Properties

Label 6-7098e3-1.1-c1e3-0-6
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 − 3·5-s + 9·6-s + 3·7-s − 10·8-s + 6·9-s + 9·10-s + 3·11-s − 18·12-s − 9·14-s + 9·15-s + 15·16-s − 9·17-s − 18·18-s − 11·19-s − 18·20-s − 9·21-s − 9·22-s + 11·23-s + 30·24-s − 2·25-s − 10·27-s + 18·28-s + 8·29-s − 27·30-s + ⋯
L(s)  = 1  − 2.12·2-s − 1.73·3-s + 3·4-s − 1.34·5-s + 3.67·6-s + 1.13·7-s − 3.53·8-s + 2·9-s + 2.84·10-s + 0.904·11-s − 5.19·12-s − 2.40·14-s + 2.32·15-s + 15/4·16-s − 2.18·17-s − 4.24·18-s − 2.52·19-s − 4.02·20-s − 1.96·21-s − 1.91·22-s + 2.29·23-s + 6.12·24-s − 2/5·25-s − 1.92·27-s + 3.40·28-s + 1.48·29-s − 4.92·30-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$A_4\times C_2$ \( 1 + 3 T + 11 T^{2} + 31 T^{3} + 11 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
11$A_4\times C_2$ \( 1 - 3 T + 15 T^{2} - 53 T^{3} + 15 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \)
17$A_4\times C_2$ \( 1 + 9 T + 57 T^{2} + 277 T^{3} + 57 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \)
19$A_4\times C_2$ \( 1 + 11 T + 53 T^{2} + 207 T^{3} + 53 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \)
23$A_4\times C_2$ \( 1 - 11 T + 93 T^{2} - 519 T^{3} + 93 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \)
29$A_4\times C_2$ \( 1 - 8 T + 71 T^{2} - 400 T^{3} + 71 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
31$A_4\times C_2$ \( 1 + 11 T + 89 T^{2} + 471 T^{3} + 89 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \)
37$A_4\times C_2$ \( 1 + 3 T - 19 T^{2} - 211 T^{3} - 19 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
41$A_4\times C_2$ \( 1 - 21 T + 249 T^{2} - 1925 T^{3} + 249 p T^{4} - 21 p^{2} T^{5} + p^{3} T^{6} \)
43$A_4\times C_2$ \( 1 - 10 T + 97 T^{2} - 532 T^{3} + 97 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
47$A_4\times C_2$ \( 1 - 14 T + 3 p T^{2} - 924 T^{3} + 3 p^{2} T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \)
53$A_4\times C_2$ \( 1 - 2 T + 39 T^{2} - 540 T^{3} + 39 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
59$A_4\times C_2$ \( 1 + 32 T + 509 T^{2} + 4888 T^{3} + 509 p T^{4} + 32 p^{2} T^{5} + p^{3} T^{6} \)
61$A_4\times C_2$ \( 1 + 6 T - T^{2} - 380 T^{3} - p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
67$A_4\times C_2$ \( 1 + 10 T + 197 T^{2} + 1236 T^{3} + 197 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
71$A_4\times C_2$ \( 1 - 21 T + 143 T^{2} - 623 T^{3} + 143 p T^{4} - 21 p^{2} T^{5} + p^{3} T^{6} \)
73$A_4\times C_2$ \( 1 + 16 T + 239 T^{2} + 2328 T^{3} + 239 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \)
79$A_4\times C_2$ \( 1 - 4 T + 233 T^{2} - 624 T^{3} + 233 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
83$A_4\times C_2$ \( 1 + 22 T + 401 T^{2} + 3980 T^{3} + 401 p T^{4} + 22 p^{2} T^{5} + p^{3} T^{6} \)
89$A_4\times C_2$ \( 1 - 21 T + 365 T^{2} - 3689 T^{3} + 365 p T^{4} - 21 p^{2} T^{5} + p^{3} T^{6} \)
97$A_4\times C_2$ \( 1 + 36 T + 695 T^{2} + 8432 T^{3} + 695 p T^{4} + 36 p^{2} T^{5} + p^{3} T^{6} \)
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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

−7.41398168829002998366486834258, −7.08857840997328315886271952872, −6.98236423818887164975589258544, −6.87397414212527089686602359988, −6.51832245713235615884307887774, −6.26204329766499321445636992326, −6.14097658052542788362709031694, −5.73114904267701630817968365503, −5.64742534427441715404395395920, −5.53498866978288669005766059712, −4.72212620838503450351031735674, −4.69609791820562719735919245814, −4.54198332281166913382168497575, −4.21384417937094046376613281758, −4.13647589826749507063462039709, −3.93384977244362960011396599043, −3.42003462255525348509387762446, −2.92096599066984843548267945902, −2.77434688956749138652460658513, −2.24329291388236527287887303178, −2.05059746865616231182288180898, −1.95865328028201742720106199816, −1.19906902808632053387626735729, −1.08806988962437790297298111951, −0.990672632756559878822397403482, 0, 0, 0, 0.990672632756559878822397403482, 1.08806988962437790297298111951, 1.19906902808632053387626735729, 1.95865328028201742720106199816, 2.05059746865616231182288180898, 2.24329291388236527287887303178, 2.77434688956749138652460658513, 2.92096599066984843548267945902, 3.42003462255525348509387762446, 3.93384977244362960011396599043, 4.13647589826749507063462039709, 4.21384417937094046376613281758, 4.54198332281166913382168497575, 4.69609791820562719735919245814, 4.72212620838503450351031735674, 5.53498866978288669005766059712, 5.64742534427441715404395395920, 5.73114904267701630817968365503, 6.14097658052542788362709031694, 6.26204329766499321445636992326, 6.51832245713235615884307887774, 6.87397414212527089686602359988, 6.98236423818887164975589258544, 7.08857840997328315886271952872, 7.41398168829002998366486834258

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