Properties

Label 6-7938e3-1.1-c1e3-0-5
Degree $6$
Conductor $500188017672$
Sign $-1$
Analytic cond. $254662.$
Root an. cond. $7.96148$
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 + 6·4-s − 5·5-s + 10·8-s − 15·10-s + 11-s − 2·13-s + 15·16-s − 4·17-s − 3·19-s − 30·20-s + 3·22-s + 7·23-s + 6·25-s − 6·26-s + 5·29-s − 14·31-s + 21·32-s − 12·34-s + 9·37-s − 9·38-s − 50·40-s − 12·41-s − 18·43-s + 6·44-s + 21·46-s + 3·47-s + ⋯
L(s)  = 1  + 2.12·2-s + 3·4-s − 2.23·5-s + 3.53·8-s − 4.74·10-s + 0.301·11-s − 0.554·13-s + 15/4·16-s − 0.970·17-s − 0.688·19-s − 6.70·20-s + 0.639·22-s + 1.45·23-s + 6/5·25-s − 1.17·26-s + 0.928·29-s − 2.51·31-s + 3.71·32-s − 2.05·34-s + 1.47·37-s − 1.45·38-s − 7.90·40-s − 1.87·41-s − 2.74·43-s + 0.904·44-s + 3.09·46-s + 0.437·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{12} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{12} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{3} \cdot 3^{12} \cdot 7^{6}\)
Sign: $-1$
Analytic conductor: \(254662.\)
Root analytic conductor: \(7.96148\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 2^{3} \cdot 3^{12} \cdot 7^{6} ,\ ( \ : 1/2, 1/2, 1/2 ),\ -1 )\)

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 \( 1 \)
7 \( 1 \)
good5$S_4\times C_2$ \( 1 + p T + 19 T^{2} + 47 T^{3} + 19 p T^{4} + p^{3} T^{5} + p^{3} T^{6} \)
11$S_4\times C_2$ \( 1 - T + 7 T^{2} - 5 p T^{3} + 7 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
13$S_4\times C_2$ \( 1 + 2 T + 36 T^{2} + 49 T^{3} + 36 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 + 4 T + 7 T^{2} - 32 T^{3} + 7 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
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$S_4\times C_2$ \( 1 - 7 T + 73 T^{2} - 319 T^{3} + 73 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 - 5 T + 55 T^{2} - 323 T^{3} + 55 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 + 14 T + 138 T^{2} + 841 T^{3} + 138 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 - 9 T + 102 T^{2} - 593 T^{3} + 102 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 + 12 T + 162 T^{2} + 1011 T^{3} + 162 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 + 18 T + 210 T^{2} + 1549 T^{3} + 210 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 - 3 T + 117 T^{2} - 309 T^{3} + 117 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 + 9 T + 117 T^{2} + 963 T^{3} + 117 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 - 4 T + 76 T^{2} - 11 p T^{3} + 76 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 - 4 T + 48 T^{2} + 229 T^{3} + 48 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 + 5 T + 143 T^{2} + 521 T^{3} + 143 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 - 7 T + 163 T^{2} - 895 T^{3} + 163 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 + 25 T + 371 T^{2} + 3601 T^{3} + 371 p T^{4} + 25 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 + 7 T + 93 T^{2} + 335 T^{3} + 93 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 - 8 T + 244 T^{2} - 1235 T^{3} + 244 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 + 9 T + 261 T^{2} + 1539 T^{3} + 261 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 + 28 T + 527 T^{2} + 5968 T^{3} + 527 p T^{4} + 28 p^{2} T^{5} + p^{3} T^{6} \)
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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

−7.19606043006391944896064971058, −7.16887213449714497488629000064, −6.85317344050322828518476338189, −6.48300864667596884728128424038, −6.24426212953151985828412780323, −6.21278527564529166064887957910, −5.95292438892158823661206157958, −5.30854271828597946489078942322, −5.26724757173210650754298956763, −5.23072977470452052331606820807, −4.66832802371027117807892567809, −4.65763285270517864550699856225, −4.56280973880038910559521995565, −4.07598416189474023705873018885, −3.84199466312935727543853423258, −3.76344554664785058407217297356, −3.56892389892465679867061488907, −3.30366588790049453212321153100, −3.00332103354705334753163365077, −2.58685017450413322915969489030, −2.45542247259939700265859002191, −2.24764112835169548083672397649, −1.49503848677633322268982248389, −1.46111242121715447322044230913, −1.21366501683653714453657033846, 0, 0, 0, 1.21366501683653714453657033846, 1.46111242121715447322044230913, 1.49503848677633322268982248389, 2.24764112835169548083672397649, 2.45542247259939700265859002191, 2.58685017450413322915969489030, 3.00332103354705334753163365077, 3.30366588790049453212321153100, 3.56892389892465679867061488907, 3.76344554664785058407217297356, 3.84199466312935727543853423258, 4.07598416189474023705873018885, 4.56280973880038910559521995565, 4.65763285270517864550699856225, 4.66832802371027117807892567809, 5.23072977470452052331606820807, 5.26724757173210650754298956763, 5.30854271828597946489078942322, 5.95292438892158823661206157958, 6.21278527564529166064887957910, 6.24426212953151985828412780323, 6.48300864667596884728128424038, 6.85317344050322828518476338189, 7.16887213449714497488629000064, 7.19606043006391944896064971058

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