Properties

Label 6-1859e3-1.1-c1e3-0-3
Degree $6$
Conductor $6424482779$
Sign $-1$
Analytic cond. $3270.91$
Root an. cond. $3.85281$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $3$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s + 4-s + 5-s + 6-s − 5·7-s − 2·8-s − 2·9-s + 10-s − 3·11-s + 12-s − 5·14-s + 15-s − 5·16-s − 7·17-s − 2·18-s − 2·19-s + 20-s − 5·21-s − 3·22-s + 3·23-s − 2·24-s − 8·25-s − 6·27-s − 5·28-s − 4·29-s + 30-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 1.88·7-s − 0.707·8-s − 2/3·9-s + 0.316·10-s − 0.904·11-s + 0.288·12-s − 1.33·14-s + 0.258·15-s − 5/4·16-s − 1.69·17-s − 0.471·18-s − 0.458·19-s + 0.223·20-s − 1.09·21-s − 0.639·22-s + 0.625·23-s − 0.408·24-s − 8/5·25-s − 1.15·27-s − 0.944·28-s − 0.742·29-s + 0.182·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(11^{3} \cdot 13^{6}\)
Sign: $-1$
Analytic conductor: \(3270.91\)
Root analytic conductor: \(3.85281\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1859} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 11^{3} \cdot 13^{6} ,\ ( \ : 1/2, 1/2, 1/2 ),\ -1 )\)

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)$
bad11$C_1$ \( ( 1 + T )^{3} \)
13 \( 1 \)
good2$A_4\times C_2$ \( 1 - T + 3 T^{3} - p^{2} T^{5} + p^{3} T^{6} \)
3$A_4\times C_2$ \( 1 - T + p T^{2} + T^{3} + p^{2} T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
5$A_4\times C_2$ \( 1 - T + 9 T^{2} - 3 T^{3} + 9 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
7$A_4\times C_2$ \( 1 + 5 T + 23 T^{2} + 69 T^{3} + 23 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \)
17$A_4\times C_2$ \( 1 + 7 T + 61 T^{2} + 231 T^{3} + 61 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \)
19$A_4\times C_2$ \( 1 + 2 T + 14 T^{2} + 153 T^{3} + 14 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
23$A_4\times C_2$ \( 1 - 3 T + 53 T^{2} - 139 T^{3} + 53 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \)
29$A_4\times C_2$ \( 1 + 4 T + 10 T^{2} + 183 T^{3} + 10 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
31$A_4\times C_2$ \( 1 + T + 68 T^{2} + 3 p T^{3} + 68 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
37$A_4\times C_2$ \( 1 + 12 T + 140 T^{2} + 857 T^{3} + 140 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \)
41$A_4\times C_2$ \( 1 + T + p T^{2} + 113 T^{3} + p^{2} T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
43$A_4\times C_2$ \( 1 + 4 T + 14 T^{2} - 237 T^{3} + 14 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
47$A_4\times C_2$ \( 1 - 8 T + 80 T^{2} - 493 T^{3} + 80 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
53$A_4\times C_2$ \( 1 + 9 T + 167 T^{2} + 905 T^{3} + 167 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \)
59$A_4\times C_2$ \( 1 + 30 T + 458 T^{2} + 4331 T^{3} + 458 p T^{4} + 30 p^{2} T^{5} + p^{3} T^{6} \)
61$A_4\times C_2$ \( 1 + 18 T + 234 T^{2} + 2203 T^{3} + 234 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \)
67$A_4\times C_2$ \( 1 + 15 T + 257 T^{2} + 2021 T^{3} + 257 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \)
71$A_4\times C_2$ \( 1 - 11 T + 247 T^{2} - 1593 T^{3} + 247 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{3} \)
79$A_4\times C_2$ \( 1 - 12 T + 209 T^{2} - 1808 T^{3} + 209 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \)
83$A_4\times C_2$ \( 1 - 14 T + 232 T^{2} - 1983 T^{3} + 232 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \)
89$A_4\times C_2$ \( 1 - 17 T + 357 T^{2} - 3177 T^{3} + 357 p T^{4} - 17 p^{2} T^{5} + p^{3} T^{6} \)
97$A_4\times C_2$ \( 1 + 11 T + 230 T^{2} + 2127 T^{3} + 230 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \)
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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

−8.833033730251007430788972250219, −8.283346070774721540305903936990, −7.918021529241501660646281982119, −7.77679022687989322394908832577, −7.75280016704423017239488884086, −7.13995672966184154119811162490, −6.83051911991716502682174580394, −6.61648830143272313699901820048, −6.31139480197183728210332762726, −6.28541197371515858869159902160, −6.07633850420625022011302226494, −5.55613804825973596348048282179, −5.44285261731920590721213538170, −5.14915686582588892105789225291, −4.59852399366341908751771238867, −4.50599636747869374283443191384, −4.12000906399521823034943651220, −3.61082324702614947203909139225, −3.29501434698583934126098022932, −3.21851029691190361073973325581, −2.99297796205471142203649578613, −2.47788355565979142086204206179, −2.28286188556339518328089312824, −1.85121565698763246165816699171, −1.56425551608615543001246249097, 0, 0, 0, 1.56425551608615543001246249097, 1.85121565698763246165816699171, 2.28286188556339518328089312824, 2.47788355565979142086204206179, 2.99297796205471142203649578613, 3.21851029691190361073973325581, 3.29501434698583934126098022932, 3.61082324702614947203909139225, 4.12000906399521823034943651220, 4.50599636747869374283443191384, 4.59852399366341908751771238867, 5.14915686582588892105789225291, 5.44285261731920590721213538170, 5.55613804825973596348048282179, 6.07633850420625022011302226494, 6.28541197371515858869159902160, 6.31139480197183728210332762726, 6.61648830143272313699901820048, 6.83051911991716502682174580394, 7.13995672966184154119811162490, 7.75280016704423017239488884086, 7.77679022687989322394908832577, 7.918021529241501660646281982119, 8.283346070774721540305903936990, 8.833033730251007430788972250219

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