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Properties

Label 12-1875e6-1.1-c1e6-0-1
Degree $12$
Conductor $4.345\times 10^{19}$
Sign $1$
Analytic cond. $1.12634\times 10^{7}$
Root an. cond. $3.86936$
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-s + 6·3-s − 6·6-s + 2·7-s − 8-s + 21·9-s − 2·14-s − 2·17-s − 21·18-s − 2·19-s + 12·21-s + 23-s − 6·24-s + 56·27-s + 31·29-s − 2·31-s − 3·32-s + 2·34-s + 22·37-s + 2·38-s + 33·41-s − 12·42-s + 3·43-s − 46-s + 6·47-s − 17·49-s − 12·51-s + ⋯
L(s)  = 1  − 0.707·2-s + 3.46·3-s − 2.44·6-s + 0.755·7-s − 0.353·8-s + 7·9-s − 0.534·14-s − 0.485·17-s − 4.94·18-s − 0.458·19-s + 2.61·21-s + 0.208·23-s − 1.22·24-s + 10.7·27-s + 5.75·29-s − 0.359·31-s − 0.530·32-s + 0.342·34-s + 3.61·37-s + 0.324·38-s + 5.15·41-s − 1.85·42-s + 0.457·43-s − 0.147·46-s + 0.875·47-s − 2.42·49-s − 1.68·51-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(3^{6} \cdot 5^{24}\)
Sign: $1$
Analytic conductor: \(1.12634\times 10^{7}\)
Root analytic conductor: \(3.86936\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 3^{6} \cdot 5^{24} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(46.38719411\)
\(L(\frac12)\) \(\approx\) \(46.38719411\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( ( 1 - T )^{6} \)
5 \( 1 \)
good2 \( 1 + T + T^{2} + p T^{3} + 3 T^{4} + 7 T^{5} + 11 T^{6} + 7 p T^{7} + 3 p^{2} T^{8} + p^{4} T^{9} + p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
7 \( 1 - 2 T + 3 p T^{2} - 4 p T^{3} + 248 T^{4} - 258 T^{5} + 2000 T^{6} - 258 p T^{7} + 248 p^{2} T^{8} - 4 p^{4} T^{9} + 3 p^{5} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 + 37 T^{2} - 8 T^{3} + 723 T^{4} - 168 T^{5} + 9470 T^{6} - 168 p T^{7} + 723 p^{2} T^{8} - 8 p^{3} T^{9} + 37 p^{4} T^{10} + p^{6} T^{12} \)
13 \( 1 + 22 T^{2} - 74 T^{3} + 447 T^{4} - 874 T^{5} + 8929 T^{6} - 874 p T^{7} + 447 p^{2} T^{8} - 74 p^{3} T^{9} + 22 p^{4} T^{10} + p^{6} T^{12} \)
17 \( 1 + 2 T + 47 T^{2} + 90 T^{3} + 75 p T^{4} + 2852 T^{5} + 25434 T^{6} + 2852 p T^{7} + 75 p^{3} T^{8} + 90 p^{3} T^{9} + 47 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 + 2 T + 40 T^{2} + 70 T^{3} + 1201 T^{4} + 1530 T^{5} + 24751 T^{6} + 1530 p T^{7} + 1201 p^{2} T^{8} + 70 p^{3} T^{9} + 40 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 - T + 3 p T^{2} + 29 T^{3} + 2123 T^{4} + 4406 T^{5} + 48270 T^{6} + 4406 p T^{7} + 2123 p^{2} T^{8} + 29 p^{3} T^{9} + 3 p^{5} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 - 31 T + 553 T^{2} - 235 p T^{3} + 63999 T^{4} - 474070 T^{5} + 2837054 T^{6} - 474070 p T^{7} + 63999 p^{2} T^{8} - 235 p^{4} T^{9} + 553 p^{4} T^{10} - 31 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 2 T + 100 T^{2} + 338 T^{3} + 5477 T^{4} + 17234 T^{5} + 210111 T^{6} + 17234 p T^{7} + 5477 p^{2} T^{8} + 338 p^{3} T^{9} + 100 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 - 22 T + 281 T^{2} - 2288 T^{3} + 13168 T^{4} - 55738 T^{5} + 260260 T^{6} - 55738 p T^{7} + 13168 p^{2} T^{8} - 2288 p^{3} T^{9} + 281 p^{4} T^{10} - 22 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 - 33 T + 645 T^{2} - 8957 T^{3} + 96287 T^{4} - 830106 T^{5} + 5865606 T^{6} - 830106 p T^{7} + 96287 p^{2} T^{8} - 8957 p^{3} T^{9} + 645 p^{4} T^{10} - 33 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 - 3 T + 182 T^{2} - 624 T^{3} + 15607 T^{4} - 52904 T^{5} + 826891 T^{6} - 52904 p T^{7} + 15607 p^{2} T^{8} - 624 p^{3} T^{9} + 182 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 - 6 T + 98 T^{2} - 6 p T^{3} + 5007 T^{4} - 26388 T^{5} + 325404 T^{6} - 26388 p T^{7} + 5007 p^{2} T^{8} - 6 p^{4} T^{9} + 98 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 14 T + 154 T^{2} + 654 T^{3} + 823 T^{4} - 55684 T^{5} - 465780 T^{6} - 55684 p T^{7} + 823 p^{2} T^{8} + 654 p^{3} T^{9} + 154 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 + 8 T + 285 T^{2} + 1760 T^{3} + 36611 T^{4} + 179960 T^{5} + 2743806 T^{6} + 179960 p T^{7} + 36611 p^{2} T^{8} + 1760 p^{3} T^{9} + 285 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 34 T + 772 T^{2} - 12098 T^{3} + 152713 T^{4} - 1545558 T^{5} + 13267723 T^{6} - 1545558 p T^{7} + 152713 p^{2} T^{8} - 12098 p^{3} T^{9} + 772 p^{4} T^{10} - 34 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 2 T + 292 T^{2} + 130 T^{3} + 37645 T^{4} - 17358 T^{5} + 3024439 T^{6} - 17358 p T^{7} + 37645 p^{2} T^{8} + 130 p^{3} T^{9} + 292 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 + 3 T + 201 T^{2} + 1225 T^{3} + 17495 T^{4} + 189438 T^{5} + 1160814 T^{6} + 189438 p T^{7} + 17495 p^{2} T^{8} + 1225 p^{3} T^{9} + 201 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 - 36 T + 869 T^{2} - 14736 T^{3} + 202708 T^{4} - 2250494 T^{5} + 21131980 T^{6} - 2250494 p T^{7} + 202708 p^{2} T^{8} - 14736 p^{3} T^{9} + 869 p^{4} T^{10} - 36 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 - 25 T + 624 T^{2} - 120 p T^{3} + 137585 T^{4} - 1471760 T^{5} + 14938465 T^{6} - 1471760 p T^{7} + 137585 p^{2} T^{8} - 120 p^{4} T^{9} + 624 p^{4} T^{10} - 25 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 - 12 T + 369 T^{2} - 3356 T^{3} + 61283 T^{4} - 441504 T^{5} + 6208854 T^{6} - 441504 p T^{7} + 61283 p^{2} T^{8} - 3356 p^{3} T^{9} + 369 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 - 18 T + 315 T^{2} - 3670 T^{3} + 38351 T^{4} - 311400 T^{5} + 3203706 T^{6} - 311400 p T^{7} + 38351 p^{2} T^{8} - 3670 p^{3} T^{9} + 315 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 + 7 T + 272 T^{2} + 1810 T^{3} + 40125 T^{4} + 194182 T^{5} + 4458809 T^{6} + 194182 p T^{7} + 40125 p^{2} T^{8} + 1810 p^{3} T^{9} + 272 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} \)
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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

