Properties

Label 12-35e12-1.1-c3e6-0-1
Degree $12$
Conductor $3.379\times 10^{18}$
Sign $1$
Analytic cond. $1.42565\times 10^{11}$
Root an. cond. $8.50160$
Motivic weight $3$
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  + 2·2-s − 16·3-s − 15·4-s − 32·6-s − 16·8-s + 82·9-s − 16·11-s + 240·12-s − 168·13-s + 191·16-s + 4·17-s + 164·18-s + 308·19-s − 32·22-s + 336·23-s + 256·24-s − 336·26-s − 124·27-s + 176·29-s + 392·31-s + 158·32-s + 256·33-s + 8·34-s − 1.23e3·36-s + 140·37-s + 616·38-s + 2.68e3·39-s + ⋯
L(s)  = 1  + 0.707·2-s − 3.07·3-s − 1.87·4-s − 2.17·6-s − 0.707·8-s + 3.03·9-s − 0.438·11-s + 5.77·12-s − 3.58·13-s + 2.98·16-s + 0.0570·17-s + 2.14·18-s + 3.71·19-s − 0.310·22-s + 3.04·23-s + 2.17·24-s − 2.53·26-s − 0.883·27-s + 1.12·29-s + 2.27·31-s + 0.872·32-s + 1.35·33-s + 0.0403·34-s − 5.69·36-s + 0.622·37-s + 2.62·38-s + 11.0·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{12} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{12} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(5^{12} \cdot 7^{12}\)
Sign: $1$
Analytic conductor: \(1.42565\times 10^{11}\)
Root analytic conductor: \(8.50160\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 5^{12} \cdot 7^{12} ,\ ( \ : [3/2]^{6} ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(3.332741391\)
\(L(\frac12)\) \(\approx\) \(3.332741391\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
good2 \( 1 - p T + 19 T^{2} - 13 p^{2} T^{3} + 83 p T^{4} - 73 p^{3} T^{5} + 305 p^{2} T^{6} - 73 p^{6} T^{7} + 83 p^{7} T^{8} - 13 p^{11} T^{9} + 19 p^{12} T^{10} - p^{16} T^{11} + p^{18} T^{12} \)
3 \( 1 + 16 T + 58 p T^{2} + 532 p T^{3} + 11876 T^{4} + 2800 p^{3} T^{5} + 425492 T^{6} + 2800 p^{6} T^{7} + 11876 p^{6} T^{8} + 532 p^{10} T^{9} + 58 p^{13} T^{10} + 16 p^{15} T^{11} + p^{18} T^{12} \)
11 \( 1 + 16 T + 812 T^{2} - 15300 T^{3} + 2567184 T^{4} - 56858680 T^{5} - 552711974 T^{6} - 56858680 p^{3} T^{7} + 2567184 p^{6} T^{8} - 15300 p^{9} T^{9} + 812 p^{12} T^{10} + 16 p^{15} T^{11} + p^{18} T^{12} \)
13 \( 1 + 168 T + 18956 T^{2} + 116156 p T^{3} + 104466904 T^{4} + 6040563440 T^{5} + 309756529486 T^{6} + 6040563440 p^{3} T^{7} + 104466904 p^{6} T^{8} + 116156 p^{10} T^{9} + 18956 p^{12} T^{10} + 168 p^{15} T^{11} + p^{18} T^{12} \)
17 \( 1 - 4 T + 14086 T^{2} + 160336 T^{3} + 115923452 T^{4} + 1228657188 T^{5} + 692428906716 T^{6} + 1228657188 p^{3} T^{7} + 115923452 p^{6} T^{8} + 160336 p^{9} T^{9} + 14086 p^{12} T^{10} - 4 p^{15} T^{11} + p^{18} T^{12} \)
19 \( 1 - 308 T + 73404 T^{2} - 12077372 T^{3} + 1625488783 T^{4} - 176451707096 T^{5} + 16038736148248 T^{6} - 176451707096 p^{3} T^{7} + 1625488783 p^{6} T^{8} - 12077372 p^{9} T^{9} + 73404 p^{12} T^{10} - 308 p^{15} T^{11} + p^{18} T^{12} \)
23 \( 1 - 336 T + 98630 T^{2} - 19185056 T^{3} + 3349928211 T^{4} - 19757853136 p T^{5} + 105705477964 p^{2} T^{6} - 19757853136 p^{4} T^{7} + 3349928211 p^{6} T^{8} - 19185056 p^{9} T^{9} + 98630 p^{12} T^{10} - 336 p^{15} T^{11} + p^{18} T^{12} \)
