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Properties

Label 12-2873e6-1.1-c1e6-0-1
Degree $12$
Conductor $5.624\times 10^{20}$
Sign $1$
Analytic cond. $1.45772\times 10^{8}$
Root an. cond. $4.78967$
Motivic weight $1$
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  − 2-s + 3-s − 2·4-s + 2·5-s − 6-s − 7·7-s + 8-s − 6·9-s − 2·10-s + 11-s − 2·12-s + 7·14-s + 2·15-s + 2·16-s − 6·17-s + 6·18-s − 23·19-s − 4·20-s − 7·21-s − 22-s − 10·23-s + 24-s − 11·25-s − 10·27-s + 14·28-s − 4·29-s − 2·30-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s − 4-s + 0.894·5-s − 0.408·6-s − 2.64·7-s + 0.353·8-s − 2·9-s − 0.632·10-s + 0.301·11-s − 0.577·12-s + 1.87·14-s + 0.516·15-s + 1/2·16-s − 1.45·17-s + 1.41·18-s − 5.27·19-s − 0.894·20-s − 1.52·21-s − 0.213·22-s − 2.08·23-s + 0.204·24-s − 2.19·25-s − 1.92·27-s + 2.64·28-s − 0.742·29-s − 0.365·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(13^{12} \cdot 17^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(13^{12} \cdot 17^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(13^{12} \cdot 17^{6}\)
Sign: $1$
Analytic conductor: \(1.45772\times 10^{8}\)
Root analytic conductor: \(4.78967\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(6\)
Selberg data: \((12,\ 13^{12} \cdot 17^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
17 \( ( 1 + T )^{6} \)
good2 \( 1 + T + 3 T^{2} + p^{2} T^{3} + 7 T^{4} + 9 T^{5} + 17 T^{6} + 9 p T^{7} + 7 p^{2} T^{8} + p^{5} T^{9} + 3 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
3 \( 1 - T + 7 T^{2} - p T^{3} + 31 T^{4} - 2 p^{2} T^{5} + 118 T^{6} - 2 p^{3} T^{7} + 31 p^{2} T^{8} - p^{4} T^{9} + 7 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
5 \( 1 - 2 T + 3 p T^{2} - 34 T^{3} + 27 p T^{4} - 244 T^{5} + 838 T^{6} - 244 p T^{7} + 27 p^{3} T^{8} - 34 p^{3} T^{9} + 3 p^{5} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
7 \( 1 + p T + 5 p T^{2} + 19 p T^{3} + 69 p T^{4} + 1478 T^{5} + 4226 T^{6} + 1478 p T^{7} + 69 p^{3} T^{8} + 19 p^{4} T^{9} + 5 p^{5} T^{10} + p^{6} T^{11} + p^{6} T^{12} \)
11 \( 1 - T + 47 T^{2} - 47 T^{3} + 97 p T^{4} - 962 T^{5} + 14714 T^{6} - 962 p T^{7} + 97 p^{3} T^{8} - 47 p^{3} T^{9} + 47 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 + 23 T + 281 T^{2} + 2361 T^{3} + 15395 T^{4} + 83094 T^{5} + 387318 T^{6} + 83094 p T^{7} + 15395 p^{2} T^{8} + 2361 p^{3} T^{9} + 281 p^{4} T^{10} + 23 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 10 T + 94 T^{2} + 526 T^{3} + 2739 T^{4} + 11948 T^{5} + 55820 T^{6} + 11948 p T^{7} + 2739 p^{2} T^{8} + 526 p^{3} T^{9} + 94 p^{4} T^{10} + 10 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + 4 T + 149 T^{2} + 500 T^{3} + 9883 T^{4} + 27000 T^{5} + 371422 T^{6} + 27000 p T^{7} + 9883 p^{2} T^{8} + 500 p^{3} T^{9} + 149 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 16 T + 213 T^{2} + 1952 T^{3} + 16411 T^{4} + 109520 T^{5} + 674382 T^{6} + 109520 p T^{7} + 16411 p^{2} T^{8} + 1952 p^{3} T^{9} + 213 p^{4} T^{10} + 16 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 + 4 T + 173 T^{2} + 572 T^{3} + 13487 T^{4} + 36184 T^{5} + 625618 T^{6} + 36184 p T^{7} + 13487 p^{2} T^{8} + 572 p^{3} T^{9} + 173 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 - 4 T + 174 T^{2} - 668 T^{3} + 14483 T^{4} - 47568 T^{5} + 740268 T^{6} - 47568 p T^{7} + 14483 p^{2} T^{8} - 668 p^{3} T^{9} + 174 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 - 10 T + 195 T^{2} - 1486 T^{3} + 17315 T^{4} - 109868 T^{5} + 941906 T^{6} - 109868 p T^{7} + 17315 p^{2} T^{8} - 1486 p^{3} T^{9} + 195 p^{4} T^{10} - 10 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 - 6 T + 118 T^{2} - 946 T^{3} + 8239 T^{4} - 55052 T^{5} + 466740 T^{6} - 55052 p T^{7} + 8239 p^{2} T^{8} - 946 p^{3} T^{9} + 118 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 27 T + 499 T^{2} + 6795 T^{3} + 76827 T^{4} + 710342 T^{5} + 5632850 T^{6} + 710342 p T^{7} + 76827 p^{2} T^{8} + 6795 p^{3} T^{9} + 499 p^{4} T^{10} + 27 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 + 10 T + 183 T^{2} + 1166 T^{3} + 15371 T^{4} + 68556 T^{5} + 893802 T^{6} + 68556 p T^{7} + 15371 p^{2} T^{8} + 1166 p^{3} T^{9} + 183 p^{4} T^{10} + 10 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 11 T + 189 T^{2} - 2195 T^{3} + 23259 T^{4} - 199062 T^{5} + 1880910 T^{6} - 199062 p T^{7} + 23259 p^{2} T^{8} - 2195 p^{3} T^{9} + 189 p^{4} T^{10} - 11 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 18 T + 398 T^{2} + 4750 T^{3} + 63127 T^{4} + 568276 T^{5} + 5525444 T^{6} + 568276 p T^{7} + 63127 p^{2} T^{8} + 4750 p^{3} T^{9} + 398 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 + 126 T^{2} + 1024 T^{3} + 12415 T^{4} + 61952 T^{5} + 1477412 T^{6} + 61952 p T^{7} + 12415 p^{2} T^{8} + 1024 p^{3} T^{9} + 126 p^{4} T^{10} + p^{6} T^{12} \)
73 \( 1 + 12 T + 261 T^{2} + 2788 T^{3} + 35927 T^{4} + 297928 T^{5} + 3223570 T^{6} + 297928 p T^{7} + 35927 p^{2} T^{8} + 2788 p^{3} T^{9} + 261 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 6 T + 343 T^{2} + 1542 T^{3} + 51335 T^{4} + 180248 T^{5} + 4818870 T^{6} + 180248 p T^{7} + 51335 p^{2} T^{8} + 1542 p^{3} T^{9} + 343 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 + 26 T + 327 T^{2} + 3238 T^{3} + 31075 T^{4} + 208996 T^{5} + 1286090 T^{6} + 208996 p T^{7} + 31075 p^{2} T^{8} + 3238 p^{3} T^{9} + 327 p^{4} T^{10} + 26 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 + 21 T + 517 T^{2} + 7089 T^{3} + 105075 T^{4} + 1081714 T^{5} + 11978126 T^{6} + 1081714 p T^{7} + 105075 p^{2} T^{8} + 7089 p^{3} T^{9} + 517 p^{4} T^{10} + 21 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 + 3 T + 371 T^{2} + 951 T^{3} + 68055 T^{4} + 142714 T^{5} + 8005550 T^{6} + 142714 p T^{7} + 68055 p^{2} T^{8} + 951 p^{3} T^{9} + 371 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
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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

