↑

Properties

Label 12-13e12-1.1-c1e6-0-2
Degree $12$
Conductor $2.330\times 10^{13}$
Sign $1$
Analytic cond. $6.03924$
Root an. cond. $1.16166$
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  + 2·2-s + 2·3-s + 5·4-s − 8·5-s + 4·6-s + 3·7-s + 8·8-s + 8·9-s − 16·10-s + 8·11-s + 10·12-s + 6·14-s − 16·15-s + 12·16-s + 2·17-s + 16·18-s + 4·19-s − 40·20-s + 6·21-s + 16·22-s + 5·23-s + 16·24-s + 12·25-s + 18·27-s + 15·28-s + 29-s − 32·30-s + ⋯
L(s)  = 1  + 1.41·2-s + 1.15·3-s + 5/2·4-s − 3.57·5-s + 1.63·6-s + 1.13·7-s + 2.82·8-s + 8/3·9-s − 5.05·10-s + 2.41·11-s + 2.88·12-s + 1.60·14-s − 4.13·15-s + 3·16-s + 0.485·17-s + 3.77·18-s + 0.917·19-s − 8.94·20-s + 1.30·21-s + 3.41·22-s + 1.04·23-s + 3.26·24-s + 12/5·25-s + 3.46·27-s + 2.83·28-s + 0.185·29-s − 5.84·30-s + ⋯

Functional equation

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

Invariants

Degree: \(12\)
Conductor: \(13^{12}\)
Sign: $1$
Analytic conductor: \(6.03924\)
Root analytic conductor: \(1.16166\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 13^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(5.966378967\)
\(L(\frac12)\) \(\approx\) \(5.966378967\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
good2 \( 1 - p T - T^{2} + p^{2} T^{3} + T^{4} - 3 T^{5} - T^{6} - 3 p T^{7} + p^{2} T^{8} + p^{5} T^{9} - p^{4} T^{10} - p^{6} T^{11} + p^{6} T^{12} \)
3 \( 1 - 2 T - 4 T^{2} + 2 p T^{3} + 2 p^{2} T^{4} - 10 T^{5} - 53 T^{6} - 10 p T^{7} + 2 p^{4} T^{8} + 2 p^{4} T^{9} - 4 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
5 \( ( 1 + 4 T + 18 T^{2} + 39 T^{3} + 18 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
7 \( 1 - 3 T - 8 T^{2} + p T^{3} + 83 T^{4} + 74 T^{5} - 937 T^{6} + 74 p T^{7} + 83 p^{2} T^{8} + p^{4} T^{9} - 8 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 - 8 T + 12 T^{2} - 38 T^{3} + 620 T^{4} - 1644 T^{5} + 895 T^{6} - 1644 p T^{7} + 620 p^{2} T^{8} - 38 p^{3} T^{9} + 12 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 - 2 T - 32 T^{2} + 90 T^{3} + 522 T^{4} - 1046 T^{5} - 6801 T^{6} - 1046 p T^{7} + 522 p^{2} T^{8} + 90 p^{3} T^{9} - 32 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 - 4 T - 30 T^{2} + 118 T^{3} + 638 T^{4} - 1398 T^{5} - 9461 T^{6} - 1398 p T^{7} + 638 p^{2} T^{8} + 118 p^{3} T^{9} - 30 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 - 5 T - 43 T^{2} + 94 T^{3} + 1975 T^{4} - 1761 T^{5} - 48154 T^{6} - 1761 p T^{7} + 1975 p^{2} T^{8} + 94 p^{3} T^{9} - 43 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 - T - 42 T^{2} + 239 T^{3} + 461 T^{4} - 4410 T^{5} + 14197 T^{6} - 4410 p T^{7} + 461 p^{2} T^{8} + 239 p^{3} T^{9} - 42 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
31 \( ( 1 + 5 T + 57 T^{2} + 143 T^{3} + 57 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
37 \( 1 + 12 T - 8 T^{2} - 10 T^{3} + 6476 T^{4} + 20836 T^{5} - 109789 T^{6} + 20836 p T^{7} + 6476 p^{2} T^{8} - 10 p^{3} T^{9} - 8 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 - 7 T - 25 T^{2} + 728 T^{3} - 1919 T^{4} - 15393 T^{5} + 219086 T^{6} - 15393 p T^{7} - 1919 p^{2} T^{8} + 728 p^{3} T^{9} - 25 p^{4} T^{10} - 7 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 + 13 T - 13 T^{3} + 6929 T^{4} + 21840 T^{5} - 160565 T^{6} + 21840 p T^{7} + 6929 p^{2} T^{8} - 13 p^{3} T^{9} + 13 p^{5} T^{11} + p^{6} T^{12} \)
47 \( ( 1 + 18 T + 242 T^{2} + 1859 T^{3} + 242 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
53 \( ( 1 - T + 73 T^{2} + 231 T^{3} + 73 p T^{4} - p^{2} T^{5} + p^{3} T^{6} )^{2} \)
59 \( 1 - 19 T + 101 T^{2} - 454 T^{3} + 9643 T^{4} - 66399 T^{5} + 196766 T^{6} - 66399 p T^{7} + 9643 p^{2} T^{8} - 454 p^{3} T^{9} + 101 p^{4} T^{10} - 19 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 4 T - 100 T^{2} - 34 T^{3} + 5384 T^{4} - 12840 T^{5} - 364389 T^{6} - 12840 p T^{7} + 5384 p^{2} T^{8} - 34 p^{3} T^{9} - 100 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - T - 128 T^{2} + 221 T^{3} + 7823 T^{4} - 9778 T^{5} - 487285 T^{6} - 9778 p T^{7} + 7823 p^{2} T^{8} + 221 p^{3} T^{9} - 128 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 - 27 T + 294 T^{2} - 2983 T^{3} + 40053 T^{4} - 374052 T^{5} + 2799119 T^{6} - 374052 p T^{7} + 40053 p^{2} T^{8} - 2983 p^{3} T^{9} + 294 p^{4} T^{10} - 27 p^{5} T^{11} + p^{6} T^{12} \)
73 \( ( 1 - 9 T + 99 T^{2} - 403 T^{3} + 99 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
79 \( ( 1 + 5 T + 75 T^{2} + 917 T^{3} + 75 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
83 \( ( 1 + 7 T + 109 T^{2} + 1365 T^{3} + 109 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
89 \( 1 - 11 T - 72 T^{2} + 1231 T^{3} + 1625 T^{4} - 32898 T^{5} - 51335 T^{6} - 32898 p T^{7} + 1625 p^{2} T^{8} + 1231 p^{3} T^{9} - 72 p^{4} T^{10} - 11 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 + 7 T - 158 T^{2} - 665 T^{3} + 15371 T^{4} + 3556 T^{5} - 1674799 T^{6} + 3556 p T^{7} + 15371 p^{2} T^{8} - 665 p^{3} T^{9} - 158 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} \)
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

