Properties

Label 6-7098e3-1.1-c1e3-0-8
Degree $6$
Conductor $357608625192$
Sign $-1$
Analytic cond. $182070.$
Root an. cond. $7.52846$
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  + 3·2-s − 3·3-s + 6·4-s − 4·5-s − 9·6-s + 3·7-s + 10·8-s + 6·9-s − 12·10-s + 3·11-s − 18·12-s + 9·14-s + 12·15-s + 15·16-s − 3·17-s + 18·18-s − 7·19-s − 24·20-s − 9·21-s + 9·22-s − 5·23-s − 30·24-s − 2·25-s − 10·27-s + 18·28-s − 2·29-s + 36·30-s + ⋯
L(s)  = 1  + 2.12·2-s − 1.73·3-s + 3·4-s − 1.78·5-s − 3.67·6-s + 1.13·7-s + 3.53·8-s + 2·9-s − 3.79·10-s + 0.904·11-s − 5.19·12-s + 2.40·14-s + 3.09·15-s + 15/4·16-s − 0.727·17-s + 4.24·18-s − 1.60·19-s − 5.36·20-s − 1.96·21-s + 1.91·22-s − 1.04·23-s − 6.12·24-s − 2/5·25-s − 1.92·27-s + 3.40·28-s − 0.371·29-s + 6.57·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 7^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 7^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{3} \cdot 3^{3} \cdot 7^{3} \cdot 13^{6}\)
Sign: $-1$
Analytic conductor: \(182070.\)
Root analytic conductor: \(7.52846\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{7098} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 2^{3} \cdot 3^{3} \cdot 7^{3} \cdot 13^{6} ,\ ( \ : 1/2, 1/2, 1/2 ),\ -1 )\)

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$C_1$ \( ( 1 - T )^{3} \)
3$C_1$ \( ( 1 + T )^{3} \)
7$C_1$ \( ( 1 - T )^{3} \)
13 \( 1 \)
good5$A_4\times C_2$ \( 1 + 4 T + 18 T^{2} + 39 T^{3} + 18 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
11$A_4\times C_2$ \( 1 - 3 T + 15 T^{2} - 39 T^{3} + 15 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \)
17$A_4\times C_2$ \( 1 + 3 T + 47 T^{2} + 103 T^{3} + 47 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
19$A_4\times C_2$ \( 1 + 7 T + 71 T^{2} + 273 T^{3} + 71 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \)
23$A_4\times C_2$ \( 1 + 5 T + 33 T^{2} + 273 T^{3} + 33 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \)
29$A_4\times C_2$ \( 1 + 2 T + 44 T^{2} - 11 T^{3} + 44 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
31$A_4\times C_2$ \( 1 + 10 T + 96 T^{2} + 607 T^{3} + 96 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
37$A_4\times C_2$ \( 1 + 104 T^{2} + 7 T^{3} + 104 p T^{4} + p^{3} T^{6} \)
41$A_4\times C_2$ \( 1 - 4 T + 126 T^{2} - 327 T^{3} + 126 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
43$A_4\times C_2$ \( 1 - 7 T + 101 T^{2} - 399 T^{3} + 101 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \)
47$A_4\times C_2$ \( 1 + 3 T + 116 T^{2} + 199 T^{3} + 116 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
53$A_4\times C_2$ \( 1 + 2 T + 88 T^{2} + 99 T^{3} + 88 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
59$A_4\times C_2$ \( 1 - 2 T + 148 T^{2} - 249 T^{3} + 148 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
61$C_2$ \( ( 1 + 7 T + p T^{2} )^{3} \)
67$A_4\times C_2$ \( 1 + 6 T + 122 T^{2} + 343 T^{3} + 122 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
71$A_4\times C_2$ \( 1 - 2 T + 177 T^{2} - 276 T^{3} + 177 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
73$A_4\times C_2$ \( 1 + 11 T + 243 T^{2} + 1577 T^{3} + 243 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \)
79$A_4\times C_2$ \( 1 + 22 T + 396 T^{2} + 3853 T^{3} + 396 p T^{4} + 22 p^{2} T^{5} + p^{3} T^{6} \)
83$A_4\times C_2$ \( 1 + 31 T + 567 T^{2} + 75 p T^{3} + 567 p T^{4} + 31 p^{2} T^{5} + p^{3} T^{6} \)
89$A_4\times C_2$ \( 1 - 6 T^{2} - 889 T^{3} - 6 p T^{4} + p^{3} T^{6} \)
97$A_4\times C_2$ \( 1 + 28 T + 536 T^{2} + 63 p T^{3} + 536 p T^{4} + 28 p^{2} T^{5} + p^{3} T^{6} \)
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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

−7.25468150654849710085992417355, −7.13350262342909721225610605075, −6.88614571631345116358310357259, −6.53231168941198109931448455183, −6.24134687254243288913979807974, −5.99837811627705000907029954961, −5.97116503712679348692788581028, −5.58527297937232432189809376816, −5.51107817631095605846922680387, −5.43589173504791265760953049906, −4.68508952431407788741268322472, −4.57021646113106550374955043495, −4.47905128903564765895006370180, −4.23062313281700466384723449318, −4.21076836877037232574123237924, −4.06271069906223762711966767591, −3.49462064273127470557564974422, −3.43237550627539072188821035241, −3.20584042931195142935833802741, −2.46090204217238709961918218358, −2.36127461030491929744758856487, −2.03443820849303010942486625168, −1.62032934137852590389601881329, −1.29354048399680351041744980482, −1.26431061078488894406595958599, 0, 0, 0, 1.26431061078488894406595958599, 1.29354048399680351041744980482, 1.62032934137852590389601881329, 2.03443820849303010942486625168, 2.36127461030491929744758856487, 2.46090204217238709961918218358, 3.20584042931195142935833802741, 3.43237550627539072188821035241, 3.49462064273127470557564974422, 4.06271069906223762711966767591, 4.21076836877037232574123237924, 4.23062313281700466384723449318, 4.47905128903564765895006370180, 4.57021646113106550374955043495, 4.68508952431407788741268322472, 5.43589173504791265760953049906, 5.51107817631095605846922680387, 5.58527297937232432189809376816, 5.97116503712679348692788581028, 5.99837811627705000907029954961, 6.24134687254243288913979807974, 6.53231168941198109931448455183, 6.88614571631345116358310357259, 7.13350262342909721225610605075, 7.25468150654849710085992417355

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