Properties

Label 6-260e3-1.1-c1e3-0-0
Degree $6$
Conductor $17576000$
Sign $1$
Analytic cond. $8.94852$
Root an. cond. $1.44087$
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  + 2·3-s + 3·5-s − 2·7-s + 3·9-s + 3·13-s + 6·15-s + 2·17-s + 8·19-s − 4·21-s − 10·23-s + 6·25-s + 4·27-s + 10·29-s − 12·31-s − 6·35-s − 2·37-s + 6·39-s − 2·41-s − 2·43-s + 9·45-s − 10·47-s + 3·49-s + 4·51-s − 18·53-s + 16·57-s − 16·59-s + 14·61-s + ⋯
L(s)  = 1  + 1.15·3-s + 1.34·5-s − 0.755·7-s + 9-s + 0.832·13-s + 1.54·15-s + 0.485·17-s + 1.83·19-s − 0.872·21-s − 2.08·23-s + 6/5·25-s + 0.769·27-s + 1.85·29-s − 2.15·31-s − 1.01·35-s − 0.328·37-s + 0.960·39-s − 0.312·41-s − 0.304·43-s + 1.34·45-s − 1.45·47-s + 3/7·49-s + 0.560·51-s − 2.47·53-s + 2.11·57-s − 2.08·59-s + 1.79·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 17576000 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17576000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(17576000\)    =    \(2^{6} \cdot 5^{3} \cdot 13^{3}\)
Sign: $1$
Analytic conductor: \(8.94852\)
Root analytic conductor: \(1.44087\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 17576000,\ (\ :1/2, 1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.161073795\)
\(L(\frac12)\) \(\approx\) \(3.161073795\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{3} \)
13$C_1$ \( ( 1 - T )^{3} \)
good3$D_{6}$ \( 1 - 2 T + T^{2} + p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
7$S_4\times C_2$ \( 1 + 2 T + T^{2} + 4 T^{3} + p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
11$S_4\times C_2$ \( 1 + 9 T^{2} + 36 T^{3} + 9 p T^{4} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 - 2 T + 15 T^{2} - 92 T^{3} + 15 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
19$S_4\times C_2$ \( 1 - 8 T + 41 T^{2} - 140 T^{3} + 41 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 + 10 T + 93 T^{2} + 472 T^{3} + 93 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 - 10 T + 99 T^{2} - 556 T^{3} + 99 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 + 12 T + 117 T^{2} + 748 T^{3} + 117 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 + 2 T + 67 T^{2} + 76 T^{3} + 67 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 + 2 T + 87 T^{2} + 188 T^{3} + 87 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 + 2 T + 121 T^{2} + 160 T^{3} + 121 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 + 10 T + 153 T^{2} + 916 T^{3} + 153 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 + 18 T + 171 T^{2} + 1260 T^{3} + 171 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 + 16 T + 3 p T^{2} + 1324 T^{3} + 3 p^{2} T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 - 14 T + 227 T^{2} - 1700 T^{3} + 227 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 - 14 T + 221 T^{2} - 1724 T^{3} + 221 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 + 189 T^{2} + 36 T^{3} + 189 p T^{4} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 - 14 T + 95 T^{2} - 260 T^{3} + 95 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 - 8 T + 221 T^{2} - 1232 T^{3} + 221 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 + 6 T + 117 T^{2} + 60 T^{3} + 117 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 + 2 T + 87 T^{2} + 572 T^{3} + 87 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 - 26 T + 431 T^{2} - 5036 T^{3} + 431 p T^{4} - 26 p^{2} T^{5} + p^{3} T^{6} \)
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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

−10.66052080806835428498984529416, −10.12618381118772996790069729210, −10.02221034842396284484249048409, −9.635120035701252929684526255163, −9.588896029624823785480645195865, −9.223965415667874847638279434336, −8.887667282686898560660287045257, −8.386265277994660403410791482109, −8.117327293029160699599557318312, −7.919297841251960545534826261088, −7.45394762840944555530108600175, −7.01140185229377012753031027012, −6.61159627597652009595990559614, −6.34777905280032026341403151687, −5.97942623479865784588510214274, −5.62432157327863760926255597836, −5.05923654679930043929837227162, −4.90138635797032216441928587546, −4.15995559979417781398737110037, −3.56958656239750513137218875214, −3.25659101273538783395765151540, −3.13786347330253335042143179697, −2.17160924833122862384495920329, −1.91481339471822433626357895927, −1.18090226205146643611297427368, 1.18090226205146643611297427368, 1.91481339471822433626357895927, 2.17160924833122862384495920329, 3.13786347330253335042143179697, 3.25659101273538783395765151540, 3.56958656239750513137218875214, 4.15995559979417781398737110037, 4.90138635797032216441928587546, 5.05923654679930043929837227162, 5.62432157327863760926255597836, 5.97942623479865784588510214274, 6.34777905280032026341403151687, 6.61159627597652009595990559614, 7.01140185229377012753031027012, 7.45394762840944555530108600175, 7.919297841251960545534826261088, 8.117327293029160699599557318312, 8.386265277994660403410791482109, 8.887667282686898560660287045257, 9.223965415667874847638279434336, 9.588896029624823785480645195865, 9.635120035701252929684526255163, 10.02221034842396284484249048409, 10.12618381118772996790069729210, 10.66052080806835428498984529416

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