−7.20795336139019911466617046865, −6.90642977775259276233257201906, −6.76071270683026384646641741446, −6.75766010586965231820362229274, −6.54325330090846404155250528903, −6.35887132005009761752043214444, −5.68057615919087888106374482128, −5.63052577988177386537391055901, −5.58180037649658538309998755809, −4.93754204437224256221149326810, −4.92524254680606482944363140589, −4.74425255097343803165234389124, −4.51962889787279237510927532994, −4.10040853589825648168981381951, −4.02043471144667868502011658411, −3.81035473216552247170653363370, −3.70093089774374010984788368137, −3.38547592000766681510887012689, −3.37132537688323264529287661625, −3.37016327596816426240421462671, −2.53560482098662809713636931378, −2.03022052234198399630783108219, −1.92326429849570522147302022287, −1.52689883213530529365582765022, −1.22231433313255160634233390875, 1.22231433313255160634233390875, 1.52689883213530529365582765022, 1.92326429849570522147302022287, 2.03022052234198399630783108219, 2.53560482098662809713636931378, 3.37016327596816426240421462671, 3.37132537688323264529287661625, 3.38547592000766681510887012689, 3.70093089774374010984788368137, 3.81035473216552247170653363370, 4.02043471144667868502011658411, 4.10040853589825648168981381951, 4.51962889787279237510927532994, 4.74425255097343803165234389124, 4.92524254680606482944363140589, 4.93754204437224256221149326810, 5.58180037649658538309998755809, 5.63052577988177386537391055901, 5.68057615919087888106374482128, 6.35887132005009761752043214444, 6.54325330090846404155250528903, 6.75766010586965231820362229274, 6.76071270683026384646641741446, 6.90642977775259276233257201906, 7.20795336139019911466617046865

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

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