Properties

Label 12-273e6-1.1-c1e6-0-4
Degree $12$
Conductor $4.140\times 10^{14}$
Sign $1$
Analytic cond. $107.309$
Root an. cond. $1.47645$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 3·3-s + 3·4-s + 6·6-s + 3·7-s + 4·8-s + 3·9-s + 8·11-s + 9·12-s + 6·14-s + 8·16-s − 4·17-s + 6·18-s + 7·19-s + 9·21-s + 16·22-s − 9·23-s + 12·24-s − 4·25-s − 2·27-s + 9·28-s + 7·29-s − 14·31-s + 11·32-s + 24·33-s − 8·34-s + 9·36-s + ⋯
L(s)  = 1  + 1.41·2-s + 1.73·3-s + 3/2·4-s + 2.44·6-s + 1.13·7-s + 1.41·8-s + 9-s + 2.41·11-s + 2.59·12-s + 1.60·14-s + 2·16-s − 0.970·17-s + 1.41·18-s + 1.60·19-s + 1.96·21-s + 3.41·22-s − 1.87·23-s + 2.44·24-s − 4/5·25-s − 0.384·27-s + 1.70·28-s + 1.29·29-s − 2.51·31-s + 1.94·32-s + 4.17·33-s − 1.37·34-s + 3/2·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 7^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 7^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(3^{6} \cdot 7^{6} \cdot 13^{6}\)
Sign: $1$
Analytic conductor: \(107.309\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{273} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 3^{6} \cdot 7^{6} \cdot 13^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(15.48511716\)
\(L(\frac12)\) \(\approx\) \(15.48511716\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( ( 1 - T + T^{2} )^{3} \)
7 \( ( 1 - T + T^{2} )^{3} \)
13 \( 1 + 5 p T^{3} + p^{3} T^{6} \)
good2 \( 1 - p T + T^{2} - 3 T^{4} + 7 T^{5} - 9 T^{6} + 7 p T^{7} - 3 p^{2} T^{8} + p^{4} T^{10} - p^{6} T^{11} + p^{6} T^{12} \)
5 \( ( 1 + 2 T^{2} + 13 T^{3} + 2 p T^{4} + p^{3} T^{6} )^{2} \)
11 \( 1 - 8 T + 14 T^{2} - 38 T^{3} + 502 T^{4} - 1218 T^{5} + 179 T^{6} - 1218 p T^{7} + 502 p^{2} T^{8} - 38 p^{3} T^{9} + 14 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 + 4 T - 36 T^{2} - 62 T^{3} + 1280 T^{4} + 1104 T^{5} - 22541 T^{6} + 1104 p T^{7} + 1280 p^{2} T^{8} - 62 p^{3} T^{9} - 36 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 - 7 T - 7 T^{2} + 46 T^{3} + 539 T^{4} + 105 T^{5} - 15594 T^{6} + 105 p T^{7} + 539 p^{2} T^{8} + 46 p^{3} T^{9} - 7 p^{4} T^{10} - 7 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 9 T - 2 T^{2} - 83 T^{3} + 1245 T^{4} + 4844 T^{5} - 4737 T^{6} + 4844 p T^{7} + 1245 p^{2} T^{8} - 83 p^{3} T^{9} - 2 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 - 7 T - 24 T^{2} + 311 T^{3} + 335 T^{4} - 6252 T^{5} + 21949 T^{6} - 6252 p T^{7} + 335 p^{2} T^{8} + 311 p^{3} T^{9} - 24 p^{4} T^{10} - 7 p^{5} T^{11} + p^{6} T^{12} \)
31 \( ( 1 + 7 T + 53 T^{2} + 153 T^{3} + 53 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
37 \( 1 - 98 T^{2} - 26 T^{3} + 5978 T^{4} + 1274 T^{5} - 253873 T^{6} + 1274 p T^{7} + 5978 p^{2} T^{8} - 26 p^{3} T^{9} - 98 p^{4} T^{10} + p^{6} T^{12} \)
41 \( 1 + 2 T - 51 T^{2} + 182 T^{3} + 842 T^{4} - 7638 T^{5} + 8389 T^{6} - 7638 p T^{7} + 842 p^{2} T^{8} + 182 p^{3} T^{9} - 51 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 - 19 T + 116 T^{2} - 929 T^{3} + 14093 T^{4} - 91236 T^{5} + 381219 T^{6} - 91236 p T^{7} + 14093 p^{2} T^{8} - 929 p^{3} T^{9} + 116 p^{4} T^{10} - 19 p^{5} T^{11} + p^{6} T^{12} \)
47 \( ( 1 + 17 T + 155 T^{2} + 1051 T^{3} + 155 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
53 \( ( 1 - 13 T + 198 T^{2} - 1365 T^{3} + 198 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
59 \( 1 - 3 T - 158 T^{2} + 157 T^{3} + 17049 T^{4} - 6268 T^{5} - 1157781 T^{6} - 6268 p T^{7} + 17049 p^{2} T^{8} + 157 p^{3} T^{9} - 158 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 13 T - 53 T^{2} - 260 T^{3} + 15293 T^{4} + 45487 T^{5} - 706882 T^{6} + 45487 p T^{7} + 15293 p^{2} T^{8} - 260 p^{3} T^{9} - 53 p^{4} T^{10} + 13 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 5 T - 102 T^{2} + 85 T^{3} + 6245 T^{4} - 27720 T^{5} - 515717 T^{6} - 27720 p T^{7} + 6245 p^{2} T^{8} + 85 p^{3} T^{9} - 102 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 + 8 T - 36 T^{2} - 442 T^{3} - 2068 T^{4} - 11172 T^{5} + 68839 T^{6} - 11172 p T^{7} - 2068 p^{2} T^{8} - 442 p^{3} T^{9} - 36 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
73 \( ( 1 + 2 T + 138 T^{2} + 5 p T^{3} + 138 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
79 \( ( 1 + T - 53 T^{2} - 179 T^{3} - 53 p T^{4} + p^{2} T^{5} + p^{3} T^{6} )^{2} \)
83 \( ( 1 - 2 T + 64 T^{2} - 561 T^{3} + 64 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
89 \( 1 - 19 T + 95 T^{2} - 328 T^{3} + 2185 T^{4} + 120555 T^{5} - 2163058 T^{6} + 120555 p T^{7} + 2185 p^{2} T^{8} - 328 p^{3} T^{9} + 95 p^{4} T^{10} - 19 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 + 27 T + 208 T^{2} + 2393 T^{3} + 63305 T^{4} + 566122 T^{5} + 2706977 T^{6} + 566122 p T^{7} + 63305 p^{2} T^{8} + 2393 p^{3} T^{9} + 208 p^{4} T^{10} + 27 p^{5} T^{11} + p^{6} T^{12} \)
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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

