Properties

Label 6-3380e3-1.1-c1e3-0-4
Degree $6$
Conductor $38614472000$
Sign $-1$
Analytic cond. $19659.9$
Root an. cond. $5.19513$
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-s + 3·5-s + 7-s − 6·9-s − 5·11-s − 3·15-s − 17-s − 21-s − 4·23-s + 6·25-s + 8·27-s − 2·29-s + 7·31-s + 5·33-s + 3·35-s − 9·37-s − 8·41-s − 43-s − 18·45-s − 10·47-s − 18·49-s + 51-s − 8·53-s − 15·55-s + 17·59-s − 29·61-s − 6·63-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.34·5-s + 0.377·7-s − 2·9-s − 1.50·11-s − 0.774·15-s − 0.242·17-s − 0.218·21-s − 0.834·23-s + 6/5·25-s + 1.53·27-s − 0.371·29-s + 1.25·31-s + 0.870·33-s + 0.507·35-s − 1.47·37-s − 1.24·41-s − 0.152·43-s − 2.68·45-s − 1.45·47-s − 2.57·49-s + 0.140·51-s − 1.09·53-s − 2.02·55-s + 2.21·59-s − 3.71·61-s − 0.755·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{6} \cdot 5^{3} \cdot 13^{6}\)
Sign: $-1$
Analytic conductor: \(19659.9\)
Root analytic conductor: \(5.19513\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3380} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 2^{6} \cdot 5^{3} \cdot 13^{6} ,\ ( \ : 1/2, 1/2, 1/2 ),\ -1 )\)

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 \( 1 \)
5$C_1$ \( ( 1 - T )^{3} \)
13 \( 1 \)
good3$A_4\times C_2$ \( 1 + T + 7 T^{2} + 5 T^{3} + 7 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
7$A_4\times C_2$ \( 1 - T + 19 T^{2} - 13 T^{3} + 19 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
11$A_4\times C_2$ \( 1 + 5 T + 39 T^{2} + 111 T^{3} + 39 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \)
17$A_4\times C_2$ \( 1 + T + 35 T^{2} + 47 T^{3} + 35 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
19$A_4\times C_2$ \( 1 + 50 T^{2} - 7 T^{3} + 50 p T^{4} + p^{3} T^{6} \)
23$A_4\times C_2$ \( 1 + 4 T + 72 T^{2} + 183 T^{3} + 72 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
29$A_4\times C_2$ \( 1 + 2 T + 72 T^{2} + 129 T^{3} + 72 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
31$A_4\times C_2$ \( 1 - 7 T + 3 p T^{2} - 427 T^{3} + 3 p^{2} T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \)
37$C_2$ \( ( 1 + 3 T + p T^{2} )^{3} \)
41$A_4\times C_2$ \( 1 + 8 T + 114 T^{2} + 543 T^{3} + 114 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
43$A_4\times C_2$ \( 1 + T + 85 T^{2} + 169 T^{3} + 85 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
47$A_4\times C_2$ \( 1 + 10 T + 88 T^{2} + 381 T^{3} + 88 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
53$A_4\times C_2$ \( 1 + 8 T + 80 T^{2} + 679 T^{3} + 80 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
59$A_4\times C_2$ \( 1 - 17 T + 208 T^{2} - 1965 T^{3} + 208 p T^{4} - 17 p^{2} T^{5} + p^{3} T^{6} \)
61$A_4\times C_2$ \( 1 + 29 T + 461 T^{2} + 4419 T^{3} + 461 p T^{4} + 29 p^{2} T^{5} + p^{3} T^{6} \)
67$A_4\times C_2$ \( 1 + 17 T + 260 T^{2} + 2265 T^{3} + 260 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6} \)
71$A_4\times C_2$ \( 1 - 3 T + 195 T^{2} - 413 T^{3} + 195 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \)
73$A_4\times C_2$ \( 1 + 16 T + 239 T^{2} + 2328 T^{3} + 239 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \)
79$A_4\times C_2$ \( 1 + 13 T + 179 T^{2} + 1257 T^{3} + 179 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \)
83$A_4\times C_2$ \( 1 - 17 T + 259 T^{2} - 2235 T^{3} + 259 p T^{4} - 17 p^{2} T^{5} + p^{3} T^{6} \)
89$A_4\times C_2$ \( 1 + 9 T + 98 T^{2} + 649 T^{3} + 98 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \)
97$A_4\times C_2$ \( 1 + 6 T + 170 T^{2} + 997 T^{3} + 170 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
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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

−8.120932380730899726464100600824, −7.67136566676092620644416657765, −7.62942552666571055108448276727, −7.42143503371270736411940878808, −6.65464932004131972100472104488, −6.60119951860150371260790089181, −6.57313247131949902715385825857, −6.21660791780946572705888083799, −5.88545569162775808917498006668, −5.78427838099588684973266631367, −5.38756689776330052663480407378, −5.22478052761812439407099729601, −5.14889997614206993723859099882, −4.71922467943821447371810909860, −4.55154015160109677187183609995, −4.32013231961836052914529360586, −3.54843799702503170132519463516, −3.28278666410211767091212967150, −3.28169615236916236210787198856, −2.64905121103709793669959372772, −2.62658691245796937738339615495, −2.39428561416103794863585868480, −1.71552400392558046669511645146, −1.45593528063494355925412779508, −1.36839303892555848008040630391, 0, 0, 0, 1.36839303892555848008040630391, 1.45593528063494355925412779508, 1.71552400392558046669511645146, 2.39428561416103794863585868480, 2.62658691245796937738339615495, 2.64905121103709793669959372772, 3.28169615236916236210787198856, 3.28278666410211767091212967150, 3.54843799702503170132519463516, 4.32013231961836052914529360586, 4.55154015160109677187183609995, 4.71922467943821447371810909860, 5.14889997614206993723859099882, 5.22478052761812439407099729601, 5.38756689776330052663480407378, 5.78427838099588684973266631367, 5.88545569162775808917498006668, 6.21660791780946572705888083799, 6.57313247131949902715385825857, 6.60119951860150371260790089181, 6.65464932004131972100472104488, 7.42143503371270736411940878808, 7.62942552666571055108448276727, 7.67136566676092620644416657765, 8.120932380730899726464100600824

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