1 - 5 T + 28 T^{2} - 4 p T^{3} + 28 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 - 2 T + 7 T^{2} - 212 T^{3} + 7 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 - 5 T + 44 T^{2} - 386 T^{3} + 44 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 - 2 T - 5 T^{2} - 172 T^{3} - 5 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 - 5 T + 124 T^{2} - 404 T^{3} + 124 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 + 2 T + 49 T^{2} + 268 T^{3} + 49 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 + 4 T + 25 T^{2} - 152 T^{3} + 25 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 + 12 T + 135 T^{2} + 1200 T^{3} + 135 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 + 16 T + 181 T^{2} + 1456 T^{3} + 181 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 + 9 T + 42 T^{2} - 180 T^{3} + 42 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 - 8 T + 193 T^{2} - 1000 T^{3} + 193 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 + 13 T + 262 T^{2} + 1894 T^{3} + 262 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 + 22 T + 299 T^{2} + 2860 T^{3} + 299 p T^{4} + 22 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 + 4 T + 121 T^{2} + 104 T^{3} + 121 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 - 8 T + 121 T^{2} - 1400 T^{3} + 121 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 + 6 T + 207 T^{2} + 780 T^{3} + 207 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 - 33 T + 588 T^{2} - 7050 T^{3} + 588 p T^{4} - 33 p^{2} T^{5} + p^{3} T^{6} \)
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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

−7.41788628347096620481516008106, −7.31187067804661166158739015584, −7.07813616535049958922285038028, −6.73727297102760109122138180987, −6.38480758613488357047713353199, −6.37808478109772162192608334779, −6.33596908134203565563182809644, −5.77899586591440957168400356729, −5.72921375183965423439269528734, −5.25787114561030642735006405671, −4.98185948304660248890110788684, −4.96789176554309532420247198927, −4.56879590447990468666404956097, −4.49000056674169312895275447553, −4.17107172911167583839091045844, −4.04898200582555408655478874385, −3.59247245033896319884606821920, −3.16795434838478300119454357167, −3.12879978799141244783620735686, −2.71320047095695757106163890873, −2.58464313245200089056499169978, −2.54979589165270434288103267782, −1.88312300423817918535834081321, −1.49995593021040348811320882232, −1.03102244215558555122025674850, 0, 0, 0, 1.03102244215558555122025674850, 1.49995593021040348811320882232, 1.88312300423817918535834081321, 2.54979589165270434288103267782, 2.58464313245200089056499169978, 2.71320047095695757106163890873, 3.12879978799141244783620735686, 3.16795434838478300119454357167, 3.59247245033896319884606821920, 4.04898200582555408655478874385, 4.17107172911167583839091045844, 4.49000056674169312895275447553, 4.56879590447990468666404956097, 4.96789176554309532420247198927, 4.98185948304660248890110788684, 5.25787114561030642735006405671, 5.72921375183965423439269528734, 5.77899586591440957168400356729, 6.33596908134203565563182809644, 6.37808478109772162192608334779, 6.38480758613488357047713353199, 6.73727297102760109122138180987, 7.07813616535049958922285038028, 7.31187067804661166158739015584, 7.41788628347096620481516008106

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