−5.12759845740952560607992794716, −4.65016093775827729196532339580, −4.45956706129104896436809994540, −4.44850029079234959607144335262, −4.25511063922486225864697198226, −4.22363453042846484654576276130, −4.01593247033937267724920312408, −4.01385076761935471910606548619, −3.90086313974225109519452117511, −3.78639619603899604492945901411, −3.50515250213053888612731713682, −3.16966258927823710340529676582, −3.11262652515806481460767585792, −3.07066712065965310221851855304, −2.91082402159791592902780979894, −2.64843179339418203296976149808, −2.51670573974260286171896616158, −2.36977272785508931294266327711, −2.18073342571864006393519077243, −2.03035798788668909460259899452, −1.74967235793195584617338000474, −1.73022122296182528561867853165, −1.60538467171246556676460544995, −1.46425216443824708091352840824, −0.70694674604211342595138073878, 0, 0, 0, 0, 0, 0, 0.70694674604211342595138073878, 1.46425216443824708091352840824, 1.60538467171246556676460544995, 1.73022122296182528561867853165, 1.74967235793195584617338000474, 2.03035798788668909460259899452, 2.18073342571864006393519077243, 2.36977272785508931294266327711, 2.51670573974260286171896616158, 2.64843179339418203296976149808, 2.91082402159791592902780979894, 3.07066712065965310221851855304, 3.11262652515806481460767585792, 3.16966258927823710340529676582, 3.50515250213053888612731713682, 3.78639619603899604492945901411, 3.90086313974225109519452117511, 4.01385076761935471910606548619, 4.01593247033937267724920312408, 4.22363453042846484654576276130, 4.25511063922486225864697198226, 4.44850029079234959607144335262, 4.45956706129104896436809994540, 4.65016093775827729196532339580, 5.12759845740952560607992794716

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

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