Properties

Label 6-1859e3-1.1-c1e3-0-1
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 $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·2-s + 3·5-s − 14·8-s − 3·9-s + 9·10-s + 3·11-s − 21·16-s − 3·17-s − 9·18-s + 6·19-s + 9·22-s + 3·25-s + 2·27-s − 3·29-s + 6·31-s + 21·32-s − 9·34-s + 3·37-s + 18·38-s − 42·40-s + 3·41-s − 6·43-s − 9·45-s − 6·47-s − 9·49-s + 9·50-s + 3·53-s + ⋯
L(s)  = 1  + 2.12·2-s + 1.34·5-s − 4.94·8-s − 9-s + 2.84·10-s + 0.904·11-s − 5.25·16-s − 0.727·17-s − 2.12·18-s + 1.37·19-s + 1.91·22-s + 3/5·25-s + 0.384·27-s − 0.557·29-s + 1.07·31-s + 3.71·32-s − 1.54·34-s + 0.493·37-s + 2.91·38-s − 6.64·40-s + 0.468·41-s − 0.914·43-s − 1.34·45-s − 0.875·47-s − 9/7·49-s + 1.27·50-s + 0.412·53-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: \(0\)
Selberg data: \((6,\ 11^{3} \cdot 13^{6} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(4.438333989\)
\(L(\frac12)\) \(\approx\) \(4.438333989\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - T + p T^{2} )^{3} \)
3$S_4\times C_2$ \( 1 + p T^{2} - 2 T^{3} + p^{2} T^{4} + p^{3} T^{6} \)
5$S_4\times C_2$ \( 1 - 3 T + 6 T^{2} - 7 T^{3} + 6 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \)
7$S_4\times C_2$ \( 1 + 9 T^{2} + 2 T^{3} + 9 p T^{4} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 + 3 T + 30 T^{2} + 53 T^{3} + 30 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
19$S_4\times C_2$ \( 1 - 6 T + 45 T^{2} - 162 T^{3} + 45 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 + 45 T^{2} + 16 T^{3} + 45 p T^{4} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 + 3 T + 48 T^{2} + 231 T^{3} + 48 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 - 6 T + 9 T^{2} + 164 T^{3} + 9 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 - 3 T + 30 T^{2} + 57 T^{3} + 30 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 - 3 T + 60 T^{2} - 319 T^{3} + 60 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 + 6 T + 117 T^{2} + 492 T^{3} + 117 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 + 6 T + 87 T^{2} + 442 T^{3} + 87 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 - 3 T + 114 T^{2} - 255 T^{3} + 114 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 + 18 T + 219 T^{2} + 1942 T^{3} + 219 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 - 9 T + 156 T^{2} - 909 T^{3} + 156 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 - 12 T + 183 T^{2} - 1406 T^{3} + 183 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \)
71$C_2$ \( ( 1 - 6 T + p T^{2} )^{3} \)
73$S_4\times C_2$ \( 1 - 9 T - 30 T^{2} + 1169 T^{3} - 30 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 + 12 T + 201 T^{2} + 1918 T^{3} + 201 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 + 165 T^{2} + 294 T^{3} + 165 p T^{4} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 - 18 T + 3 p T^{2} - 2448 T^{3} + 3 p^{2} T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 - 18 T + 375 T^{2} - 3608 T^{3} + 375 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \)
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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

−8.415223831717115387485642193811, −8.062442496978053488398885439127, −7.63695046355354644934752246336, −7.45170764337181489237579460172, −6.66324414536037001479459077793, −6.58302359384078998077300083996, −6.49803109170442741828055835556, −6.02327506898658858389252976494, −6.00420939013043536816905355584, −5.72793952772154734029994259383, −5.17950351876856038900869941581, −5.09044550278763172614169077777, −5.08378612631823373345260527427, −4.74197369643940032784104159827, −4.20376448758559288954064020820, −4.18464834660147873618149790435, −3.63230809694465598780711437982, −3.44362640868295960425022357228, −3.22029146313498483787054389142, −2.85377984765603891606613705573, −2.50825995326670552262391988283, −2.15229335217131980244979435527, −1.50818878855248461912936282843, −0.894329385019462081369376727628, −0.42758339549553748542860686006, 0.42758339549553748542860686006, 0.894329385019462081369376727628, 1.50818878855248461912936282843, 2.15229335217131980244979435527, 2.50825995326670552262391988283, 2.85377984765603891606613705573, 3.22029146313498483787054389142, 3.44362640868295960425022357228, 3.63230809694465598780711437982, 4.18464834660147873618149790435, 4.20376448758559288954064020820, 4.74197369643940032784104159827, 5.08378612631823373345260527427, 5.09044550278763172614169077777, 5.17950351876856038900869941581, 5.72793952772154734029994259383, 6.00420939013043536816905355584, 6.02327506898658858389252976494, 6.49803109170442741828055835556, 6.58302359384078998077300083996, 6.66324414536037001479459077793, 7.45170764337181489237579460172, 7.63695046355354644934752246336, 8.062442496978053488398885439127, 8.415223831717115387485642193811

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