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Properties

Label 12-304e6-1.1-c1e6-0-5
Degree $12$
Conductor $7.893\times 10^{14}$
Sign $1$
Analytic cond. $204.599$
Root an. cond. $1.55802$
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  + 6·3-s + 6·5-s − 3·7-s + 21·9-s + 3·11-s + 6·13-s + 36·15-s + 6·17-s − 18·21-s + 12·23-s + 21·25-s + 55·27-s − 6·29-s − 3·31-s + 18·33-s − 18·35-s − 12·37-s + 36·39-s − 6·41-s − 12·43-s + 126·45-s − 6·47-s + 12·49-s + 36·51-s − 30·53-s + 18·55-s − 30·59-s + ⋯
L(s)  = 1  + 3.46·3-s + 2.68·5-s − 1.13·7-s + 7·9-s + 0.904·11-s + 1.66·13-s + 9.29·15-s + 1.45·17-s − 3.92·21-s + 2.50·23-s + 21/5·25-s + 10.5·27-s − 1.11·29-s − 0.538·31-s + 3.13·33-s − 3.04·35-s − 1.97·37-s + 5.76·39-s − 0.937·41-s − 1.82·43-s + 18.7·45-s − 0.875·47-s + 12/7·49-s + 5.04·51-s − 4.12·53-s + 2.42·55-s − 3.90·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{24} \cdot 19^{6}\)
Sign: $1$
Analytic conductor: \(204.599\)
Root analytic conductor: \(1.55802\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{24} \cdot 19^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(20.19250632\)
\(L(\frac12)\) \(\approx\) \(20.19250632\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 + 9 T^{2} + 64 T^{3} + 9 p T^{4} + p^{3} T^{6} \)
good3 \( 1 - 2 p T + 5 p T^{2} - 19 T^{3} + p T^{4} + 17 p T^{5} - 134 T^{6} + 17 p^{2} T^{7} + p^{3} T^{8} - 19 p^{3} T^{9} + 5 p^{5} T^{10} - 2 p^{6} T^{11} + p^{6} T^{12} \)
5 \( 1 - 6 T + 3 p T^{2} - 19 T^{3} - 3 T^{4} + 117 T^{5} - 394 T^{6} + 117 p T^{7} - 3 p^{2} T^{8} - 19 p^{3} T^{9} + 3 p^{5} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
7 \( 1 + 3 T - 3 T^{2} - 6 p T^{3} - 57 T^{4} + 111 T^{5} + 758 T^{6} + 111 p T^{7} - 57 p^{2} T^{8} - 6 p^{4} T^{9} - 3 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 - 3 T - 15 T^{2} + 54 T^{3} + 123 T^{4} - 291 T^{5} - 794 T^{6} - 291 p T^{7} + 123 p^{2} T^{8} + 54 p^{3} T^{9} - 15 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 - 6 T + 15 T^{2} - 19 T^{3} - 27 T^{4} + 621 T^{5} - 3834 T^{6} + 621 p T^{7} - 27 p^{2} T^{8} - 19 p^{3} T^{9} + 15 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 - 6 T + 15 T^{2} - 19 T^{3} - 39 T^{4} + 1017 T^{5} - 7666 T^{6} + 1017 p T^{7} - 39 p^{2} T^{8} - 19 p^{3} T^{9} + 15 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 - 12 T + 99 T^{2} - 675 T^{3} + 4455 T^{4} - 24051 T^{5} + 121258 T^{6} - 24051 p T^{7} + 4455 p^{2} T^{8} - 675 p^{3} T^{9} + 99 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + 6 T + 111 T^{2} + 641 T^{3} + 6501 T^{4} + 31149 T^{5} + 238982 T^{6} + 31149 p T^{7} + 6501 p^{2} T^{8} + 641 p^{3} T^{9} + 111 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 3 T - 51 T^{2} + 22 T^{3} + 1503 T^{4} - 3537 T^{5} - 54426 T^{6} - 3537 p T^{7} + 1503 p^{2} T^{8} + 22 p^{3} T^{9} - 51 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
37 \( ( 1 + 6 T + 75 T^{2} + 292 T^{3} + 75 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
41 \( 1 + 6 T - 9 T^{2} + 9 T^{3} - 1647 T^{4} - 10407 T^{5} + 28270 T^{6} - 10407 p T^{7} - 1647 p^{2} T^{8} + 9 p^{3} T^{9} - 9 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 + 12 T + 39 T^{2} - 257 T^{3} - 2061 T^{4} + 10323 T^{5} + 155562 T^{6} + 10323 p T^{7} - 2061 p^{2} T^{8} - 257 p^{3} T^{9} + 39 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 + 6 T + 9 T^{2} - 225 T^{3} - 1017 T^{4} + 18069 T^{5} + 214174 T^{6} + 18069 p T^{7} - 1017 p^{2} T^{8} - 225 p^{3} T^{9} + 9 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 30 T + 543 T^{2} + 7297 T^{3} + 80301 T^{4} + 739485 T^{5} + 5819702 T^{6} + 739485 p T^{7} + 80301 p^{2} T^{8} + 7297 p^{3} T^{9} + 543 p^{4} T^{10} + 30 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 + 30 T + 345 T^{2} + 1429 T^{3} - 7485 T^{4} - 150705 T^{5} - 1342138 T^{6} - 150705 p T^{7} - 7485 p^{2} T^{8} + 1429 p^{3} T^{9} + 345 p^{4} T^{10} + 30 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 6 T + 3 T^{2} + 229 T^{3} - 1323 T^{4} + 22437 T^{5} - 58818 T^{6} + 22437 p T^{7} - 1323 p^{2} T^{8} + 229 p^{3} T^{9} + 3 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 18 T + 189 T^{2} + 2151 T^{3} + 17523 T^{4} + 135045 T^{5} + 1262942 T^{6} + 135045 p T^{7} + 17523 p^{2} T^{8} + 2151 p^{3} T^{9} + 189 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 + 36 T + 711 T^{2} + 10179 T^{3} + 117063 T^{4} + 1149867 T^{5} + 10108450 T^{6} + 1149867 p T^{7} + 117063 p^{2} T^{8} + 10179 p^{3} T^{9} + 711 p^{4} T^{10} + 36 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 - 6 T - 153 T^{2} + 561 T^{3} + 12969 T^{4} - 20067 T^{5} - 836002 T^{6} - 20067 p T^{7} + 12969 p^{2} T^{8} + 561 p^{3} T^{9} - 153 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 18 T + 207 T^{2} + 2761 T^{3} + 26055 T^{4} + 254043 T^{5} + 2717202 T^{6} + 254043 p T^{7} + 26055 p^{2} T^{8} + 2761 p^{3} T^{9} + 207 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 + 15 T + 153 T^{2} + 1106 T^{3} - 597 T^{4} - 76881 T^{5} - 836554 T^{6} - 76881 p T^{7} - 597 p^{2} T^{8} + 1106 p^{3} T^{9} + 153 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 + 18 T + 135 T^{2} + 513 T^{3} - 1431 T^{4} - 92187 T^{5} - 1410722 T^{6} - 92187 p T^{7} - 1431 p^{2} T^{8} + 513 p^{3} T^{9} + 135 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 30 T + 375 T^{2} - 2375 T^{3} + 225 T^{4} + 240885 T^{5} - 3559698 T^{6} + 240885 p T^{7} + 225 p^{2} T^{8} - 2375 p^{3} T^{9} + 375 p^{4} T^{10} - 30 p^{5} T^{11} + p^{6} T^{12} \)
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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

