Properties

Degree $6$
Conductor $182284263$
Sign $-1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $3$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 4-s − 5·5-s − 3·7-s + 5·10-s − 2·11-s + 3·13-s + 3·14-s + 16-s − 12·17-s − 3·19-s + 5·20-s + 2·22-s + 8·25-s − 3·26-s + 3·28-s + 29-s − 3·31-s + 2·32-s + 12·34-s + 15·35-s − 3·37-s + 3·38-s − 22·41-s − 3·43-s + 2·44-s − 9·47-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s − 2.23·5-s − 1.13·7-s + 1.58·10-s − 0.603·11-s + 0.832·13-s + 0.801·14-s + 1/4·16-s − 2.91·17-s − 0.688·19-s + 1.11·20-s + 0.426·22-s + 8/5·25-s − 0.588·26-s + 0.566·28-s + 0.185·29-s − 0.538·31-s + 0.353·32-s + 2.05·34-s + 2.53·35-s − 0.493·37-s + 0.486·38-s − 3.43·41-s − 0.457·43-s + 0.301·44-s − 1.31·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(3^{12} \cdot 7^{3}\)
Sign: $-1$
Motivic weight: \(1\)
Character: induced by $\chi_{567} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 3^{12} \cdot 7^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ -1 )\)

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)$
bad3 \( 1 \)
7$C_1$ \( ( 1 + T )^{3} \)
good2$S_4\times C_2$ \( 1 + T + p T^{2} + 3 T^{3} + p^{2} T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
5$S_4\times C_2$ \( 1 + p T + 17 T^{2} + 39 T^{3} + 17 p T^{4} + p^{3} T^{5} + p^{3} T^{6} \)
11$S_4\times C_2$ \( 1 + 2 T + 14 T^{2} - 3 T^{3} + 14 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
13$C_2$ \( ( 1 - T + p T^{2} )^{3} \)
17$S_4\times C_2$ \( 1 + 12 T + 90 T^{2} + 435 T^{3} + 90 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \)
19$S_4\times C_2$ \( 1 + 3 T + 51 T^{2} + 107 T^{3} + 51 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 + 36 T^{2} - 9 T^{3} + 36 p T^{4} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 - T + 83 T^{2} - 57 T^{3} + 83 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 + 3 T + 69 T^{2} + 213 T^{3} + 69 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 + 3 T + 57 T^{2} + 303 T^{3} + 57 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 + 22 T + 278 T^{2} + 2157 T^{3} + 278 p T^{4} + 22 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 + 3 T + 63 T^{2} + 379 T^{3} + 63 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 + 9 T + 87 T^{2} + 657 T^{3} + 87 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 + 18 T + 234 T^{2} + 1917 T^{3} + 234 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 + 9 T + 171 T^{2} + 999 T^{3} + 171 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 + 6 T + 162 T^{2} + 665 T^{3} + 162 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 - 6 T^{2} + 683 T^{3} - 6 p T^{4} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 - 9 T + 207 T^{2} - 1197 T^{3} + 207 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 - 3 T + 51 T^{2} - 681 T^{3} + 51 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 - 15 T + 189 T^{2} - 1601 T^{3} + 189 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 + 12 T + 288 T^{2} + 2019 T^{3} + 288 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 + 2 T + 116 T^{2} + 735 T^{3} + 116 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 - 3 T + 177 T^{2} + 21 T^{3} + 177 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \)
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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

−10.13716391300802609309052136648, −9.478926114819564418637865086677, −9.331916306062282560231081080998, −9.273928989831878405783571273698, −8.611894876496353828936787676140, −8.498973145566492954161623228048, −8.465855119925237173302695739346, −7.992616401197181024289997434457, −7.85211843023724863786930854203, −7.44934434299483394062740077289, −6.84096789915199637976782712864, −6.60980171432291819252986663250, −6.60711651848748690250196402689, −6.40550248589223874724653859772, −5.58760781409360007139260328445, −5.38607150054507828403370041555, −4.71364151121870565129588476827, −4.48419964124764514322003497109, −4.40533551715497340336436394663, −3.70353282541754078266402613965, −3.55591448621153857250248519602, −3.25023180569680189903146223395, −2.78746966287949503706406316513, −2.07072477954214546651559024617, −1.56542435205385803237484305376, 0, 0, 0, 1.56542435205385803237484305376, 2.07072477954214546651559024617, 2.78746966287949503706406316513, 3.25023180569680189903146223395, 3.55591448621153857250248519602, 3.70353282541754078266402613965, 4.40533551715497340336436394663, 4.48419964124764514322003497109, 4.71364151121870565129588476827, 5.38607150054507828403370041555, 5.58760781409360007139260328445, 6.40550248589223874724653859772, 6.60711651848748690250196402689, 6.60980171432291819252986663250, 6.84096789915199637976782712864, 7.44934434299483394062740077289, 7.85211843023724863786930854203, 7.992616401197181024289997434457, 8.465855119925237173302695739346, 8.498973145566492954161623228048, 8.611894876496353828936787676140, 9.273928989831878405783571273698, 9.331916306062282560231081080998, 9.478926114819564418637865086677, 10.13716391300802609309052136648

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