Properties

Label 6-950e3-1.1-c1e3-0-4
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 + 4·7-s + 10·8-s + 12·12-s + 8·13-s + 12·14-s + 15·16-s − 2·17-s + 3·19-s + 8·21-s + 20·24-s + 24·26-s − 4·27-s + 24·28-s − 8·29-s + 4·31-s + 21·32-s − 6·34-s + 14·37-s + 9·38-s + 16·39-s + 2·41-s + 24·42-s + 18·43-s + ⋯
L(s)  = 1  + 2.12·2-s + 1.15·3-s + 3·4-s + 2.44·6-s + 1.51·7-s + 3.53·8-s + 3.46·12-s + 2.21·13-s + 3.20·14-s + 15/4·16-s − 0.485·17-s + 0.688·19-s + 1.74·21-s + 4.08·24-s + 4.70·26-s − 0.769·27-s + 4.53·28-s − 1.48·29-s + 0.718·31-s + 3.71·32-s − 1.02·34-s + 2.30·37-s + 1.45·38-s + 2.56·39-s + 0.312·41-s + 3.70·42-s + 2.74·43-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

\(L(1)\) \(\approx\) \(25.37941039\)
\(L(\frac12)\) \(\approx\) \(25.37941039\)
\(L(\frac{3}{2})\) 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$S_4\times C_2$ \( 1 - 2 T + 4 T^{2} - 4 T^{3} + 4 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
7$S_4\times C_2$ \( 1 - 4 T + 20 T^{2} - 54 T^{3} + 20 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
11$S_4\times C_2$ \( 1 + 23 T^{2} - 8 T^{3} + 23 p T^{4} + p^{3} T^{6} \)
13$S_4\times C_2$ \( 1 - 8 T + 4 p T^{2} - 206 T^{3} + 4 p^{2} T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 + 2 T + 44 T^{2} + 64 T^{3} + 44 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 + 20 T^{2} + 122 T^{3} + 20 p T^{4} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 + 8 T + 36 T^{2} + 54 T^{3} + 36 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 - 4 T + p T^{2} - 16 T^{3} + p^{2} T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 - 14 T + 151 T^{2} - 1052 T^{3} + 151 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 - 2 T + 73 T^{2} - 264 T^{3} + 73 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 - 18 T + 227 T^{2} - 1696 T^{3} + 227 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 + 14 T + 173 T^{2} + 1252 T^{3} + 173 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 - 16 T + 236 T^{2} - 34 p T^{3} + 236 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 - 2 T + 148 T^{2} - 156 T^{3} + 148 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 + 30 T + 473 T^{2} + 4552 T^{3} + 473 p T^{4} + 30 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 - 2 T + 140 T^{2} - 204 T^{3} + 140 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 + 8 T + 91 T^{2} + 120 T^{3} + 91 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 - 10 T + 124 T^{2} - 1624 T^{3} + 124 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 + 9 T^{2} - 880 T^{3} + 9 p T^{4} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 + 6 T + 221 T^{2} + 988 T^{3} + 221 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 + 14 T + 313 T^{2} + 2472 T^{3} + 313 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 - 10 T + 191 T^{2} - 1452 T^{3} + 191 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
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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

−9.088954514169329491538434544461, −8.400727658299860311479107987103, −8.310643863562817307567122523703, −8.066584754341174477718301840495, −7.70112990065426159796451958898, −7.47989366033095291378484096825, −7.47863101195266106039433056515, −6.60771309481624430154368613679, −6.59993178550752668897699915371, −6.26737111533225019899374025043, −5.78483499119591474567777319996, −5.63849646536998177428560158018, −5.56426547275458976874290920043, −4.93123364380885842918339897804, −4.65421671376345175047536336500, −4.29964974453753053496762080771, −4.12689746501296188096492043279, −3.69723985435682161960086704932, −3.58574044521159132003033228842, −2.84996055664434429722477949435, −2.76153651521125170547512577970, −2.54368697698038401537038750792, −1.75092734376408264942781665029, −1.51482443109946751895611695575, −1.12642417778193919113120049963, 1.12642417778193919113120049963, 1.51482443109946751895611695575, 1.75092734376408264942781665029, 2.54368697698038401537038750792, 2.76153651521125170547512577970, 2.84996055664434429722477949435, 3.58574044521159132003033228842, 3.69723985435682161960086704932, 4.12689746501296188096492043279, 4.29964974453753053496762080771, 4.65421671376345175047536336500, 4.93123364380885842918339897804, 5.56426547275458976874290920043, 5.63849646536998177428560158018, 5.78483499119591474567777319996, 6.26737111533225019899374025043, 6.59993178550752668897699915371, 6.60771309481624430154368613679, 7.47863101195266106039433056515, 7.47989366033095291378484096825, 7.70112990065426159796451958898, 8.066584754341174477718301840495, 8.310643863562817307567122523703, 8.400727658299860311479107987103, 9.088954514169329491538434544461

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