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Properties

Label 12-1690e6-1.1-c1e6-0-0
Degree $12$
Conductor $2.330\times 10^{19}$
Sign $1$
Analytic cond. $6.03924\times 10^{6}$
Root an. cond. $3.67351$
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  + 3·2-s + 3-s + 3·4-s + 6·5-s + 3·6-s + 2·7-s − 2·8-s + 7·9-s + 18·10-s − 13·11-s + 3·12-s + 6·14-s + 6·15-s − 9·16-s + 17-s + 21·18-s − 3·19-s + 18·20-s + 2·21-s − 39·22-s + 10·23-s − 2·24-s + 21·25-s + 10·27-s + 6·28-s + 8·29-s + 18·30-s + ⋯
L(s)  = 1  + 2.12·2-s + 0.577·3-s + 3/2·4-s + 2.68·5-s + 1.22·6-s + 0.755·7-s − 0.707·8-s + 7/3·9-s + 5.69·10-s − 3.91·11-s + 0.866·12-s + 1.60·14-s + 1.54·15-s − 9/4·16-s + 0.242·17-s + 4.94·18-s − 0.688·19-s + 4.02·20-s + 0.436·21-s − 8.31·22-s + 2.08·23-s − 0.408·24-s + 21/5·25-s + 1.92·27-s + 1.13·28-s + 1.48·29-s + 3.28·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{6} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{6} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{6} \cdot 5^{6} \cdot 13^{12}\)
Sign: $1$
Analytic conductor: \(6.03924\times 10^{6}\)
Root analytic conductor: \(3.67351\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{6} \cdot 5^{6} \cdot 13^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.680895690\)
\(L(\frac12)\) \(\approx\) \(1.680895690\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 - T + T^{2} )^{3} \)
5 \( ( 1 - T )^{6} \)
13 \( 1 \)
good3 \( 1 - T - 2 p T^{2} + p T^{3} + 23 T^{4} - 2 T^{5} - 77 T^{6} - 2 p T^{7} + 23 p^{2} T^{8} + p^{4} T^{9} - 2 p^{5} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
7 \( 1 - 2 T - 9 T^{2} + 2 p T^{3} + 38 T^{4} + 6 T^{5} - 293 T^{6} + 6 p T^{7} + 38 p^{2} T^{8} + 2 p^{4} T^{9} - 9 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 + 13 T + 82 T^{2} + 417 T^{3} + 1971 T^{4} + 7750 T^{5} + 26403 T^{6} + 7750 p T^{7} + 1971 p^{2} T^{8} + 417 p^{3} T^{9} + 82 p^{4} T^{10} + 13 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 - T - 2 p T^{2} - 25 T^{3} + 625 T^{4} + 726 T^{5} - 11263 T^{6} + 726 p T^{7} + 625 p^{2} T^{8} - 25 p^{3} T^{9} - 2 p^{5} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 + 3 T - 44 T^{2} - 71 T^{3} + 1457 T^{4} + 1136 T^{5} - 29677 T^{6} + 1136 p T^{7} + 1457 p^{2} T^{8} - 71 p^{3} T^{9} - 44 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 - 10 T + 7 T^{2} + 6 T^{3} + 1830 T^{4} - 6034 T^{5} - 8469 T^{6} - 6034 p T^{7} + 1830 p^{2} T^{8} + 6 p^{3} T^{9} + 7 p^{4} T^{10} - 10 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 - 8 T - 35 T^{2} + 120 T^{3} + 3282 T^{4} - 5936 T^{5} - 81339 T^{6} - 5936 p T^{7} + 3282 p^{2} T^{8} + 120 p^{3} T^{9} - 35 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
31 \( ( 1 - 4 T + 33 T^{2} - 16 T^{3} + 33 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
37 \( 1 + 6 T - 3 T^{2} - 6 p T^{3} - 1290 T^{4} - 210 T^{5} + 47729 T^{6} - 210 p T^{7} - 1290 p^{2} T^{8} - 6 p^{4} T^{9} - 3 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 + 19 T + 148 T^{2} + 705 T^{3} + 4887 T^{4} + 56008 T^{5} + 463113 T^{6} + 56008 p T^{7} + 4887 p^{2} T^{8} + 705 p^{3} T^{9} + 148 p^{4} T^{10} + 19 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 + 5 T + 30 T^{2} + 793 T^{3} + 2285 T^{4} + 13440 T^{5} + 278995 T^{6} + 13440 p T^{7} + 2285 p^{2} T^{8} + 793 p^{3} T^{9} + 30 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} \)
47 \( ( 1 - 8 T + 97 T^{2} - 744 T^{3} + 97 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
53 \( ( 1 + 8 T + 59 T^{2} + 280 T^{3} + 59 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
59 \( 1 + 5 T - 18 T^{2} + 713 T^{3} + 557 T^{4} - 18672 T^{5} + 276643 T^{6} - 18672 p T^{7} + 557 p^{2} T^{8} + 713 p^{3} T^{9} - 18 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 8 T - 19 T^{2} + 152 T^{3} - 1438 T^{4} + 17376 T^{5} - 37947 T^{6} + 17376 p T^{7} - 1438 p^{2} T^{8} + 152 p^{3} T^{9} - 19 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + T - 58 T^{2} - 1327 T^{3} - 1165 T^{4} + 37470 T^{5} + 844059 T^{6} + 37470 p T^{7} - 1165 p^{2} T^{8} - 1327 p^{3} T^{9} - 58 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 + 12 T - 33 T^{2} - 628 T^{3} + 2250 T^{4} + 2436 T^{5} - 341065 T^{6} + 2436 p T^{7} + 2250 p^{2} T^{8} - 628 p^{3} T^{9} - 33 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
73 \( ( 1 - 5 T + 113 T^{2} - 87 T^{3} + 113 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
79 \( ( 1 - 2 T + 173 T^{2} - 420 T^{3} + 173 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
83 \( ( 1 + 7 T + 151 T^{2} + 1365 T^{3} + 151 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
89 \( 1 + 21 T + 76 T^{2} + 287 T^{3} + 23271 T^{4} + 148456 T^{5} - 331527 T^{6} + 148456 p T^{7} + 23271 p^{2} T^{8} + 287 p^{3} T^{9} + 76 p^{4} T^{10} + 21 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 + 21 T + 10 T^{2} + 329 T^{3} + 52373 T^{4} + 321286 T^{5} - 1324519 T^{6} + 321286 p T^{7} + 52373 p^{2} T^{8} + 329 p^{3} T^{9} + 10 p^{4} T^{10} + 21 p^{5} T^{11} + p^{6} T^{12} \)
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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