29 \( 1 - 176 T + 92656 T^{2} - 12546380 T^{3} + 4288608312 T^{4} - 478053343680 T^{5} + 127431913139622 T^{6} - 478053343680 p^{3} T^{7} + 4288608312 p^{6} T^{8} - 12546380 p^{9} T^{9} + 92656 p^{12} T^{10} - 176 p^{15} T^{11} + p^{18} T^{12} \)
31 \( 1 - 392 T + 153030 T^{2} - 32287912 T^{3} + 7677507203 T^{4} - 1249077738464 T^{5} + 255230995151820 T^{6} - 1249077738464 p^{3} T^{7} + 7677507203 p^{6} T^{8} - 32287912 p^{9} T^{9} + 153030 p^{12} T^{10} - 392 p^{15} T^{11} + p^{18} T^{12} \)
37 \( 1 - 140 T + 213942 T^{2} - 12843948 T^{3} + 18729025499 T^{4} - 132721017192 T^{5} + 1059724116139756 T^{6} - 132721017192 p^{3} T^{7} + 18729025499 p^{6} T^{8} - 12843948 p^{9} T^{9} + 213942 p^{12} T^{10} - 140 p^{15} T^{11} + p^{18} T^{12} \)
41 \( 1 - 16 p T + 421976 T^{2} - 166159344 T^{3} + 65879968003 T^{4} - 19715169399936 T^{5} + 5871622216065200 T^{6} - 19715169399936 p^{3} T^{7} + 65879968003 p^{6} T^{8} - 166159344 p^{9} T^{9} + 421976 p^{12} T^{10} - 16 p^{16} T^{11} + p^{18} T^{12} \)
43 \( 1 - 388 T + 421158 T^{2} - 116550300 T^{3} + 73177187783 T^{4} - 15517478343240 T^{5} + 7317237698378164 T^{6} - 15517478343240 p^{3} T^{7} + 73177187783 p^{6} T^{8} - 116550300 p^{9} T^{9} + 421158 p^{12} T^{10} - 388 p^{15} T^{11} + p^{18} T^{12} \)
47 \( 1 + 628 T + 589452 T^{2} + 246437960 T^{3} + 139029029008 T^{4} + 44711775028180 T^{5} + 18650987083950686 T^{6} + 44711775028180 p^{3} T^{7} + 139029029008 p^{6} T^{8} + 246437960 p^{9} T^{9} + 589452 p^{12} T^{10} + 628 p^{15} T^{11} + p^{18} T^{12} \)
53 \( 1 - 676 T + 756806 T^{2} - 367769844 T^{3} + 249160429851 T^{4} - 95287278237352 T^{5} + 47830207170007020 T^{6} - 95287278237352 p^{3} T^{7} + 249160429851 p^{6} T^{8} - 367769844 p^{9} T^{9} + 756806 p^{12} T^{10} - 676 p^{15} T^{11} + p^{18} T^{12} \)
59 \( 1 - 996 T + 1036536 T^{2} - 686090796 T^{3} + 452936806907 T^{4} - 228269279812616 T^{5} + 117358818568794736 T^{6} - 228269279812616 p^{3} T^{7} + 452936806907 p^{6} T^{8} - 686090796 p^{9} T^{9} + 1036536 p^{12} T^{10} - 996 p^{15} T^{11} + p^{18} T^{12} \)
61 \( 1 - 740 T + 970482 T^{2} - 661891780 T^{3} + 495713522631 T^{4} - 257223694985640 T^{5} + 147582202110218748 T^{6} - 257223694985640 p^{3} T^{7} + 495713522631 p^{6} T^{8} - 661891780 p^{9} T^{9} + 970482 p^{12} T^{10} - 740 p^{15} T^{11} + p^{18} T^{12} \)
67 \( 1 + 1768 T + 2799902 T^{2} + 2855569976 T^{3} + 2569585021227 T^{4} + 1777311287531536 T^{5} + 1093536943175084732 T^{6} + 1777311287531536 p^{3} T^{7} + 2569585021227 p^{6} T^{8} + 2855569976 p^{9} T^{9} + 2799902 p^{12} T^{10} + 1768 p^{15} T^{11} + p^{18} T^{12} \)
71 \( 1 + 224 T + 735802 T^{2} - 33746048 T^{3} + 362779099663 T^{4} + 5782399589792 T^{5} + 175695286389376652 T^{6} + 5782399589792 p^{3} T^{7} + 362779099663 p^{6} T^{8} - 33746048 p^{9} T^{9} + 735802 p^{12} T^{10} + 224 p^{15} T^{11} + p^{18} T^{12} \)
73 \( 1 + 2640 T + 4127428 T^{2} + 4281599440 T^{3} + 3477156414455 T^{4} + 2335723050987360 T^{5} + 1487916107494354120 T^{6} + 2335723050987360 p^{3} T^{7} + 3477156414455 p^{6} T^{8} + 4281599440 p^{9} T^{9} + 4127428 p^{12} T^{10} + 2640 p^{15} T^{11} + p^{18} T^{12} \)
