Properties

Degree 6
Conductor $ 23^{3} $
Sign $1$
Motivic weight 2
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 7·8-s − 69·23-s + 75·25-s − 38·27-s + 147·49-s + 78·59-s − 15·64-s − 498·101-s + 363·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 483·184-s + 191-s + 193-s + 197-s + 199-s − 525·200-s + ⋯
L(s)  = 1  − 7/8·8-s − 3·23-s + 3·25-s − 1.40·27-s + 3·49-s + 1.32·59-s − 0.234·64-s − 4.93·101-s + 3·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 21/8·184-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s − 2.62·200-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 12167 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 12167 ^{s/2} \, \Gamma_{\C}(s+1)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(6\)
\( N \)  =  \(12167\)    =    \(23^{3}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(2\)
character  :  induced by $\chi_{23} (22, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(6,\ 12167,\ (\ :1, 1, 1),\ 1)$
$L(\frac{3}{2})$  $\approx$  $0.703809$
$L(\frac12)$  $\approx$  $0.703809$
$L(2)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 23$, \(F_p\) is a polynomial of degree 6. If $p = 23$, then $F_p$ is a polynomial of degree at most 5.
$p$$\Gal(F_p)$$F_p$
bad23$C_1$ \( ( 1 + p T )^{3} \)
good2$D_{6}$ \( 1 + 7 T^{3} + p^{6} T^{6} \)
3$D_{6}$ \( 1 + 38 T^{3} + p^{6} T^{6} \)
5$C_1$$\times$$C_1$ \( ( 1 - p T )^{3}( 1 + p T )^{3} \)
7$C_1$$\times$$C_1$ \( ( 1 - p T )^{3}( 1 + p T )^{3} \)
11$C_1$$\times$$C_1$ \( ( 1 - p T )^{3}( 1 + p T )^{3} \)
13$D_{6}$ \( 1 - 1082 T^{3} + p^{6} T^{6} \)
17$C_1$$\times$$C_1$ \( ( 1 - p T )^{3}( 1 + p T )^{3} \)
19$C_1$$\times$$C_1$ \( ( 1 - p T )^{3}( 1 + p T )^{3} \)
29$D_{6}$ \( 1 - 30746 T^{3} + p^{6} T^{6} \)
31$D_{6}$ \( 1 - 58754 T^{3} + p^{6} T^{6} \)
37$C_1$$\times$$C_1$ \( ( 1 - p T )^{3}( 1 + p T )^{3} \)
41$D_{6}$ \( 1 - 43634 T^{3} + p^{6} T^{6} \)
43$C_1$$\times$$C_1$ \( ( 1 - p T )^{3}( 1 + p T )^{3} \)
47$D_{6}$ \( 1 + 205342 T^{3} + p^{6} T^{6} \)
53$C_1$$\times$$C_1$ \( ( 1 - p T )^{3}( 1 + p T )^{3} \)
59$C_2$ \( ( 1 - 26 T + p^{2} T^{2} )^{3} \)
61$C_1$$\times$$C_1$ \( ( 1 - p T )^{3}( 1 + p T )^{3} \)
67$C_1$$\times$$C_1$ \( ( 1 - p T )^{3}( 1 + p T )^{3} \)
71$D_{6}$ \( 1 - 667154 T^{3} + p^{6} T^{6} \)
73$D_{6}$ \( 1 - 725042 T^{3} + p^{6} T^{6} \)
79$C_1$$\times$$C_1$ \( ( 1 - p T )^{3}( 1 + p T )^{3} \)
83$C_1$$\times$$C_1$ \( ( 1 - p T )^{3}( 1 + p T )^{3} \)
89$C_1$$\times$$C_1$ \( ( 1 - p T )^{3}( 1 + p T )^{3} \)
97$C_1$$\times$$C_1$ \( ( 1 - p T )^{3}( 1 + p T )^{3} \)
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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

−16.25266090259699302544844085089, −15.38379949405769162888300515781, −15.25045957856847525792594451566, −14.82823558841975961978977417582, −14.30448934002337246511345373331, −13.87709638210915675759205960405, −13.54348074276356119184964203974, −12.88326648893801043752994094227, −12.34454102311100891116481500791, −12.18652506141296656813498136593, −11.66311862169944965457286273302, −11.12595221769137640392299636957, −10.35740039286693641585425892371, −10.33664492004809423462870333956, −9.319600292021350002126565316276, −9.255683549229959107838401341587, −8.254627656874356514129115296000, −8.249320660797426125572295542019, −7.15878206375237697440477144189, −6.81039803586467718459107985552, −5.79533420479259215020430426704, −5.67469663404156687062790353124, −4.46007887250445186823029671800, −3.69917499700457382202237665871, −2.50316094689835549482408964468, 2.50316094689835549482408964468, 3.69917499700457382202237665871, 4.46007887250445186823029671800, 5.67469663404156687062790353124, 5.79533420479259215020430426704, 6.81039803586467718459107985552, 7.15878206375237697440477144189, 8.249320660797426125572295542019, 8.254627656874356514129115296000, 9.255683549229959107838401341587, 9.319600292021350002126565316276, 10.33664492004809423462870333956, 10.35740039286693641585425892371, 11.12595221769137640392299636957, 11.66311862169944965457286273302, 12.18652506141296656813498136593, 12.34454102311100891116481500791, 12.88326648893801043752994094227, 13.54348074276356119184964203974, 13.87709638210915675759205960405, 14.30448934002337246511345373331, 14.82823558841975961978977417582, 15.25045957856847525792594451566, 15.38379949405769162888300515781, 16.25266090259699302544844085089

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