Properties

Label 6-950e3-1.1-c1e3-0-0
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 + 4·9-s + 2·11-s − 12·12-s + 2·13-s + 6·14-s + 15·16-s + 4·17-s − 12·18-s − 3·19-s + 4·21-s − 6·22-s − 14·23-s + 20·24-s − 6·26-s − 27-s − 12·28-s + 14·29-s − 4·31-s − 21·32-s − 4·33-s − 12·34-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 + 4/3·9-s + 0.603·11-s − 3.46·12-s + 0.554·13-s + 1.60·14-s + 15/4·16-s + 0.970·17-s − 2.82·18-s − 0.688·19-s + 0.872·21-s − 1.27·22-s − 2.91·23-s + 4.08·24-s − 1.17·26-s − 0.192·27-s − 2.26·28-s + 2.59·29-s − 0.718·31-s − 3.71·32-s − 0.696·33-s − 2.05·34-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: induced by $\chi_{950} (1, \cdot )$
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

\(L(1)\) \(\approx\) \(0.4316766307\)
\(L(\frac12)\) \(\approx\) \(0.4316766307\)
\(L(\frac{3}{2})\) 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$S_4\times C_2$ \( 1 + 2 T - 7 T^{3} + 2 p^{2} T^{5} + p^{3} T^{6} \)
7$S_4\times C_2$ \( 1 + 2 T - 3 p T^{3} + 2 p^{2} T^{5} + p^{3} T^{6} \)
11$S_4\times C_2$ \( 1 - 2 T + T^{2} - 20 T^{3} + p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
13$S_4\times C_2$ \( 1 - 2 T + 16 T^{2} - 73 T^{3} + 16 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 - 4 T + 36 T^{2} - 143 T^{3} + 36 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 + 14 T + 126 T^{2} + 707 T^{3} + 126 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 - 14 T + 128 T^{2} - 837 T^{3} + 128 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 + 4 T + 57 T^{2} + 96 T^{3} + 57 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 - 17 T + 174 T^{2} - 1249 T^{3} + 174 p T^{4} - 17 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 + 8 T + 75 T^{2} + 592 T^{3} + 75 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 + 2 T + 89 T^{2} + 244 T^{3} + 89 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 + 13 T + 116 T^{2} + 697 T^{3} + 116 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 - 10 T + 170 T^{2} - 1057 T^{3} + 170 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 + 6 T + 148 T^{2} + 533 T^{3} + 148 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 - 22 T + 255 T^{2} - 2148 T^{3} + 255 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 + 70 T^{2} + 7 p T^{3} + 70 p T^{4} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 + 2 T + 197 T^{2} + 260 T^{3} + 197 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 - 12 T + 192 T^{2} - 1509 T^{3} + 192 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 - 24 T + 349 T^{2} - 3472 T^{3} + 349 p T^{4} - 24 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 + 12 T + 85 T^{2} + 448 T^{3} + 85 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \)
89$C_2$ \( ( 1 + 10 T + p T^{2} )^{3} \)
97$S_4\times C_2$ \( 1 + 111 T^{2} - 648 T^{3} + 111 p T^{4} + p^{3} T^{6} \)
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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

−9.105715558906536678433583184720, −8.394426162959089550157129719986, −8.343265806475094067947076399705, −8.282540130943472470637899407321, −8.031710433744774355856835215129, −7.42610218019810107613377226861, −7.42406151617227104596017565192, −6.71419546895230966046710993368, −6.62229112629908940967412365005, −6.59494829848030558341751140422, −6.23488625279696781589515800554, −5.86890229701844753394653297861, −5.73763677719512194073549486951, −5.22847686210246455263254895493, −4.82610909478823080427932893358, −4.28496907792576111862592985338, −4.12784535879209137338614423943, −3.57201579756469169734144714353, −3.39413889247916756875369629856, −2.63166424789398613292078802838, −2.44082091680374607224470303204, −1.88315830857719771936136713808, −1.32777611572376336861192585493, −0.968573678160421751939388915416, −0.41536246201856860465777332556, 0.41536246201856860465777332556, 0.968573678160421751939388915416, 1.32777611572376336861192585493, 1.88315830857719771936136713808, 2.44082091680374607224470303204, 2.63166424789398613292078802838, 3.39413889247916756875369629856, 3.57201579756469169734144714353, 4.12784535879209137338614423943, 4.28496907792576111862592985338, 4.82610909478823080427932893358, 5.22847686210246455263254895493, 5.73763677719512194073549486951, 5.86890229701844753394653297861, 6.23488625279696781589515800554, 6.59494829848030558341751140422, 6.62229112629908940967412365005, 6.71419546895230966046710993368, 7.42406151617227104596017565192, 7.42610218019810107613377226861, 8.031710433744774355856835215129, 8.282540130943472470637899407321, 8.343265806475094067947076399705, 8.394426162959089550157129719986, 9.105715558906536678433583184720

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