Properties

Label 6-7098e3-1.1-c1e3-0-9
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

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·2-s − 3·3-s + 6·4-s + 2·5-s − 9·6-s − 3·7-s + 10·8-s + 6·9-s + 6·10-s + 3·11-s − 18·12-s − 9·14-s − 6·15-s + 15·16-s − 7·17-s + 18·18-s + 3·19-s + 12·20-s + 9·21-s + 9·22-s − 7·23-s − 30·24-s − 10·25-s − 10·27-s − 18·28-s − 12·29-s − 18·30-s + ⋯
L(s)  = 1  + 2.12·2-s − 1.73·3-s + 3·4-s + 0.894·5-s − 3.67·6-s − 1.13·7-s + 3.53·8-s + 2·9-s + 1.89·10-s + 0.904·11-s − 5.19·12-s − 2.40·14-s − 1.54·15-s + 15/4·16-s − 1.69·17-s + 4.24·18-s + 0.688·19-s + 2.68·20-s + 1.96·21-s + 1.91·22-s − 1.45·23-s − 6.12·24-s − 2·25-s − 1.92·27-s − 3.40·28-s − 2.22·29-s − 3.28·30-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$A_4\times C_2$ \( 1 - 2 T + 14 T^{2} - 19 T^{3} + 14 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
11$A_4\times C_2$ \( 1 - 3 T + 29 T^{2} - 67 T^{3} + 29 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \)
17$A_4\times C_2$ \( 1 + 7 T + 65 T^{2} + 245 T^{3} + 65 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \)
19$A_4\times C_2$ \( 1 - 3 T + 39 T^{2} - 101 T^{3} + 39 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \)
23$A_4\times C_2$ \( 1 + 7 T + 83 T^{2} + 329 T^{3} + 83 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \)
29$A_4\times C_2$ \( 1 + 12 T + 114 T^{2} + 683 T^{3} + 114 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \)
31$A_4\times C_2$ \( 1 + 72 T^{2} + 7 T^{3} + 72 p T^{4} + p^{3} T^{6} \)
37$A_4\times C_2$ \( 1 + 8 T + 102 T^{2} + 479 T^{3} + 102 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
41$A_4\times C_2$ \( 1 + 2 T + 80 T^{2} + 37 T^{3} + 80 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
43$A_4\times C_2$ \( 1 + 9 T + 149 T^{2} + 787 T^{3} + 149 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \)
47$A_4\times C_2$ \( 1 + 7 T + 148 T^{2} + 651 T^{3} + 148 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \)
53$A_4\times C_2$ \( 1 + 152 T^{2} - 7 T^{3} + 152 p T^{4} + p^{3} T^{6} \)
59$A_4\times C_2$ \( 1 + 2 T + 148 T^{2} + 165 T^{3} + 148 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
61$A_4\times C_2$ \( 1 - 3 T + 158 T^{2} - 283 T^{3} + 158 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \)
67$A_4\times C_2$ \( 1 + 32 T + 526 T^{2} + 5351 T^{3} + 526 p T^{4} + 32 p^{2} T^{5} + p^{3} T^{6} \)
71$A_4\times C_2$ \( 1 - 6 T + 113 T^{2} - 188 T^{3} + 113 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
73$A_4\times C_2$ \( 1 + 13 T + 133 T^{2} + 709 T^{3} + 133 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \)
79$A_4\times C_2$ \( 1 - 12 T + 236 T^{2} - 1855 T^{3} + 236 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \)
83$A_4\times C_2$ \( 1 - 7 T + 123 T^{2} - 189 T^{3} + 123 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \)
89$A_4\times C_2$ \( 1 - 10 T + 298 T^{2} - 1809 T^{3} + 298 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
97$A_4\times C_2$ \( 1 + 10 T + 294 T^{2} + 19 p T^{3} + 294 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
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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

−7.32712859037496336430154832223, −6.78853367155120123597976858349, −6.72424136306248245161010785238, −6.51716860082565932167643268317, −6.23820194652244149235027527758, −6.11729602317828646721622645859, −6.00934759650516386315192299300, −5.56965376723155833151075930337, −5.55184323085577536378974258414, −5.47085664785600040572254088755, −4.87675568874690410867041446476, −4.80294716237811291468055938445, −4.67584007102615375864792178436, −4.09247443070998543862701885464, −4.03309464453287665316344101915, −3.87119923335918470657531533710, −3.57078884783953226666904571460, −3.28189637712399890644047787118, −3.18717598740439187574770905744, −2.33795864867247457927094410700, −2.27550500066033274202694183598, −2.24076148305167287093340309090, −1.55825808918224414579285535785, −1.41401729874899893507491471304, −1.37524699536699559647608680968, 0, 0, 0, 1.37524699536699559647608680968, 1.41401729874899893507491471304, 1.55825808918224414579285535785, 2.24076148305167287093340309090, 2.27550500066033274202694183598, 2.33795864867247457927094410700, 3.18717598740439187574770905744, 3.28189637712399890644047787118, 3.57078884783953226666904571460, 3.87119923335918470657531533710, 4.03309464453287665316344101915, 4.09247443070998543862701885464, 4.67584007102615375864792178436, 4.80294716237811291468055938445, 4.87675568874690410867041446476, 5.47085664785600040572254088755, 5.55184323085577536378974258414, 5.56965376723155833151075930337, 6.00934759650516386315192299300, 6.11729602317828646721622645859, 6.23820194652244149235027527758, 6.51716860082565932167643268317, 6.72424136306248245161010785238, 6.78853367155120123597976858349, 7.32712859037496336430154832223

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