Properties

Label 6-8470e3-1.1-c1e3-0-2
Degree $6$
Conductor $607645423000$
Sign $1$
Analytic cond. $309372.$
Root an. cond. $8.22394$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·2-s + 2·3-s + 6·4-s + 3·5-s − 6·6-s + 3·7-s − 10·8-s + 9-s − 9·10-s + 12·12-s − 2·13-s − 9·14-s + 6·15-s + 15·16-s − 2·17-s − 3·18-s + 6·19-s + 18·20-s + 6·21-s + 10·23-s − 20·24-s + 6·25-s + 6·26-s + 18·28-s − 18·30-s − 12·31-s − 21·32-s + ⋯
L(s)  = 1  − 2.12·2-s + 1.15·3-s + 3·4-s + 1.34·5-s − 2.44·6-s + 1.13·7-s − 3.53·8-s + 1/3·9-s − 2.84·10-s + 3.46·12-s − 0.554·13-s − 2.40·14-s + 1.54·15-s + 15/4·16-s − 0.485·17-s − 0.707·18-s + 1.37·19-s + 4.02·20-s + 1.30·21-s + 2.08·23-s − 4.08·24-s + 6/5·25-s + 1.17·26-s + 3.40·28-s − 3.28·30-s − 2.15·31-s − 3.71·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 5^{3} \cdot 7^{3} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 5^{3} \cdot 7^{3} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{3} \cdot 5^{3} \cdot 7^{3} \cdot 11^{6}\)
Sign: $1$
Analytic conductor: \(309372.\)
Root analytic conductor: \(8.22394\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 2^{3} \cdot 5^{3} \cdot 7^{3} \cdot 11^{6} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(4.792149395\)
\(L(\frac12)\) \(\approx\) \(4.792149395\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{3} \)
5$C_1$ \( ( 1 - T )^{3} \)
7$C_1$ \( ( 1 - T )^{3} \)
11 \( 1 \)
good3$S_4\times C_2$ \( 1 - 2 T + p T^{2} - 4 T^{3} + p^{2} T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
13$S_4\times C_2$ \( 1 + 2 T + 23 T^{2} + 36 T^{3} + 23 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 + 2 T + 23 T^{2} + 76 T^{3} + 23 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
19$S_4\times C_2$ \( 1 - 6 T + 11 T^{2} + 4 T^{3} + 11 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 - 10 T + 95 T^{2} - 476 T^{3} + 95 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 + 21 T^{2} + 92 T^{3} + 21 p T^{4} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 + 12 T + 113 T^{2} + 712 T^{3} + 113 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 - 4 T + 109 T^{2} - 292 T^{3} + 109 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 + 4 T + 33 T^{2} + 12 p T^{3} + 33 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 + 8 T + 21 T^{2} - 160 T^{3} + 21 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 + 29 T^{2} - 128 T^{3} + 29 p T^{4} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 - 12 T + 141 T^{2} - 980 T^{3} + 141 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 + 4 T + 113 T^{2} + 344 T^{3} + 113 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 - 6 T + 83 T^{2} - 388 T^{3} + 83 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 - 4 T + 89 T^{2} - 600 T^{3} + 89 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 + 28 T + 413 T^{2} + 4040 T^{3} + 413 p T^{4} + 28 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 - 14 T + 255 T^{2} - 2036 T^{3} + 255 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 + 18 T + 287 T^{2} + 2588 T^{3} + 287 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 - 8 T + 241 T^{2} - 1296 T^{3} + 241 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 - 18 T + 239 T^{2} - 1996 T^{3} + 239 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 + 4 T + 137 T^{2} + 1092 T^{3} + 137 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
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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

−7.19558674327580512944838775717, −6.90300425525271161204259166193, −6.52140951385644753733343816993, −6.40966989065172094313639560424, −5.84907127756862277030340796421, −5.74906769180087884411473950460, −5.72905950411484301571223813369, −5.19462346015292754778413417257, −5.03160986931992406868887951308, −5.01786326332934573911360804580, −4.55204984954478532663162877535, −4.26606229740453695291208601117, −3.87622110000924825929186332213, −3.42654582609784171412374442967, −3.21912130602949394407059928559, −3.17613585520047066847966027986, −2.73791213988939382077415955212, −2.40910225422251583507326352964, −2.34779885784852404045892139007, −1.99427337453769251688043633450, −1.55945187206749899373916071934, −1.53216886223004944863851438552, −1.25236157293659147558882679445, −0.63596818524233915662653421223, −0.49752383748670777740194383243, 0.49752383748670777740194383243, 0.63596818524233915662653421223, 1.25236157293659147558882679445, 1.53216886223004944863851438552, 1.55945187206749899373916071934, 1.99427337453769251688043633450, 2.34779885784852404045892139007, 2.40910225422251583507326352964, 2.73791213988939382077415955212, 3.17613585520047066847966027986, 3.21912130602949394407059928559, 3.42654582609784171412374442967, 3.87622110000924825929186332213, 4.26606229740453695291208601117, 4.55204984954478532663162877535, 5.01786326332934573911360804580, 5.03160986931992406868887951308, 5.19462346015292754778413417257, 5.72905950411484301571223813369, 5.74906769180087884411473950460, 5.84907127756862277030340796421, 6.40966989065172094313639560424, 6.52140951385644753733343816993, 6.90300425525271161204259166193, 7.19558674327580512944838775717

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