Properties

Label 6-950e3-1.1-c1e3-0-5
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 $3$

Origins

Origins of factors

Downloads

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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: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 2^{3} \cdot 5^{6} \cdot 19^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ -1 )\)

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$C_1$ \( ( 1 + T )^{3} \)
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

−9.570129306125444829423161640153, −8.928598727444693045189016575900, −8.918176324831190525046078950281, −8.893607415941541781459788532898, −8.060076202098181060261049926615, −7.993329020850616265206428203782, −7.67320077953484857098807346540, −7.37961330546576687399460410872, −7.16067638001083096200682895369, −6.91417056560953214042847626485, −6.45297168503771135047305077366, −6.33009541371317790535834993373, −6.21227652886072284894188099584, −5.49942535587784043290781602513, −5.39178419415701584695387620667, −5.24993733857020100458888328404, −4.73324696949202920321370441690, −4.28342306821144394432530696198, −3.74402461492789290157897642506, −3.04008517123061672221722669610, −2.98904386941154966480214527286, −2.96927646352108528622151749854, −2.14600112983315418433596422853, −1.55564356243632533712808947973, −1.37919515776564355708143992288, 0, 0, 0, 1.37919515776564355708143992288, 1.55564356243632533712808947973, 2.14600112983315418433596422853, 2.96927646352108528622151749854, 2.98904386941154966480214527286, 3.04008517123061672221722669610, 3.74402461492789290157897642506, 4.28342306821144394432530696198, 4.73324696949202920321370441690, 5.24993733857020100458888328404, 5.39178419415701584695387620667, 5.49942535587784043290781602513, 6.21227652886072284894188099584, 6.33009541371317790535834993373, 6.45297168503771135047305077366, 6.91417056560953214042847626485, 7.16067638001083096200682895369, 7.37961330546576687399460410872, 7.67320077953484857098807346540, 7.993329020850616265206428203782, 8.060076202098181060261049926615, 8.893607415941541781459788532898, 8.918176324831190525046078950281, 8.928598727444693045189016575900, 9.570129306125444829423161640153

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