Properties

Label 6-70e6-1.1-c1e3-0-2
Degree $6$
Conductor $117649000000$
Sign $-1$
Analytic cond. $59898.9$
Root an. cond. $6.25513$
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-s + 4·11-s − 8·13-s − 10·17-s + 2·19-s − 3·23-s − 2·27-s + 3·31-s − 4·33-s + 6·37-s + 8·39-s − 11·41-s − 11·43-s − 4·47-s + 10·51-s + 14·53-s − 2·57-s + 5·59-s − 17·61-s + 20·67-s + 3·69-s − 19·71-s − 12·73-s − 79-s + 2·81-s − 28·83-s + 14·89-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.20·11-s − 2.21·13-s − 2.42·17-s + 0.458·19-s − 0.625·23-s − 0.384·27-s + 0.538·31-s − 0.696·33-s + 0.986·37-s + 1.28·39-s − 1.71·41-s − 1.67·43-s − 0.583·47-s + 1.40·51-s + 1.92·53-s − 0.264·57-s + 0.650·59-s − 2.17·61-s + 2.44·67-s + 0.361·69-s − 2.25·71-s − 1.40·73-s − 0.112·79-s + 2/9·81-s − 3.07·83-s + 1.48·89-s + ⋯

Functional equation

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

Invariants

Degree: \(6\)
Conductor: \(2^{6} \cdot 5^{6} \cdot 7^{6}\)
Sign: $-1$
Analytic conductor: \(59898.9\)
Root analytic conductor: \(6.25513\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 2^{6} \cdot 5^{6} \cdot 7^{6} ,\ ( \ : 1/2, 1/2, 1/2 ),\ -1 )\)

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 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good3$S_4\times C_2$ \( 1 + T + T^{2} + p T^{3} + p T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
11$S_4\times C_2$ \( 1 - 4 T + 30 T^{2} - 79 T^{3} + 30 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
13$S_4\times C_2$ \( 1 + 8 T + 4 p T^{2} + 205 T^{3} + 4 p^{2} T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 + 10 T + 60 T^{2} + 259 T^{3} + 60 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
19$S_4\times C_2$ \( 1 - 2 T + 34 T^{2} - 97 T^{3} + 34 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 + 3 T + 33 T^{2} + 57 T^{3} + 33 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 + 42 T^{2} + 81 T^{3} + 42 p T^{4} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 - 3 T + 51 T^{2} - 223 T^{3} + 51 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 - 6 T + 48 T^{2} - 295 T^{3} + 48 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 + 11 T + 45 T^{2} + 29 T^{3} + 45 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 + 11 T + 143 T^{2} + 875 T^{3} + 143 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 + 4 T + 120 T^{2} + 367 T^{3} + 120 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 - 14 T + 144 T^{2} - 935 T^{3} + 144 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 - 5 T + 3 p T^{2} - 581 T^{3} + 3 p^{2} T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 + 17 T + 209 T^{2} + 2075 T^{3} + 209 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 - 20 T + 301 T^{2} - 2752 T^{3} + 301 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 + 19 T + 300 T^{2} + 2743 T^{3} + 300 p T^{4} + 19 p^{2} T^{5} + p^{3} T^{6} \)
73$C_2$ \( ( 1 + 4 T + p T^{2} )^{3} \)
79$S_4\times C_2$ \( 1 + T + 119 T^{2} + 607 T^{3} + 119 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 + 28 T + 378 T^{2} + 3667 T^{3} + 378 p T^{4} + 28 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 - 14 T + 306 T^{2} - 2447 T^{3} + 306 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 + 15 T + 273 T^{2} + 2905 T^{3} + 273 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \)
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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

−7.71920191484672283863743356444, −7.18707751540595310388673389020, −7.16658950971811917340978676884, −6.87939662545077235872532903585, −6.72195279051675364985770820241, −6.50668535483035169722619410280, −6.45599400572719204882499138864, −5.86837094520446926586770848278, −5.64934180254994674669913687923, −5.56440975021499183724389494499, −5.19685020958603945136686937170, −4.93764865244718069834355621296, −4.65623057378925454794560210844, −4.39789712028435498181854159752, −4.21111345721352070324153816680, −4.12354757703870222598903484522, −3.66088547707806972914988030384, −3.27114606555612804085389851130, −2.98700096528412121908527402699, −2.56925213437852162186480735690, −2.49221441059160069332541343509, −2.06514728224337786948031699021, −1.78569232748833117148105030716, −1.30534389583557870960338611247, −1.12572333962308657853215940228, 0, 0, 0, 1.12572333962308657853215940228, 1.30534389583557870960338611247, 1.78569232748833117148105030716, 2.06514728224337786948031699021, 2.49221441059160069332541343509, 2.56925213437852162186480735690, 2.98700096528412121908527402699, 3.27114606555612804085389851130, 3.66088547707806972914988030384, 4.12354757703870222598903484522, 4.21111345721352070324153816680, 4.39789712028435498181854159752, 4.65623057378925454794560210844, 4.93764865244718069834355621296, 5.19685020958603945136686937170, 5.56440975021499183724389494499, 5.64934180254994674669913687923, 5.86837094520446926586770848278, 6.45599400572719204882499138864, 6.50668535483035169722619410280, 6.72195279051675364985770820241, 6.87939662545077235872532903585, 7.16658950971811917340978676884, 7.18707751540595310388673389020, 7.71920191484672283863743356444

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