79 \( 1 - 1636 T + 3307008 T^{2} - 41764072 p T^{3} + 3796696094800 T^{4} - 2753755692366396 T^{5} + 2363360511013187334 T^{6} - 2753755692366396 p^{3} T^{7} + 3796696094800 p^{6} T^{8} - 41764072 p^{10} T^{9} + 3307008 p^{12} T^{10} - 1636 p^{15} T^{11} + p^{18} T^{12} \)
83 \( 1 + 140 T + 1262996 T^{2} - 466318380 T^{3} + 677854086191 T^{4} - 649780965410440 T^{5} + 346141233175430792 T^{6} - 649780965410440 p^{3} T^{7} + 677854086191 p^{6} T^{8} - 466318380 p^{9} T^{9} + 1262996 p^{12} T^{10} + 140 p^{15} T^{11} + p^{18} T^{12} \)
89 \( 1 + 1904 T + 3824692 T^{2} + 4929406096 T^{3} + 6160110405943 T^{4} + 5994022973389568 T^{5} + 5629689575835156904 T^{6} + 5994022973389568 p^{3} T^{7} + 6160110405943 p^{6} T^{8} + 4929406096 p^{9} T^{9} + 3824692 p^{12} T^{10} + 1904 p^{15} T^{11} + p^{18} T^{12} \)
97 \( 1 + 516 T + 3952326 T^{2} + 925377376 T^{3} + 6580438313612 T^{4} + 355370807466588 T^{5} + 6947174177634861996 T^{6} + 355370807466588 p^{3} T^{7} + 6580438313612 p^{6} T^{8} + 925377376 p^{9} T^{9} + 3952326 p^{12} T^{10} + 516 p^{15} T^{11} + p^{18} T^{12} \)
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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

−4.95216768326983884010718940896, −4.63836806820666409382779166302, −4.43635663819386642082709148160, −4.37292101345767634640064946993, −4.26161839242828653981305804756, −4.11870755244066184986582619141, −4.08653988054015793670117507008, −3.61147939222381342795592231257, −3.28383813907457420087770639570, −3.11249020094540300291158147337, −3.04980353578761495428201239968, −3.04853528192477598243290769731, −2.67203049278395694197154159790, −2.64903046175087497124700479832, −2.50331825335118487129376897344, −2.18942327824191924712059272771, −1.79311881155461495280885654205, −1.42677120765331638156853528758, −1.32226627112145691779050048774, −1.07681179736508049259330664310, −0.77502464694909192650427165752, −0.56497139314284805606595603077, −0.53427308550926807525572715806, −0.51381402720665545207744093757, −0.33304368055917984196650067721, 0.33304368055917984196650067721, 0.51381402720665545207744093757, 0.53427308550926807525572715806, 0.56497139314284805606595603077, 0.77502464694909192650427165752, 1.07681179736508049259330664310, 1.32226627112145691779050048774, 1.42677120765331638156853528758, 1.79311881155461495280885654205, 2.18942327824191924712059272771, 2.50331825335118487129376897344, 2.64903046175087497124700479832, 2.67203049278395694197154159790, 3.04853528192477598243290769731, 3.04980353578761495428201239968, 3.11249020094540300291158147337, 3.28383813907457420087770639570, 3.61147939222381342795592231257, 4.08653988054015793670117507008, 4.11870755244066184986582619141, 4.26161839242828653981305804756, 4.37292101345767634640064946993, 4.43635663819386642082709148160, 4.63836806820666409382779166302, 4.95216768326983884010718940896

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

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