Properties

Label 12-1323e6-1.1-c3e6-0-5
Degree $12$
Conductor $5.362\times 10^{18}$
Sign $1$
Analytic cond. $2.26232\times 10^{11}$
Root an. cond. $8.83513$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $6$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·4-s − 108·13-s + 51·16-s − 198·19-s − 165·25-s + 90·31-s − 402·37-s − 660·43-s + 648·52-s − 1.15e3·61-s − 300·64-s + 924·67-s − 1.26e3·73-s + 1.18e3·76-s − 1.50e3·79-s − 3.31e3·97-s + 990·100-s − 2.79e3·103-s + 1.49e3·109-s − 5.52e3·121-s − 540·124-s + 127-s + 131-s + 137-s + 139-s + 2.41e3·148-s + 149-s + ⋯
L(s)  = 1  − 3/4·4-s − 2.30·13-s + 0.796·16-s − 2.39·19-s − 1.31·25-s + 0.521·31-s − 1.78·37-s − 2.34·43-s + 1.72·52-s − 2.41·61-s − 0.585·64-s + 1.68·67-s − 2.02·73-s + 1.79·76-s − 2.13·79-s − 3.46·97-s + 0.989·100-s − 2.66·103-s + 1.31·109-s − 4.15·121-s − 0.391·124-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 1.33·148-s + 0.000549·149-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{18} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{18} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(3^{18} \cdot 7^{12}\)
Sign: $1$
Analytic conductor: \(2.26232\times 10^{11}\)
Root analytic conductor: \(8.83513\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(6\)
Selberg data: \((12,\ 3^{18} \cdot 7^{12} ,\ ( \ : [3/2]^{6} ),\ 1 )\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + 3 p T^{2} - 15 T^{4} - 3 p^{5} T^{6} - 15 p^{6} T^{8} + 3 p^{13} T^{10} + p^{18} T^{12} \)
5 \( 1 + 33 p T^{2} + 27978 T^{4} + 3045013 T^{6} + 27978 p^{6} T^{8} + 33 p^{13} T^{10} + p^{18} T^{12} \)
11 \( 1 + 5529 T^{2} + 13797714 T^{4} + 21853948785 T^{6} + 13797714 p^{6} T^{8} + 5529 p^{12} T^{10} + p^{18} T^{12} \)
13 \( ( 1 + 54 T + 5907 T^{2} + 189108 T^{3} + 5907 p^{3} T^{4} + 54 p^{6} T^{5} + p^{9} T^{6} )^{2} \)
17 \( 1 + 10830 T^{2} + 97584351 T^{4} + 529226734948 T^{6} + 97584351 p^{6} T^{8} + 10830 p^{12} T^{10} + p^{18} T^{12} \)
19 \( ( 1 + 99 T + 16032 T^{2} + 907695 T^{3} + 16032 p^{3} T^{4} + 99 p^{6} T^{5} + p^{9} T^{6} )^{2} \)
23 \( 1 + 52065 T^{2} + 53811222 p T^{4} + 18288252040473 T^{6} + 53811222 p^{7} T^{8} + 52065 p^{12} T^{10} + p^{18} T^{12} \)
29 \( 1 + 26730 T^{2} + 132319767 T^{4} + 2039984262252 T^{6} + 132319767 p^{6} T^{8} + 26730 p^{12} T^{10} + p^{18} T^{12} \)
31 \( ( 1 - 45 T + 34392 T^{2} - 4357053 T^{3} + 34392 p^{3} T^{4} - 45 p^{6} T^{5} + p^{9} T^{6} )^{2} \)
37 \( ( 1 + 201 T + 108390 T^{2} + 11599089 T^{3} + 108390 p^{3} T^{4} + 201 p^{6} T^{5} + p^{9} T^{6} )^{2} \)
41 \( 1 + 220989 T^{2} + 30272507826 T^{4} + 2478677323994989 T^{6} + 30272507826 p^{6} T^{8} + 220989 p^{12} T^{10} + p^{18} T^{12} \)
43 \( ( 1 + 330 T + 238989 T^{2} + 51655332 T^{3} + 238989 p^{3} T^{4} + 330 p^{6} T^{5} + p^{9} T^{6} )^{2} \)
47 \( 1 + 450822 T^{2} + 98858883759 T^{4} + 12911874676225684 T^{6} + 98858883759 p^{6} T^{8} + 450822 p^{12} T^{10} + p^{18} T^{12} \)
53 \( 1 + 569694 T^{2} + 154045269543 T^{4} + 27174714077345028 T^{6} + 154045269543 p^{6} T^{8} + 569694 p^{12} T^{10} + p^{18} T^{12} \)
59 \( 1 + 421662 T^{2} + 144347486919 T^{4} + 29291640339385348 T^{6} + 144347486919 p^{6} T^{8} + 421662 p^{12} T^{10} + p^{18} T^{12} \)
61 \( ( 1 + 576 T + 628959 T^{2} + 251884800 T^{3} + 628959 p^{3} T^{4} + 576 p^{6} T^{5} + p^{9} T^{6} )^{2} \)
67 \( ( 1 - 462 T + 937365 T^{2} - 273377356 T^{3} + 937365 p^{3} T^{4} - 462 p^{6} T^{5} + p^{9} T^{6} )^{2} \)
71 \( 1 + 1962393 T^{2} + 1667929208394 T^{4} + 782638591804754817 T^{6} + 1667929208394 p^{6} T^{8} + 1962393 p^{12} T^{10} + p^{18} T^{12} \)
73 \( ( 1 + 630 T + 616863 T^{2} + 569158884 T^{3} + 616863 p^{3} T^{4} + 630 p^{6} T^{5} + p^{9} T^{6} )^{2} \)
79 \( ( 1 + 750 T + 1155345 T^{2} + 759728572 T^{3} + 1155345 p^{3} T^{4} + 750 p^{6} T^{5} + p^{9} T^{6} )^{2} \)
83 \( 1 + 149934 T^{2} + 622729918167 T^{4} - 3006517940898140 T^{6} + 622729918167 p^{6} T^{8} + 149934 p^{12} T^{10} + p^{18} T^{12} \)
89 \( 1 + 60645 T^{2} + 358766920770 T^{4} - 148130490720197819 T^{6} + 358766920770 p^{6} T^{8} + 60645 p^{12} T^{10} + p^{18} T^{12} \)
97 \( ( 1 + 1656 T + 2525331 T^{2} + 2109759984 T^{3} + 2525331 p^{3} T^{4} + 1656 p^{6} T^{5} + p^{9} T^{6} )^{2} \)
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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

