Properties

Degree $6$
Conductor $5545233$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s − 2·4-s − 2·5-s + 9·7-s + 8-s + 6·9-s − 2·11-s + 6·12-s + 4·13-s + 6·15-s + 3·17-s + 7·19-s + 4·20-s − 27·21-s + 23-s − 3·24-s − 6·25-s − 10·27-s − 18·28-s − 11·29-s + 13·31-s − 4·32-s + 6·33-s − 18·35-s − 12·36-s − 5·37-s − 12·39-s + ⋯
L(s)  = 1  − 1.73·3-s − 4-s − 0.894·5-s + 3.40·7-s + 0.353·8-s + 2·9-s − 0.603·11-s + 1.73·12-s + 1.10·13-s + 1.54·15-s + 0.727·17-s + 1.60·19-s + 0.894·20-s − 5.89·21-s + 0.208·23-s − 0.612·24-s − 6/5·25-s − 1.92·27-s − 3.40·28-s − 2.04·29-s + 2.33·31-s − 0.707·32-s + 1.04·33-s − 3.04·35-s − 2·36-s − 0.821·37-s − 1.92·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5545233 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5545233 ^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(5545233\)    =    \(3^{3} \cdot 59^{3}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{177} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 5545233,\ (\ :1/2, 1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.827767\)
\(L(\frac12)\) \(\approx\) \(0.827767\)
\(L(\frac{3}{2})\) 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)$
bad3$C_1$ \( ( 1 + T )^{3} \)
59$C_1$ \( ( 1 - T )^{3} \)
good2$S_4\times C_2$ \( 1 + p T^{2} - T^{3} + p^{2} T^{4} + p^{3} T^{6} \)
5$S_4\times C_2$ \( 1 + 2 T + 2 p T^{2} + 18 T^{3} + 2 p^{2} T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
7$S_4\times C_2$ \( 1 - 9 T + 44 T^{2} - 142 T^{3} + 44 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \)
11$S_4\times C_2$ \( 1 + 2 T + 2 p T^{2} + 48 T^{3} + 2 p^{2} T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
13$S_4\times C_2$ \( 1 - 4 T + 32 T^{2} - 6 p T^{3} + 32 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 - 3 T + 8 T^{2} - 4 T^{3} + 8 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \)
19$S_4\times C_2$ \( 1 - 7 T + 68 T^{2} - 270 T^{3} + 68 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 - T + 42 T^{2} + 18 T^{3} + 42 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 + 11 T + 96 T^{2} + 564 T^{3} + 96 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 - 13 T + 130 T^{2} - 778 T^{3} + 130 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 + 5 T + 92 T^{2} + 384 T^{3} + 92 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 + T + 84 T^{2} + 156 T^{3} + 84 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 - 6 T + 38 T^{2} + 76 T^{3} + 38 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 - 11 T + 104 T^{2} - 538 T^{3} + 104 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 - 2 T + 70 T^{2} - 270 T^{3} + 70 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 + T + 82 T^{2} + 220 T^{3} + 82 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 - 10 T + 82 T^{2} - 556 T^{3} + 82 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 - 26 T + 406 T^{2} - 4116 T^{3} + 406 p T^{4} - 26 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 - 7 T + 78 T^{2} - 304 T^{3} + 78 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 - 2 T + 206 T^{2} - 348 T^{3} + 206 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 + 3 T + 50 T^{2} + 350 T^{3} + 50 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 + 23 T + 358 T^{2} + 3816 T^{3} + 358 p T^{4} + 23 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 - 14 T + 266 T^{2} - 2514 T^{3} + 266 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \)
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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

−11.44093664029276226870754784110, −11.15985552852358300343642681868, −10.87566813003725267209946673304, −10.71237798456219590306607710559, −10.16212449178790756649317697097, −9.721217379621078642055548092576, −9.494066915536887870405735842648, −8.898106803057333287936314895230, −8.401406427558736789534259397578, −8.216604281349738454771958114708, −7.85009310363940692148093872438, −7.62603594227267634951765609978, −7.39228872938730447510477628252, −6.87134282413878860829818389583, −6.13976133051542895489123088615, −5.72611373648574677162227333751, −5.14592092532012328961583137997, −5.10041782871437357986933036727, −5.08367229387006211553320505309, −4.21342416017582223945039289972, −4.03097010048642152134062221981, −3.70709883862564397941779945184, −2.31017527179416978223257425965, −1.47181740843826448674411088050, −0.983967072382549312772776173028, 0.983967072382549312772776173028, 1.47181740843826448674411088050, 2.31017527179416978223257425965, 3.70709883862564397941779945184, 4.03097010048642152134062221981, 4.21342416017582223945039289972, 5.08367229387006211553320505309, 5.10041782871437357986933036727, 5.14592092532012328961583137997, 5.72611373648574677162227333751, 6.13976133051542895489123088615, 6.87134282413878860829818389583, 7.39228872938730447510477628252, 7.62603594227267634951765609978, 7.85009310363940692148093872438, 8.216604281349738454771958114708, 8.401406427558736789534259397578, 8.898106803057333287936314895230, 9.494066915536887870405735842648, 9.721217379621078642055548092576, 10.16212449178790756649317697097, 10.71237798456219590306607710559, 10.87566813003725267209946673304, 11.15985552852358300343642681868, 11.44093664029276226870754784110

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