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

Origins of factors

Downloads

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Normalization:  

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 \( ( 1 - p T^{2} )^{6} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

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.