Properties

Label 6-4025e3-1.1-c1e3-0-0
Degree $6$
Conductor $65207515625$
Sign $1$
Analytic cond. $33199.3$
Root an. cond. $5.66919$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 2·3-s − 2·6-s + 3·7-s + 2·8-s − 3·9-s + 4·11-s − 2·13-s + 3·14-s + 3·16-s − 4·17-s − 3·18-s + 8·19-s − 6·21-s + 4·22-s − 3·23-s − 4·24-s − 2·26-s + 10·27-s − 2·29-s + 16·31-s + 3·32-s − 8·33-s − 4·34-s + 6·37-s + 8·38-s + 4·39-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.15·3-s − 0.816·6-s + 1.13·7-s + 0.707·8-s − 9-s + 1.20·11-s − 0.554·13-s + 0.801·14-s + 3/4·16-s − 0.970·17-s − 0.707·18-s + 1.83·19-s − 1.30·21-s + 0.852·22-s − 0.625·23-s − 0.816·24-s − 0.392·26-s + 1.92·27-s − 0.371·29-s + 2.87·31-s + 0.530·32-s − 1.39·33-s − 0.685·34-s + 0.986·37-s + 1.29·38-s + 0.640·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{6} \cdot 7^{3} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{6} \cdot 7^{3} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(5^{6} \cdot 7^{3} \cdot 23^{3}\)
Sign: $1$
Analytic conductor: \(33199.3\)
Root analytic conductor: \(5.66919\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 5^{6} \cdot 7^{3} \cdot 23^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(3.989216684\)
\(L(\frac12)\) \(\approx\) \(3.989216684\)
\(L(\frac{3}{2})\) 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)$
bad5 \( 1 \)
7$C_1$ \( ( 1 - T )^{3} \)
23$C_1$ \( ( 1 + T )^{3} \)
good2$S_4\times C_2$ \( 1 - T + T^{2} - 3 T^{3} + p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
3$S_4\times C_2$ \( 1 + 2 T + 7 T^{2} + 10 T^{3} + 7 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
11$S_4\times C_2$ \( 1 - 4 T + 3 p T^{2} - 84 T^{3} + 3 p^{2} T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
13$D_{6}$ \( 1 + 2 T + 27 T^{2} + 44 T^{3} + 27 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 + 4 T + 53 T^{2} + 134 T^{3} + 53 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
19$S_4\times C_2$ \( 1 - 8 T + 41 T^{2} - 144 T^{3} + 41 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 + 2 T + 79 T^{2} + 120 T^{3} + 79 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 - 16 T + 119 T^{2} - 654 T^{3} + 119 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 - 6 T + 107 T^{2} - 404 T^{3} + 107 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 + 14 T + 135 T^{2} + 996 T^{3} + 135 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 - 8 T + 89 T^{2} - 384 T^{3} + 89 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 + 16 T + 203 T^{2} + 1514 T^{3} + 203 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 + 6 T + 107 T^{2} + 388 T^{3} + 107 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 + 10 T + 75 T^{2} + 210 T^{3} + 75 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 - 10 T + 137 T^{2} - 726 T^{3} + 137 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 + 16 T + 193 T^{2} + 1468 T^{3} + 193 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 - 4 T + 133 T^{2} - 504 T^{3} + 133 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 - 6 T + 111 T^{2} - 660 T^{3} + 111 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 - 16 T + 245 T^{2} - 2100 T^{3} + 245 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 + 20 T + 145 T^{2} + 648 T^{3} + 145 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 - 4 T + 153 T^{2} - 990 T^{3} + 153 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 + 6 T + 269 T^{2} + 1166 T^{3} + 269 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
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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

−7.55094611493570232224494915695, −7.14838417074017457430152474628, −6.73800115265251183416342396957, −6.63264666279038371027171497431, −6.51887004167577004271702838700, −6.11418879267833437266980956828, −6.11244313467841724458671412888, −5.51360607472759548503880074758, −5.33930079429844532811989389245, −5.28816853780363849232015189000, −5.03299347904650089768613060712, −4.61207763373888271781025574341, −4.55347115449464553874552600374, −4.28171840738164095283746032644, −3.98621684787323796951996007523, −3.72724678193834488615221403709, −3.18331514332770643042462514844, −2.95260857445131105142650594155, −2.66202331167526240277853732272, −2.56039456724400180162346232427, −1.63738074433197946410813916656, −1.58069807411594877123945982277, −1.46690919173678160397908538507, −0.57473501613142959468924361549, −0.54471909104086818357053693887, 0.54471909104086818357053693887, 0.57473501613142959468924361549, 1.46690919173678160397908538507, 1.58069807411594877123945982277, 1.63738074433197946410813916656, 2.56039456724400180162346232427, 2.66202331167526240277853732272, 2.95260857445131105142650594155, 3.18331514332770643042462514844, 3.72724678193834488615221403709, 3.98621684787323796951996007523, 4.28171840738164095283746032644, 4.55347115449464553874552600374, 4.61207763373888271781025574341, 5.03299347904650089768613060712, 5.28816853780363849232015189000, 5.33930079429844532811989389245, 5.51360607472759548503880074758, 6.11244313467841724458671412888, 6.11418879267833437266980956828, 6.51887004167577004271702838700, 6.63264666279038371027171497431, 6.73800115265251183416342396957, 7.14838417074017457430152474628, 7.55094611493570232224494915695

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