Properties

Label 6-8550e3-1.1-c1e3-0-2
Degree $6$
Conductor $625026375000$
Sign $1$
Analytic cond. $318221.$
Root an. cond. $8.26269$
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 + 6·4-s − 2·7-s − 10·8-s + 5·11-s − 4·13-s + 6·14-s + 15·16-s − 2·17-s − 3·19-s − 15·22-s + 23-s + 12·26-s − 12·28-s + 5·29-s + 5·31-s − 21·32-s + 6·34-s − 4·37-s + 9·38-s + 10·41-s − 2·43-s + 30·44-s − 3·46-s − 4·47-s − 11·49-s − 24·52-s + ⋯
L(s)  = 1  − 2.12·2-s + 3·4-s − 0.755·7-s − 3.53·8-s + 1.50·11-s − 1.10·13-s + 1.60·14-s + 15/4·16-s − 0.485·17-s − 0.688·19-s − 3.19·22-s + 0.208·23-s + 2.35·26-s − 2.26·28-s + 0.928·29-s + 0.898·31-s − 3.71·32-s + 1.02·34-s − 0.657·37-s + 1.45·38-s + 1.56·41-s − 0.304·43-s + 4.52·44-s − 0.442·46-s − 0.583·47-s − 1.57·49-s − 3.32·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{6} \cdot 5^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{6} \cdot 5^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{3} \cdot 3^{6} \cdot 5^{6} \cdot 19^{3}\)
Sign: $1$
Analytic conductor: \(318221.\)
Root analytic conductor: \(8.26269\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 2^{3} \cdot 3^{6} \cdot 5^{6} \cdot 19^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.063047797\)
\(L(\frac12)\) \(\approx\) \(1.063047797\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
19$C_1$ \( ( 1 + T )^{3} \)
good7$S_4\times C_2$ \( 1 + 2 T + 15 T^{2} + 18 T^{3} + 15 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
11$S_4\times C_2$ \( 1 - 5 T + 28 T^{2} - 83 T^{3} + 28 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \)
13$S_4\times C_2$ \( 1 + 4 T + 31 T^{2} + 98 T^{3} + 31 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 + 2 T + 45 T^{2} + 58 T^{3} + 45 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 - T + 24 T^{2} + 35 T^{3} + 24 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 - 5 T + 44 T^{2} - 93 T^{3} + 44 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 - 5 T + 72 T^{2} - 309 T^{3} + 72 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 + 4 T + 63 T^{2} + 152 T^{3} + 63 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 - 10 T + 147 T^{2} - 824 T^{3} + 147 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 + 2 T + 29 T^{2} + 334 T^{3} + 29 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 + 4 T + 53 T^{2} + 580 T^{3} + 53 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 + 5 T + 86 T^{2} + 647 T^{3} + 86 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 - 12 T + 205 T^{2} - 1366 T^{3} + 205 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 - T + 102 T^{2} + 153 T^{3} + 102 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 + 9 T + 58 T^{2} + 361 T^{3} + 58 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 - 16 T + 215 T^{2} - 1918 T^{3} + 215 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 + 15 T + 210 T^{2} + 1947 T^{3} + 210 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 + 9 T + 136 T^{2} + 713 T^{3} + 136 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 - 3 T + 214 T^{2} - 371 T^{3} + 214 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 - 9 T + 210 T^{2} - 1325 T^{3} + 210 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 - 6 T + 237 T^{2} - 894 T^{3} + 237 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
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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

−7.04621337759989106497561341425, −6.62016988197654874554367749207, −6.51402977207278268704051236237, −6.49429958039934027623896745067, −5.98918881739563516938686212340, −5.92787427100435568610545633445, −5.84742652410022585117623722722, −5.11929374338271561630850422625, −5.00094313408202638377249058336, −4.89749380861827777620818265916, −4.44488765662644566427812513868, −4.23574070896489561218798070015, −3.94144338980316184809530822640, −3.57944691097599733263945347659, −3.43755880125447030781598100798, −3.04816280047491355333267091879, −2.73813691271432471896261109695, −2.50083815329465673593787519554, −2.44334477005419258094757504904, −1.76814440530380085741961290355, −1.71640144263303283861071150557, −1.41182891108580044401013521748, −0.957304570217738444360065861434, −0.44060189563096756749391661136, −0.42748495198840934577381925340, 0.42748495198840934577381925340, 0.44060189563096756749391661136, 0.957304570217738444360065861434, 1.41182891108580044401013521748, 1.71640144263303283861071150557, 1.76814440530380085741961290355, 2.44334477005419258094757504904, 2.50083815329465673593787519554, 2.73813691271432471896261109695, 3.04816280047491355333267091879, 3.43755880125447030781598100798, 3.57944691097599733263945347659, 3.94144338980316184809530822640, 4.23574070896489561218798070015, 4.44488765662644566427812513868, 4.89749380861827777620818265916, 5.00094313408202638377249058336, 5.11929374338271561630850422625, 5.84742652410022585117623722722, 5.92787427100435568610545633445, 5.98918881739563516938686212340, 6.49429958039934027623896745067, 6.51402977207278268704051236237, 6.62016988197654874554367749207, 7.04621337759989106497561341425

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