−6.71019786842470768018950689351, −6.18865687689488090115080754141, −6.04883629938477222855105806303, −5.90183285358460665847405354335, −5.72154763964279330404249440879, −5.65078213809830347559893574763, −5.52257422488013204477100648040, −5.06811462902136490464263870260, −4.93980089316024543068307165808, −4.74678904748333587912981009639, −4.32797491109275325481141649353, −4.23993664862428089946512540646, −4.19826478399693015389396201591, −3.94604048598894512256666093342, −3.64462008834228693150872169353, −3.52235045553583954417208475145, −3.39658591620786792295248998291, −2.96596235019041213893525120197, −2.80810832438042863005868921978, −2.66517276448428684285057340624, −2.07348242146880424158847942121, −1.87450293777002994741027986699, −1.72022681880131498204131089045, −1.55789646908467410830852969784, −0.970504142698149759165400378650, 0.970504142698149759165400378650, 1.55789646908467410830852969784, 1.72022681880131498204131089045, 1.87450293777002994741027986699, 2.07348242146880424158847942121, 2.66517276448428684285057340624, 2.80810832438042863005868921978, 2.96596235019041213893525120197, 3.39658591620786792295248998291, 3.52235045553583954417208475145, 3.64462008834228693150872169353, 3.94604048598894512256666093342, 4.19826478399693015389396201591, 4.23993664862428089946512540646, 4.32797491109275325481141649353, 4.74678904748333587912981009639, 4.93980089316024543068307165808, 5.06811462902136490464263870260, 5.52257422488013204477100648040, 5.65078213809830347559893574763, 5.72154763964279330404249440879, 5.90183285358460665847405354335, 6.04883629938477222855105806303, 6.18865687689488090115080754141, 6.71019786842470768018950689351

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

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