Properties

Degree 6
Conductor $ 2^{6} \cdot 17^{3} \cdot 59^{3} $
Sign $-1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 3

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s + 5-s + 3·7-s − 3·9-s − 6·11-s + 2·13-s − 3·15-s + 3·17-s − 7·19-s − 9·21-s + 3·25-s + 26·27-s − 5·29-s + 4·31-s + 18·33-s + 3·35-s + 6·37-s − 6·39-s − 7·41-s − 8·43-s − 3·45-s − 12·47-s − 15·49-s − 9·51-s + 17·53-s − 6·55-s + 21·57-s + ⋯
L(s)  = 1  − 1.73·3-s + 0.447·5-s + 1.13·7-s − 9-s − 1.80·11-s + 0.554·13-s − 0.774·15-s + 0.727·17-s − 1.60·19-s − 1.96·21-s + 3/5·25-s + 5.00·27-s − 0.928·29-s + 0.718·31-s + 3.13·33-s + 0.507·35-s + 0.986·37-s − 0.960·39-s − 1.09·41-s − 1.21·43-s − 0.447·45-s − 1.75·47-s − 2.14·49-s − 1.26·51-s + 2.33·53-s − 0.809·55-s + 2.78·57-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{6} \cdot 17^{3} \cdot 59^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & -\,\Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{6} \cdot 17^{3} \cdot 59^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr =\mathstrut & -\,\Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(6\)
\( N \)  =  \(2^{6} \cdot 17^{3} \cdot 59^{3}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{4012} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  3
Selberg data  =  $(6,\ 2^{6} \cdot 17^{3} \cdot 59^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ -1 )$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;17,\;59\}$, \(F_p\) is a polynomial of degree 6. If $p \in \{2,\;17,\;59\}$, then $F_p$ is a polynomial of degree at most 5.
$p$$\Gal(F_p)$$F_p$
bad2 \( 1 \)
17$C_1$ \( ( 1 - T )^{3} \)
59$C_1$ \( ( 1 + T )^{3} \)
good3$C_2$ \( ( 1 + T + p T^{2} )^{3} \)
5$S_4\times C_2$ \( 1 - T - 2 T^{2} - T^{3} - 2 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
7$C_2$ \( ( 1 - T + p T^{2} )^{3} \)
11$C_2$ \( ( 1 + 2 T + p T^{2} )^{3} \)
13$S_4\times C_2$ \( 1 - 2 T + 23 T^{2} - 44 T^{3} + 23 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
19$S_4\times C_2$ \( 1 + 7 T + 56 T^{2} + 235 T^{3} + 56 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 + 33 T^{2} + 8 T^{3} + 33 p T^{4} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 + 5 T + 78 T^{2} + 269 T^{3} + 78 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 - 4 T + 49 T^{2} - 80 T^{3} + 49 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 - 6 T + 15 T^{2} + 188 T^{3} + 15 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 + 7 T + 122 T^{2} + 543 T^{3} + 122 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 + 8 T + 101 T^{2} + 680 T^{3} + 101 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 + 12 T + 153 T^{2} + 1040 T^{3} + 153 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 - 17 T + 154 T^{2} - 1085 T^{3} + 154 p T^{4} - 17 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 + 8 T + 187 T^{2} + 952 T^{3} + 187 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 + 6 T + 69 T^{2} + 460 T^{3} + 69 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 + 4 T + 117 T^{2} + 760 T^{3} + 117 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{3} \)
79$C_2$ \( ( 1 + 5 T + p T^{2} )^{3} \)
83$S_4\times C_2$ \( 1 + 6 T + 153 T^{2} + 1212 T^{3} + 153 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 - 10 T + 251 T^{2} - 1548 T^{3} + 251 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 + 12 T + 207 T^{2} + 1936 T^{3} + 207 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.029543041451572794945177254801, −7.61534479522816023643667154472, −7.42259670650967108752949488553, −6.99461622700724687408939768802, −6.59260303077055468452134603944, −6.56623538986048203928823539720, −6.32985664917928411947869130406, −5.98421873904288820088493297184, −5.82995478223402936263624602935, −5.63128930390334481218375890573, −5.32492384863894211487470495345, −5.08936022576443553844333274072, −5.03594860526762005395342643400, −4.68861124239776947928216451303, −4.56522575124683794996913070657, −4.14900301099168479685349612341, −3.62503438546142810655571113065, −3.28741556755469027254772645352, −3.15291237176619664944276582847, −2.64576878800819781500371666205, −2.51900774308071596193943062192, −2.27721176831537020118005293547, −1.64597547128874264796108915571, −1.32881466096707980999311054514, −1.05882803055996315281064102656, 0, 0, 0, 1.05882803055996315281064102656, 1.32881466096707980999311054514, 1.64597547128874264796108915571, 2.27721176831537020118005293547, 2.51900774308071596193943062192, 2.64576878800819781500371666205, 3.15291237176619664944276582847, 3.28741556755469027254772645352, 3.62503438546142810655571113065, 4.14900301099168479685349612341, 4.56522575124683794996913070657, 4.68861124239776947928216451303, 5.03594860526762005395342643400, 5.08936022576443553844333274072, 5.32492384863894211487470495345, 5.63128930390334481218375890573, 5.82995478223402936263624602935, 5.98421873904288820088493297184, 6.32985664917928411947869130406, 6.56623538986048203928823539720, 6.59260303077055468452134603944, 6.99461622700724687408939768802, 7.42259670650967108752949488553, 7.61534479522816023643667154472, 8.029543041451572794945177254801

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