Properties

Degree 12
Conductor $ 3^{6} \cdot 5^{6} \cdot 7^{6} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 4-s + 2·5-s − 3·9-s + 12·11-s + 2·16-s − 12·19-s + 2·20-s + 25-s − 4·29-s + 4·31-s − 3·36-s + 4·41-s + 12·44-s − 6·45-s − 3·49-s + 24·55-s − 32·59-s − 12·61-s + 6·64-s + 12·71-s − 12·76-s − 24·79-s + 4·80-s + 6·81-s − 28·89-s − 24·95-s − 36·99-s + ⋯
L(s)  = 1  + 1/2·4-s + 0.894·5-s − 9-s + 3.61·11-s + 1/2·16-s − 2.75·19-s + 0.447·20-s + 1/5·25-s − 0.742·29-s + 0.718·31-s − 1/2·36-s + 0.624·41-s + 1.80·44-s − 0.894·45-s − 3/7·49-s + 3.23·55-s − 4.16·59-s − 1.53·61-s + 3/4·64-s + 1.42·71-s − 1.37·76-s − 2.70·79-s + 0.447·80-s + 2/3·81-s − 2.96·89-s − 2.46·95-s − 3.61·99-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(12\)
\( N \)  =  \(3^{6} \cdot 5^{6} \cdot 7^{6}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{105} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(12,\ 3^{6} \cdot 5^{6} \cdot 7^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )$
$L(1)$  $\approx$  $1.36038$
$L(\frac12)$  $\approx$  $1.36038$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 12. If $p \in \{3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 11.
$p$$F_p(T)$
bad3 \( ( 1 + T^{2} )^{3} \)
5 \( 1 - 2 T + 3 T^{2} - 12 T^{3} + 3 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
7 \( ( 1 + T^{2} )^{3} \)
good2 \( 1 - T^{2} - T^{4} - 3 T^{6} - p^{2} T^{8} - p^{4} T^{10} + p^{6} T^{12} \)
11 \( ( 1 - 2 T + p T^{2} )^{6} \)
13 \( 1 - 34 T^{2} + 359 T^{4} - 2172 T^{6} + 359 p^{2} T^{8} - 34 p^{4} T^{10} + p^{6} T^{12} \)
17 \( 1 - 70 T^{2} + 2415 T^{4} - 51220 T^{6} + 2415 p^{2} T^{8} - 70 p^{4} T^{10} + p^{6} T^{12} \)
19 \( ( 1 + 6 T + 53 T^{2} + 188 T^{3} + 53 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
23 \( 1 - 106 T^{2} + 5183 T^{4} - 150348 T^{6} + 5183 p^{2} T^{8} - 106 p^{4} T^{10} + p^{6} T^{12} \)
29 \( ( 1 + 2 T + 35 T^{2} + 76 T^{3} + 35 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
31 \( ( 1 - 2 T + 41 T^{2} + 60 T^{3} + 41 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
37 \( 1 - 46 T^{2} + 1399 T^{4} - 74788 T^{6} + 1399 p^{2} T^{8} - 46 p^{4} T^{10} + p^{6} T^{12} \)
41 \( ( 1 - 2 T + 63 T^{2} + 36 T^{3} + 63 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
43 \( 1 + 46 T^{2} + 2839 T^{4} + 118948 T^{6} + 2839 p^{2} T^{8} + 46 p^{4} T^{10} + p^{6} T^{12} \)
47 \( 1 - 154 T^{2} + 12143 T^{4} - 652332 T^{6} + 12143 p^{2} T^{8} - 154 p^{4} T^{10} + p^{6} T^{12} \)
53 \( 1 - 146 T^{2} + 14103 T^{4} - 884828 T^{6} + 14103 p^{2} T^{8} - 146 p^{4} T^{10} + p^{6} T^{12} \)
59 \( ( 1 + 16 T + 113 T^{2} + 608 T^{3} + 113 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
61 \( ( 1 + 6 T + 131 T^{2} + 484 T^{3} + 131 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
67 \( 1 - 274 T^{2} + 36103 T^{4} - 2962972 T^{6} + 36103 p^{2} T^{8} - 274 p^{4} T^{10} + p^{6} T^{12} \)
71 \( ( 1 - 2 T + p T^{2} )^{6} \)
73 \( 1 - 298 T^{2} + 43775 T^{4} - 3982284 T^{6} + 43775 p^{2} T^{8} - 298 p^{4} T^{10} + p^{6} T^{12} \)
79 \( ( 1 + 12 T + 221 T^{2} + 1576 T^{3} + 221 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
83 \( 1 - 306 T^{2} + 47783 T^{4} - 4793948 T^{6} + 47783 p^{2} T^{8} - 306 p^{4} T^{10} + p^{6} T^{12} \)
89 \( ( 1 + 14 T + 319 T^{2} + 2532 T^{3} + 319 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
97 \( 1 - 26 T^{2} + 8719 T^{4} + 446932 T^{6} + 8719 p^{2} T^{8} - 26 p^{4} T^{10} + p^{6} T^{12} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.051970160515102683253813684499, −7.35766252171978515182272696079, −7.30837504742868795446581709238, −7.14629385346476907324518962572, −6.93928565868959444837122682974, −6.57151319130391681096079613420, −6.35057381328158010672196517790, −6.29451319876752336862472580399, −6.15715531072572172642212975156, −6.11461818596453356629325934372, −5.74893100189629764222583720633, −5.35710948013680642658277880643, −5.35059424126018785705391587969, −4.66538171949640035486101255138, −4.65243554777495446773253593720, −4.25559555092394245803359709341, −4.05511280869542593063179756039, −3.83329502351719482017700569137, −3.76884378217202253573176078288, −2.89337844562107903556268616422, −2.84094034813626633848091687740, −2.81996533810922616835865143105, −1.82985493990099968121407889957, −1.65761090081185543965036771449, −1.51994669840817750292699618157, 1.51994669840817750292699618157, 1.65761090081185543965036771449, 1.82985493990099968121407889957, 2.81996533810922616835865143105, 2.84094034813626633848091687740, 2.89337844562107903556268616422, 3.76884378217202253573176078288, 3.83329502351719482017700569137, 4.05511280869542593063179756039, 4.25559555092394245803359709341, 4.65243554777495446773253593720, 4.66538171949640035486101255138, 5.35059424126018785705391587969, 5.35710948013680642658277880643, 5.74893100189629764222583720633, 6.11461818596453356629325934372, 6.15715531072572172642212975156, 6.29451319876752336862472580399, 6.35057381328158010672196517790, 6.57151319130391681096079613420, 6.93928565868959444837122682974, 7.14629385346476907324518962572, 7.30837504742868795446581709238, 7.35766252171978515182272696079, 8.051970160515102683253813684499

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

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