Properties

Label 6-950e3-1.1-c1e3-0-1
Degree $6$
Conductor $857375000$
Sign $1$
Analytic cond. $436.517$
Root an. cond. $2.75423$
Motivic weight $1$
Arithmetic yes
Rational yes
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 − 2·3-s + 6·4-s + 6·6-s − 2·7-s − 10·8-s + 2·11-s − 12·12-s + 6·13-s + 6·14-s + 15·16-s − 12·17-s + 3·19-s + 4·21-s − 6·22-s + 2·23-s + 20·24-s − 18·26-s − 27-s − 12·28-s + 6·29-s + 8·31-s − 21·32-s − 4·33-s + 36·34-s + 37-s − 9·38-s + ⋯
L(s)  = 1  − 2.12·2-s − 1.15·3-s + 3·4-s + 2.44·6-s − 0.755·7-s − 3.53·8-s + 0.603·11-s − 3.46·12-s + 1.66·13-s + 1.60·14-s + 15/4·16-s − 2.91·17-s + 0.688·19-s + 0.872·21-s − 1.27·22-s + 0.417·23-s + 4.08·24-s − 3.53·26-s − 0.192·27-s − 2.26·28-s + 1.11·29-s + 1.43·31-s − 3.71·32-s − 0.696·33-s + 6.17·34-s + 0.164·37-s − 1.45·38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 5^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 5^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{3} \cdot 5^{6} \cdot 19^{3}\)
Sign: $1$
Analytic conductor: \(436.517\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 2^{3} \cdot 5^{6} \cdot 19^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.5274813846\)
\(L(\frac12)\) \(\approx\) \(0.5274813846\)
\(L(\frac{3}{2})\) 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 \( 1 \)
19$C_1$ \( ( 1 - T )^{3} \)
good3$S_4\times C_2$ \( 1 + 2 T + 4 T^{2} + p^{2} T^{3} + 4 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
7$S_4\times C_2$ \( 1 + 2 T + 8 T^{2} + 29 T^{3} + 8 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
11$S_4\times C_2$ \( 1 - 2 T + 9 T^{2} - 20 T^{3} + 9 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
13$S_4\times C_2$ \( 1 - 6 T + 36 T^{2} - 121 T^{3} + 36 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 + 12 T + 84 T^{2} + 399 T^{3} + 84 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 - 2 T + 18 T^{2} - 203 T^{3} + 18 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 - 6 T + 84 T^{2} - 339 T^{3} + 84 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 - 8 T + 89 T^{2} - 440 T^{3} + 89 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 - T + 86 T^{2} - 73 T^{3} + 86 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
41$C_2$ \( ( 1 + p T^{2} )^{3} \)
43$S_4\times C_2$ \( 1 - 14 T + 169 T^{2} - 1180 T^{3} + 169 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 + 3 T + 84 T^{2} + 327 T^{3} + 84 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 - 10 T + 54 T^{2} - 193 T^{3} + 54 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 - 6 T - 24 T^{2} + 723 T^{3} - 24 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 + 2 T + 127 T^{2} + 124 T^{3} + 127 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 + 4 T + 178 T^{2} + 461 T^{3} + 178 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 + 6 T + 21 T^{2} - 660 T^{3} + 21 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 - 12 T + 192 T^{2} - 1435 T^{3} + 192 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 - 20 T + 269 T^{2} - 2840 T^{3} + 269 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 - 4 T + 141 T^{2} - 832 T^{3} + 141 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 - 14 T + 231 T^{2} - 2180 T^{3} + 231 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 - 28 T + 335 T^{2} - 2992 T^{3} + 335 p T^{4} - 28 p^{2} T^{5} + p^{3} T^{6} \)
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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

−8.982276324623309472042494640925, −8.639001068839464170886260170496, −8.521448830253231235359416044607, −8.332061166287800552966024863925, −7.87090803372382901348887543614, −7.41217508186696347892222880658, −7.38488087519467905986878206804, −6.79987784882685878751611341578, −6.62475566274911169556962867734, −6.49354109200159539805895542335, −6.04700981419646254201216884446, −6.00041715106679067236547214076, −5.89352257333102002536281791366, −5.03309684466173152644774972770, −4.90734770671744497980694283616, −4.48541362263047129672427065251, −3.96201196858484717118376599700, −3.60856768550939118383302149255, −3.30471999333405805959327841466, −2.70461780836463595661692656779, −2.31306649710184920689067678148, −2.08305757445137087473678268818, −1.33426300363512606444126662482, −0.67496779432486798073596660042, −0.62032701408482285459408628652, 0.62032701408482285459408628652, 0.67496779432486798073596660042, 1.33426300363512606444126662482, 2.08305757445137087473678268818, 2.31306649710184920689067678148, 2.70461780836463595661692656779, 3.30471999333405805959327841466, 3.60856768550939118383302149255, 3.96201196858484717118376599700, 4.48541362263047129672427065251, 4.90734770671744497980694283616, 5.03309684466173152644774972770, 5.89352257333102002536281791366, 6.00041715106679067236547214076, 6.04700981419646254201216884446, 6.49354109200159539805895542335, 6.62475566274911169556962867734, 6.79987784882685878751611341578, 7.38488087519467905986878206804, 7.41217508186696347892222880658, 7.87090803372382901348887543614, 8.332061166287800552966024863925, 8.521448830253231235359416044607, 8.639001068839464170886260170496, 8.982276324623309472042494640925

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