−6.82537473069416461545284177355, −6.04400807128333806241180227752, −5.87940460290630132760276483622, −5.84988780397646688405440581318, −5.74906788041497234876407478615, −5.66609997652178681822075172657, −5.63069881013247044758825668803, −4.73049670422462457205255306697, −4.64027863500454777428445074893, −4.63425334117247911339896243327, −4.63205214210317137697661452602, −4.41347482171495054831622694887, −3.78553708729769592232839687256, −3.34011484307727595897162006318, −3.28632411852099735555284537665, −3.28403664517903785828647713578, −3.20835474087008706022776194123, −3.19733476705968351236261183651, −2.87397163399326601153781866814, −2.39748348811973948071794164679, −1.98063053887870594816886670629, −1.66699722584734515254755754282, −1.65918249475121077308990668986, −1.48114941818230087355862444908, −1.27028711462027867839972661674, 1.27028711462027867839972661674, 1.48114941818230087355862444908, 1.65918249475121077308990668986, 1.66699722584734515254755754282, 1.98063053887870594816886670629, 2.39748348811973948071794164679, 2.87397163399326601153781866814, 3.19733476705968351236261183651, 3.20835474087008706022776194123, 3.28403664517903785828647713578, 3.28632411852099735555284537665, 3.34011484307727595897162006318, 3.78553708729769592232839687256, 4.41347482171495054831622694887, 4.63205214210317137697661452602, 4.63425334117247911339896243327, 4.64027863500454777428445074893, 4.73049670422462457205255306697, 5.63069881013247044758825668803, 5.66609997652178681822075172657, 5.74906788041497234876407478615, 5.84988780397646688405440581318, 5.87940460290630132760276483622, 6.04400807128333806241180227752, 6.82537473069416461545284177355

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

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