↑

Properties

Label 12-378e6-1.1-c1e6-0-1
Degree $12$
Conductor $2.917\times 10^{15}$
Sign $1$
Analytic cond. $756.159$
Root an. cond. $1.73733$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3·2-s + 3·4-s + 2·5-s − 4·7-s + 2·8-s − 6·10-s − 2·11-s + 8·13-s + 12·14-s − 9·16-s + 4·17-s − 3·19-s + 6·20-s + 6·22-s − 14·23-s − 15·25-s − 24·26-s − 12·28-s + 5·29-s + 20·31-s + 9·32-s − 12·34-s − 8·35-s + 3·37-s + 9·38-s + 4·40-s − 6·43-s + ⋯
L(s)  = 1  − 2.12·2-s + 3/2·4-s + 0.894·5-s − 1.51·7-s + 0.707·8-s − 1.89·10-s − 0.603·11-s + 2.21·13-s + 3.20·14-s − 9/4·16-s + 0.970·17-s − 0.688·19-s + 1.34·20-s + 1.27·22-s − 2.91·23-s − 3·25-s − 4.70·26-s − 2.26·28-s + 0.928·29-s + 3.59·31-s + 1.59·32-s − 2.05·34-s − 1.35·35-s + 0.493·37-s + 1.45·38-s + 0.632·40-s − 0.914·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{18} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{18} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{6} \cdot 3^{18} \cdot 7^{6}\)
Sign: $1$
Analytic conductor: \(756.159\)
Root analytic conductor: \(1.73733\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{6} \cdot 3^{18} \cdot 7^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.5460866819\)
\(L(\frac12)\) \(\approx\) \(0.5460866819\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + T + T^{2} )^{3} \)
3 \( 1 \)
7 \( 1 + 4 T + 2 p T^{2} + 55 T^{3} + 2 p^{2} T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
good5 \( ( 1 - T + 9 T^{2} - 13 T^{3} + 9 p T^{4} - p^{2} T^{5} + p^{3} T^{6} )^{2} \)
11 \( ( 1 + T + 27 T^{2} + 25 T^{3} + 27 p T^{4} + p^{2} T^{5} + p^{3} T^{6} )^{2} \)
13 \( 1 - 8 T + 24 T^{2} - 42 T^{3} - 32 T^{4} + 1408 T^{5} - 7901 T^{6} + 1408 p T^{7} - 32 p^{2} T^{8} - 42 p^{3} T^{9} + 24 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 - 4 T - 23 T^{2} + 4 p T^{3} + 410 T^{4} - 220 T^{5} - 8111 T^{6} - 220 p T^{7} + 410 p^{2} T^{8} + 4 p^{4} T^{9} - 23 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 + 3 T - 12 T^{2} - 67 T^{3} - 153 T^{4} + 54 T^{5} + 6315 T^{6} + 54 p T^{7} - 153 p^{2} T^{8} - 67 p^{3} T^{9} - 12 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
23 \( ( 1 + 7 T + 81 T^{2} + 325 T^{3} + 81 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
29 \( 1 - 5 T + 4 T^{2} - 251 T^{3} + 197 T^{4} + 3418 T^{5} + 20293 T^{6} + 3418 p T^{7} + 197 p^{2} T^{8} - 251 p^{3} T^{9} + 4 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 - 20 T + 6 p T^{2} - 1398 T^{3} + 10342 T^{4} - 62234 T^{5} + 331987 T^{6} - 62234 p T^{7} + 10342 p^{2} T^{8} - 1398 p^{3} T^{9} + 6 p^{5} T^{10} - 20 p^{5} T^{11} + p^{6} T^{12} \)
37 \( ( 1 - 11 T + p T^{2} )^{3}( 1 + 10 T + p T^{2} )^{3} \)
41 \( 1 - 90 T^{2} - 18 T^{3} + 4410 T^{4} + 810 T^{5} - 194177 T^{6} + 810 p T^{7} + 4410 p^{2} T^{8} - 18 p^{3} T^{9} - 90 p^{4} T^{10} + p^{6} T^{12} \)
43 \( ( 1 - 12 T - 6 T^{2} + 547 T^{3} - 6 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} )( 1 + 18 T + 198 T^{2} + 1519 T^{3} + 198 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} ) \)
