Properties

Label 6-1859e3-1.1-c1e3-0-2
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

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: \(3\)
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$C_2$ \( ( 1 + T + p T^{2} )^{3} \)
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

−8.538298757838176634606157430030, −8.347228448982201812833033178597, −8.300750352023258414043192083396, −8.237362503111811048225216525344, −7.48756309134970383646291009604, −7.48030421733173809924907587768, −7.27826251990656497401388546419, −7.14964681070961041311026470575, −6.59243559512965596320987937104, −6.40973160153059417578672052315, −5.70109373622720831622957883267, −5.48133346358482224854470332919, −5.43716698325643619821817027044, −4.99600428314235092104159210905, −4.68448815619780475066717924515, −4.34202510139997404983480576011, −4.08729924007336391120061870774, −3.99411177855531016231784473950, −3.67854344988793548825357380144, −3.04576354798957517936371072780, −2.83659294524702960312553979956, −2.24999373050341079973267211389, −1.71059690174772208215974798855, −1.52211600501641223530039251974, −0.837200078331154618820739186820, 0, 0, 0, 0.837200078331154618820739186820, 1.52211600501641223530039251974, 1.71059690174772208215974798855, 2.24999373050341079973267211389, 2.83659294524702960312553979956, 3.04576354798957517936371072780, 3.67854344988793548825357380144, 3.99411177855531016231784473950, 4.08729924007336391120061870774, 4.34202510139997404983480576011, 4.68448815619780475066717924515, 4.99600428314235092104159210905, 5.43716698325643619821817027044, 5.48133346358482224854470332919, 5.70109373622720831622957883267, 6.40973160153059417578672052315, 6.59243559512965596320987937104, 7.14964681070961041311026470575, 7.27826251990656497401388546419, 7.48030421733173809924907587768, 7.48756309134970383646291009604, 8.237362503111811048225216525344, 8.300750352023258414043192083396, 8.347228448982201812833033178597, 8.538298757838176634606157430030

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