−4.99167613724954425690088766039, −4.83983106021321248774848443996, −4.64864915230093083313361348155, −4.51833344728968118482015675310, −4.37407964392701833041069803234, −4.28353701986053442137436498619, −4.04399345290508871408758547333, −3.83118129399080676165880781885, −3.58415420516584477792296107710, −3.54468674912149504677373121646, −3.15329884696549264420990599549, −3.01165635072219819559380581911, −2.97199412160979200740752017163, −2.77718338499621801457857523182, −2.58962897212533151391344702221, −2.53972693308954697014872280720, −2.36494956670407776638426101943, −2.23502131414853847159044222706, −1.87953161099560151470385105654, −1.68433454498235745611360785910, −1.39748156609658035905582335494, −1.36217339720612047455829278759, −1.05513254652658292137367092420, −0.831275141172216498488381013328, −0.06462733305825442043415864144, 0.06462733305825442043415864144, 0.831275141172216498488381013328, 1.05513254652658292137367092420, 1.36217339720612047455829278759, 1.39748156609658035905582335494, 1.68433454498235745611360785910, 1.87953161099560151470385105654, 2.23502131414853847159044222706, 2.36494956670407776638426101943, 2.53972693308954697014872280720, 2.58962897212533151391344702221, 2.77718338499621801457857523182, 2.97199412160979200740752017163, 3.01165635072219819559380581911, 3.15329884696549264420990599549, 3.54468674912149504677373121646, 3.58415420516584477792296107710, 3.83118129399080676165880781885, 4.04399345290508871408758547333, 4.28353701986053442137436498619, 4.37407964392701833041069803234, 4.51833344728968118482015675310, 4.64864915230093083313361348155, 4.83983106021321248774848443996, 4.99167613724954425690088766039

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

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