Properties

Label 6-2166e3-1.1-c1e3-0-3
Degree $6$
Conductor $10161910296$
Sign $1$
Analytic cond. $5173.76$
Root an. cond. $4.15879$
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  + 3·2-s + 3·3-s + 6·4-s + 9·6-s − 3·7-s + 10·8-s + 6·9-s + 18·12-s + 6·13-s − 9·14-s + 15·16-s + 6·17-s + 18·18-s − 9·21-s + 6·23-s + 30·24-s − 6·25-s + 18·26-s + 10·27-s − 18·28-s + 6·29-s + 9·31-s + 21·32-s + 18·34-s + 36·36-s + 9·37-s + 18·39-s + ⋯
L(s)  = 1  + 2.12·2-s + 1.73·3-s + 3·4-s + 3.67·6-s − 1.13·7-s + 3.53·8-s + 2·9-s + 5.19·12-s + 1.66·13-s − 2.40·14-s + 15/4·16-s + 1.45·17-s + 4.24·18-s − 1.96·21-s + 1.25·23-s + 6.12·24-s − 6/5·25-s + 3.53·26-s + 1.92·27-s − 3.40·28-s + 1.11·29-s + 1.61·31-s + 3.71·32-s + 3.08·34-s + 6·36-s + 1.47·37-s + 2.88·39-s + ⋯

Functional equation

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

Invariants

Degree: \(6\)
Conductor: \(2^{3} \cdot 3^{3} \cdot 19^{6}\)
Sign: $1$
Analytic conductor: \(5173.76\)
Root analytic conductor: \(4.15879\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2166} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 2^{3} \cdot 3^{3} \cdot 19^{6} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(41.60275814\)
\(L(\frac12)\) \(\approx\) \(41.60275814\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{3} \)
3$C_1$ \( ( 1 - T )^{3} \)
19 \( 1 \)
good5$A_4\times C_2$ \( 1 + 6 T^{2} + 9 T^{3} + 6 p T^{4} + p^{3} T^{6} \)
7$A_4\times C_2$ \( 1 + 3 T + 3 p T^{2} + 41 T^{3} + 3 p^{2} T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
11$A_4\times C_2$ \( 1 + 30 T^{2} - T^{3} + 30 p T^{4} + p^{3} T^{6} \)
13$A_4\times C_2$ \( 1 - 6 T + 48 T^{2} - 159 T^{3} + 48 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
17$A_4\times C_2$ \( 1 - 6 T + 42 T^{2} - 11 p T^{3} + 42 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
23$A_4\times C_2$ \( 1 - 6 T + 24 T^{2} - 7 T^{3} + 24 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
29$A_4\times C_2$ \( 1 - 6 T + 42 T^{2} - 135 T^{3} + 42 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
31$A_4\times C_2$ \( 1 - 9 T + 3 p T^{2} - 531 T^{3} + 3 p^{2} T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \)
37$C_6$ \( 1 - 9 T + 54 T^{2} - 305 T^{3} + 54 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \)
41$A_4\times C_2$ \( 1 - 9 T + 111 T^{2} - 629 T^{3} + 111 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \)
43$A_4\times C_2$ \( 1 - 15 T + 183 T^{2} - 1347 T^{3} + 183 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \)
47$A_4\times C_2$ \( 1 - 3 T + 117 T^{2} - 229 T^{3} + 117 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \)
53$A_4\times C_2$ \( 1 + 3 T + 6 T^{2} + 11 T^{3} + 6 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
59$A_4\times C_2$ \( 1 + 12 T + 144 T^{2} + 1399 T^{3} + 144 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \)
61$A_4\times C_2$ \( 1 + 33 T + 543 T^{2} + 5323 T^{3} + 543 p T^{4} + 33 p^{2} T^{5} + p^{3} T^{6} \)
67$A_4\times C_2$ \( 1 - 12 T + 210 T^{2} - 1605 T^{3} + 210 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \)
71$A_4\times C_2$ \( 1 - 15 T + 159 T^{2} - 1681 T^{3} + 159 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \)
73$A_4\times C_2$ \( 1 - 3 T + 105 T^{2} - 475 T^{3} + 105 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \)
79$A_4\times C_2$ \( 1 - 15 T + 87 T^{2} - 245 T^{3} + 87 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \)
83$A_4\times C_2$ \( 1 - 27 T + 471 T^{2} - 4985 T^{3} + 471 p T^{4} - 27 p^{2} T^{5} + p^{3} T^{6} \)
89$A_4\times C_2$ \( 1 + 15 T + 231 T^{2} + 2563 T^{3} + 231 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \)
97$A_4\times C_2$ \( 1 + 12 T + 336 T^{2} + 2381 T^{3} + 336 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \)
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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

−7.921316383429760413081432946983, −7.66855515502088286048426249841, −7.61269687974144945754820095390, −7.41933966318204320025041704707, −6.81501965247708818078862204191, −6.52999800085345835332225270461, −6.37538304332119573327121078952, −6.11253658685247986043717759249, −5.99628739186168474857195042971, −5.91563474198050230805927988420, −5.00341872275411380583833195329, −4.96909058163643420547256287185, −4.91400769963306670568693244064, −4.30934508866633252469056170567, −4.05698336335590967748236515580, −3.82244159939346923760838703742, −3.49705626998349679384245393045, −3.40556623100983644681039955843, −2.98605416111299620352314484815, −2.70141910927224720419609126593, −2.55502466899488522267943121322, −2.25886047239748081247111682439, −1.37465927691918900173775273371, −1.30546881369965573227079593671, −0.904353645007629231783193235115, 0.904353645007629231783193235115, 1.30546881369965573227079593671, 1.37465927691918900173775273371, 2.25886047239748081247111682439, 2.55502466899488522267943121322, 2.70141910927224720419609126593, 2.98605416111299620352314484815, 3.40556623100983644681039955843, 3.49705626998349679384245393045, 3.82244159939346923760838703742, 4.05698336335590967748236515580, 4.30934508866633252469056170567, 4.91400769963306670568693244064, 4.96909058163643420547256287185, 5.00341872275411380583833195329, 5.91563474198050230805927988420, 5.99628739186168474857195042971, 6.11253658685247986043717759249, 6.37538304332119573327121078952, 6.52999800085345835332225270461, 6.81501965247708818078862204191, 7.41933966318204320025041704707, 7.61269687974144945754820095390, 7.66855515502088286048426249841, 7.921316383429760413081432946983

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