47 \( 1 - 9 T - 6 T^{2} + 531 T^{3} - 2433 T^{4} - 3438 T^{5} + 104623 T^{6} - 3438 p T^{7} - 2433 p^{2} T^{8} + 531 p^{3} T^{9} - 6 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 15 T + 33 T^{3} + 13635 T^{4} + 60360 T^{5} - 225155 T^{6} + 60360 p T^{7} + 13635 p^{2} T^{8} + 33 p^{3} T^{9} + 15 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 - 14 T - 20 T^{2} + 154 T^{3} + 11666 T^{4} - 35126 T^{5} - 499301 T^{6} - 35126 p T^{7} + 11666 p^{2} T^{8} + 154 p^{3} T^{9} - 20 p^{4} T^{10} - 14 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 8 T - 114 T^{2} + 342 T^{3} + 13762 T^{4} - 13214 T^{5} - 937217 T^{6} - 13214 p T^{7} + 13762 p^{2} T^{8} + 342 p^{3} T^{9} - 114 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - T - 88 T^{2} - 243 T^{3} + 2035 T^{4} + 14290 T^{5} + 72259 T^{6} + 14290 p T^{7} + 2035 p^{2} T^{8} - 243 p^{3} T^{9} - 88 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
71 \( ( 1 + 7 T + 15 T^{2} - 599 T^{3} + 15 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
73 \( 1 - 19 T + 134 T^{2} - 27 T^{3} - 5759 T^{4} + 41986 T^{5} - 314903 T^{6} + 41986 p T^{7} - 5759 p^{2} T^{8} - 27 p^{3} T^{9} + 134 p^{4} T^{10} - 19 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 - 5 T - 138 T^{2} + 123 T^{3} + 11347 T^{4} + 21118 T^{5} - 1048937 T^{6} + 21118 p T^{7} + 11347 p^{2} T^{8} + 123 p^{3} T^{9} - 138 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 + 2 T - 182 T^{2} + 2 T^{3} + 18788 T^{4} - 13564 T^{5} - 1721225 T^{6} - 13564 p T^{7} + 18788 p^{2} T^{8} + 2 p^{3} T^{9} - 182 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 - 9 T - 144 T^{2} + 1197 T^{3} + 16101 T^{4} - 73314 T^{5} - 1141967 T^{6} - 73314 p T^{7} + 16101 p^{2} T^{8} + 1197 p^{3} T^{9} - 144 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 28 T + 281 T^{2} - 2724 T^{3} + 45178 T^{4} - 388196 T^{5} + 2169217 T^{6} - 388196 p T^{7} + 45178 p^{2} T^{8} - 2724 p^{3} T^{9} + 281 p^{4} T^{10} - 28 p^{5} T^{11} + p^{6} T^{12} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−6.33494758858488591759251296799, −6.02304779982039994760018645355, −5.89381929066827786350808604127, −5.67587743052420171466070092200, −5.65171965761350440612036963254, −5.55940084735348107065927736712, −5.06043877812093635130566172818, −4.81664585564264533858711100717, −4.81146958152562143543122396892, −4.35302998245710172781996760259, −4.19174952634560981867959741388, −3.99317401508695538031094859079, −3.86472451637733685311821579134, −3.69721634376078247122664610306, −3.60050624706311344299279063644, −2.99197892063502348043752662188, −2.97002455780179579234501584556, −2.58332120179096718550293936343, −2.55147765920002646137517570120, −1.92734219114311631730865400329, −1.80486715827586510244283793887, −1.72543276160858166446234379667, −1.21296200663693563952143453751, −0.64848224129327926610869568510, −0.47917294819184509109412475082, 0.47917294819184509109412475082, 0.64848224129327926610869568510, 1.21296200663693563952143453751, 1.72543276160858166446234379667, 1.80486715827586510244283793887, 1.92734219114311631730865400329, 2.55147765920002646137517570120, 2.58332120179096718550293936343, 2.97002455780179579234501584556, 2.99197892063502348043752662188, 3.60050624706311344299279063644, 3.69721634376078247122664610306, 3.86472451637733685311821579134, 3.99317401508695538031094859079, 4.19174952634560981867959741388, 4.35302998245710172781996760259, 4.81146958152562143543122396892, 4.81664585564264533858711100717, 5.06043877812093635130566172818, 5.55940084735348107065927736712, 5.65171965761350440612036963254, 5.67587743052420171466070092200, 5.89381929066827786350808604127, 6.02304779982039994760018645355, 6.33494758858488591759251296799

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

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