# Properties

 Label 12-69e6-1.1-c1e6-0-0 Degree $12$ Conductor $107918163081$ Sign $1$ Analytic cond. $0.0279741$ Root an. cond. $0.742272$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 30·25-s + 4·27-s + 42·49-s − 7·64-s − 66·121-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 + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯
 L(s)  = 1 − 6·25-s + 0.769·27-s + 6·49-s − 7/8·64-s − 6·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + 0.0655·233-s + 0.0646·239-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 23^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 23^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$12$$ Conductor: $$3^{6} \cdot 23^{6}$$ Sign: $1$ Analytic conductor: $$0.0279741$$ Root analytic conductor: $$0.742272$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{69} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(12,\ 3^{6} \cdot 23^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.5419140974$$ $$L(\frac12)$$ $$\approx$$ $$0.5419140974$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 - 4 T^{3} + p^{3} T^{6}$$
23 $$( 1 + p T^{2} )^{3}$$
good2 $$( 1 - 3 T^{3} + p^{3} T^{6} )( 1 + 3 T^{3} + p^{3} T^{6} )$$
5 $$( 1 + p T^{2} )^{6}$$
7 $$( 1 - p T^{2} )^{6}$$
11 $$( 1 + p T^{2} )^{6}$$
13 $$( 1 - 74 T^{3} + p^{3} T^{6} )^{2}$$
17 $$( 1 + p T^{2} )^{6}$$
19 $$( 1 - p T^{2} )^{6}$$
29 $$( 1 - 282 T^{3} + p^{3} T^{6} )( 1 + 282 T^{3} + p^{3} T^{6} )$$
31 $$( 1 - 344 T^{3} + p^{3} T^{6} )^{2}$$
37 $$( 1 - p T^{2} )^{6}$$
41 $$( 1 - 426 T^{3} + p^{3} T^{6} )( 1 + 426 T^{3} + p^{3} T^{6} )$$
43 $$( 1 - p T^{2} )^{6}$$
47 $$( 1 - 48 T^{3} + p^{3} T^{6} )( 1 + 48 T^{3} + p^{3} T^{6} )$$
53 $$( 1 + p T^{2} )^{6}$$
59 $$( 1 - 12 T + p T^{2} )^{3}( 1 + 12 T + p T^{2} )^{3}$$
61 $$( 1 - p T^{2} )^{6}$$
67 $$( 1 - p T^{2} )^{6}$$
71 $$( 1 - 1176 T^{3} + p^{3} T^{6} )( 1 + 1176 T^{3} + p^{3} T^{6} )$$
73 $$( 1 - 1226 T^{3} + p^{3} T^{6} )^{2}$$
79 $$( 1 - p T^{2} )^{6}$$
83 $$( 1 + p T^{2} )^{6}$$
89 $$( 1 + p T^{2} )^{6}$$
97 $$( 1 - p T^{2} )^{6}$$
show less
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$

## Imaginary part of the first few zeros on the critical line

−8.556597545613138012436785354598, −8.023096448523080129214174074782, −8.010252325936754420631075237797, −7.80071370771741418136223586317, −7.75392894977755580774051116033, −7.44461818042972989246570578362, −7.10634518351940479789103619501, −7.06925972431746398824871010416, −6.60575718879056758640354074663, −6.43377571319064060198559125906, −6.07970326932404353267966841481, −5.75815579350626966047076266268, −5.69670628217488235399376793487, −5.64190668724476481857015871817, −5.28722177481878344435313472929, −4.88563067272750609639119986432, −4.40022809963226505440061252615, −4.17779343243913727442041511497, −3.97771951603574081853788893948, −3.70606195478842663425752487009, −3.62212382977332336213232284029, −2.77192614700738942165334686396, −2.42077868263742505761468144597, −2.22404428139095077576025598609, −1.55299249300073336255592343752, 1.55299249300073336255592343752, 2.22404428139095077576025598609, 2.42077868263742505761468144597, 2.77192614700738942165334686396, 3.62212382977332336213232284029, 3.70606195478842663425752487009, 3.97771951603574081853788893948, 4.17779343243913727442041511497, 4.40022809963226505440061252615, 4.88563067272750609639119986432, 5.28722177481878344435313472929, 5.64190668724476481857015871817, 5.69670628217488235399376793487, 5.75815579350626966047076266268, 6.07970326932404353267966841481, 6.43377571319064060198559125906, 6.60575718879056758640354074663, 7.06925972431746398824871010416, 7.10634518351940479789103619501, 7.44461818042972989246570578362, 7.75392894977755580774051116033, 7.80071370771741418136223586317, 8.010252325936754420631075237797, 8.023096448523080129214174074782, 8.556597545613138012436785354598

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

Plot not available for L-functions of degree greater than 10.