−5.00183291225306081581359057523, −4.41580639633890287123486468101, −4.33518226542432320239393330600, −4.26370850621327521774557595857, −4.12942932164495422157477021775, −4.10061548602113697094931449378, −4.04170950207012769633248817455, −3.51141197785353861299989016366, −3.49618856125692233369878862990, −3.49478236203762653783807190131, −3.18720216355073832677997037937, −2.75826331655926814995966437831, −2.75513045792017422822301439607, −2.72658678122537410509583383682, −2.54778183378965958878234281694, −2.47512843354189512259585193224, −2.34034839444469079098821499494, −2.14156886220949796045974786182, −1.86538005554266374911877924120, −1.63739762862154629372277968437, −1.35194372062325440677981058791, −0.999439935016653283758086051659, −0.858031163400899411759154944994, −0.806568428686751634628426558246, −0.62297888921708691071820629333, 0.62297888921708691071820629333, 0.806568428686751634628426558246, 0.858031163400899411759154944994, 0.999439935016653283758086051659, 1.35194372062325440677981058791, 1.63739762862154629372277968437, 1.86538005554266374911877924120, 2.14156886220949796045974786182, 2.34034839444469079098821499494, 2.47512843354189512259585193224, 2.54778183378965958878234281694, 2.72658678122537410509583383682, 2.75513045792017422822301439607, 2.75826331655926814995966437831, 3.18720216355073832677997037937, 3.49478236203762653783807190131, 3.49618856125692233369878862990, 3.51141197785353861299989016366, 4.04170950207012769633248817455, 4.10061548602113697094931449378, 4.12942932164495422157477021775, 4.26370850621327521774557595857, 4.33518226542432320239393330600, 4.41580639633890287123486468101, 5.00183291225306081581359057523

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

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