−5.22178105931158189628059645475, −4.98638985031370177412622639434, −4.84147748973002035578153263653, −4.53657823815292542868469635494, −4.46898132550675239041061387163, −4.41464236453232697061142053021, −4.38383023217453483743719492515, −3.91176817004763677625707941149, −3.79994966979357697062035809427, −3.74404437662575211039729165193, −3.64848649802183551090742946675, −3.52630624393083522607441266943, −3.10687650502353439544300625072, −2.81380658064743074155440008777, −2.79263034457667444553670450622, −2.75855447211683108367669926884, −2.49788349514918925902341456587, −2.19263456103118420789672703579, −2.16421325523284409120256249426, −1.89511308316023603413653525526, −1.73685439566175562739123496783, −1.34102181639747164538667425451, −1.28865727498584934176123859257, −1.07128461777247974476458954715, −1.05807427934557369498020555089, 0, 0, 0, 0, 0, 0, 1.05807427934557369498020555089, 1.07128461777247974476458954715, 1.28865727498584934176123859257, 1.34102181639747164538667425451, 1.73685439566175562739123496783, 1.89511308316023603413653525526, 2.16421325523284409120256249426, 2.19263456103118420789672703579, 2.49788349514918925902341456587, 2.75855447211683108367669926884, 2.79263034457667444553670450622, 2.81380658064743074155440008777, 3.10687650502353439544300625072, 3.52630624393083522607441266943, 3.64848649802183551090742946675, 3.74404437662575211039729165193, 3.79994966979357697062035809427, 3.91176817004763677625707941149, 4.38383023217453483743719492515, 4.41464236453232697061142053021, 4.46898132550675239041061387163, 4.53657823815292542868469635494, 4.84147748973002035578153263653, 4.98638985031370177412622639434, 5.22178105931158189628059645475

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

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