Properties

Label 6-5824e3-1.1-c1e3-0-3
Degree $6$
Conductor $197544116224$
Sign $-1$
Analytic cond. $100576.$
Root an. cond. $6.81944$
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  − 2·3-s − 2·5-s + 3·7-s + 9-s + 2·11-s − 3·13-s + 4·15-s + 4·17-s − 4·19-s − 6·21-s − 10·23-s − 8·25-s − 24·29-s + 4·31-s − 4·33-s − 6·35-s + 6·39-s + 2·41-s + 10·43-s − 2·45-s + 8·47-s + 6·49-s − 8·51-s − 8·53-s − 4·55-s + 8·57-s − 4·59-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.894·5-s + 1.13·7-s + 1/3·9-s + 0.603·11-s − 0.832·13-s + 1.03·15-s + 0.970·17-s − 0.917·19-s − 1.30·21-s − 2.08·23-s − 8/5·25-s − 4.45·29-s + 0.718·31-s − 0.696·33-s − 1.01·35-s + 0.960·39-s + 0.312·41-s + 1.52·43-s − 0.298·45-s + 1.16·47-s + 6/7·49-s − 1.12·51-s − 1.09·53-s − 0.539·55-s + 1.05·57-s − 0.520·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 7^{3} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 7^{3} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{18} \cdot 7^{3} \cdot 13^{3}\)
Sign: $-1$
Analytic conductor: \(100576.\)
Root analytic conductor: \(6.81944\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 2^{18} \cdot 7^{3} \cdot 13^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ -1 )\)

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)$
bad2 \( 1 \)
7$C_1$ \( ( 1 - T )^{3} \)
13$C_1$ \( ( 1 + T )^{3} \)
good3$S_4\times C_2$ \( 1 + 2 T + p T^{2} + 4 T^{3} + p^{2} T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
5$S_4\times C_2$ \( 1 + 2 T + 12 T^{2} + 18 T^{3} + 12 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
11$S_4\times C_2$ \( 1 - 2 T + 27 T^{2} - 36 T^{3} + 27 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 - 4 T + 41 T^{2} - 140 T^{3} + 41 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
19$S_4\times C_2$ \( 1 + 4 T + 58 T^{2} + 148 T^{3} + 58 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 + 10 T + 70 T^{2} + 324 T^{3} + 70 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 + 24 T + 272 T^{2} + 1846 T^{3} + 272 p T^{4} + 24 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 - 4 T + 74 T^{2} - 264 T^{3} + 74 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 + 53 T^{2} + 124 T^{3} + 53 p T^{4} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 - 2 T + 95 T^{2} - 172 T^{3} + 95 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 - 10 T + 58 T^{2} - 232 T^{3} + 58 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 - 8 T + 62 T^{2} - 208 T^{3} + 62 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 + 8 T + 124 T^{2} + 870 T^{3} + 124 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 + 4 T + 21 T^{2} - 216 T^{3} + 21 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{3} \)
67$S_4\times C_2$ \( 1 + 12 T + 77 T^{2} + 632 T^{3} + 77 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 - 6 T + 191 T^{2} - 868 T^{3} + 191 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 + 10 T + 120 T^{2} + 1186 T^{3} + 120 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 - 14 T + 242 T^{2} - 2196 T^{3} + 242 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 + 12 T - 22 T^{2} - 1276 T^{3} - 22 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 - 2 T + 172 T^{2} + 66 T^{3} + 172 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 + 10 T + 320 T^{2} + 1962 T^{3} + 320 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
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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

−7.58925538935750885858502324469, −7.48542045473327995273580114471, −7.05337696719728820119794129583, −7.01774046625046740800830754677, −6.30897007453947294788332285930, −6.25377243316185959572864053977, −6.13511413022611470859089942669, −5.66844040369005271812771367029, −5.63812007916384627321839317999, −5.52280857193637166551129323841, −5.21189465649124702957521392597, −4.88400980258178901774248395358, −4.53881806651060782536391295355, −4.08495554083479580606151108407, −4.07810922910570282680886557712, −4.03414363815320158917673114855, −3.82364097801894604317477357943, −3.27187333993529221725359602041, −3.09063714810667577463488139834, −2.42792354907095302242987270135, −2.19998034803094493572443018027, −2.05773068506813969950072068484, −1.67369752502613725165964045198, −1.33649537350056745866492778264, −0.935715890731264421552198409937, 0, 0, 0, 0.935715890731264421552198409937, 1.33649537350056745866492778264, 1.67369752502613725165964045198, 2.05773068506813969950072068484, 2.19998034803094493572443018027, 2.42792354907095302242987270135, 3.09063714810667577463488139834, 3.27187333993529221725359602041, 3.82364097801894604317477357943, 4.03414363815320158917673114855, 4.07810922910570282680886557712, 4.08495554083479580606151108407, 4.53881806651060782536391295355, 4.88400980258178901774248395358, 5.21189465649124702957521392597, 5.52280857193637166551129323841, 5.63812007916384627321839317999, 5.66844040369005271812771367029, 6.13511413022611470859089942669, 6.25377243316185959572864053977, 6.30897007453947294788332285930, 7.01774046625046740800830754677, 7.05337696719728820119794129583, 7.48542045473327995273580114471, 7.58925538935750885858502324469

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