Properties

Label 12-912e6-1.1-c2e6-0-3
Degree $12$
Conductor $5.754\times 10^{17}$
Sign $1$
Analytic cond. $2.35493\times 10^{8}$
Root an. cond. $4.98499$
Motivic weight $2$
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  + 9·3-s + 4·5-s + 22·7-s + 45·9-s + 36·11-s − 3·13-s + 36·15-s + 38·17-s + 10·19-s + 198·21-s − 54·23-s + 35·25-s + 162·27-s − 102·29-s + 324·33-s + 88·35-s − 27·39-s + 96·41-s − 107·43-s + 180·45-s + 50·47-s + 71·49-s + 342·51-s − 90·53-s + 144·55-s + 90·57-s + 27·61-s + ⋯
L(s)  = 1  + 3·3-s + 4/5·5-s + 22/7·7-s + 5·9-s + 3.27·11-s − 0.230·13-s + 12/5·15-s + 2.23·17-s + 0.526·19-s + 66/7·21-s − 2.34·23-s + 7/5·25-s + 6·27-s − 3.51·29-s + 9.81·33-s + 2.51·35-s − 0.692·39-s + 2.34·41-s − 2.48·43-s + 4·45-s + 1.06·47-s + 1.44·49-s + 6.70·51-s − 1.69·53-s + 2.61·55-s + 1.57·57-s + 0.442·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{24} \cdot 3^{6} \cdot 19^{6}\)
Sign: $1$
Analytic conductor: \(2.35493\times 10^{8}\)
Root analytic conductor: \(4.98499\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{912} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{24} \cdot 3^{6} \cdot 19^{6} ,\ ( \ : [1]^{6} ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(73.45510705\)
\(L(\frac12)\) \(\approx\) \(73.45510705\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( ( 1 - p T + p T^{2} )^{3} \)
19 \( 1 - 10 T - 249 T^{2} + 172 p T^{3} - 249 p^{2} T^{4} - 10 p^{4} T^{5} + p^{6} T^{6} \)
good5 \( 1 - 4 T - 19 T^{2} - 84 T^{3} + 238 T^{4} + 704 p T^{5} - 8591 T^{6} + 704 p^{3} T^{7} + 238 p^{4} T^{8} - 84 p^{6} T^{9} - 19 p^{8} T^{10} - 4 p^{10} T^{11} + p^{12} T^{12} \)
7 \( ( 1 - 11 T + 146 T^{2} - 1031 T^{3} + 146 p^{2} T^{4} - 11 p^{4} T^{5} + p^{6} T^{6} )^{2} \)
11 \( ( 1 - 18 T + 267 T^{2} - 3272 T^{3} + 267 p^{2} T^{4} - 18 p^{4} T^{5} + p^{6} T^{6} )^{2} \)
13 \( 1 + 3 T + 167 T^{2} + 492 T^{3} + 101 T^{4} + 65361 T^{5} - 4911130 T^{6} + 65361 p^{2} T^{7} + 101 p^{4} T^{8} + 492 p^{6} T^{9} + 167 p^{8} T^{10} + 3 p^{10} T^{11} + p^{12} T^{12} \)
17 \( 1 - 38 T + 541 T^{2} - 4834 T^{3} - 334 p T^{4} + 3349474 T^{5} - 90714787 T^{6} + 3349474 p^{2} T^{7} - 334 p^{5} T^{8} - 4834 p^{6} T^{9} + 541 p^{8} T^{10} - 38 p^{10} T^{11} + p^{12} T^{12} \)
23 \( 1 + 54 T + 765 T^{2} + 11066 T^{3} + 651546 T^{4} + 5242626 T^{5} - 206650371 T^{6} + 5242626 p^{2} T^{7} + 651546 p^{4} T^{8} + 11066 p^{6} T^{9} + 765 p^{8} T^{10} + 54 p^{10} T^{11} + p^{12} T^{12} \)
29 \( 1 + 102 T + 6203 T^{2} + 278970 T^{3} + 9713378 T^{4} + 292520358 T^{5} + 8574179483 T^{6} + 292520358 p^{2} T^{7} + 9713378 p^{4} T^{8} + 278970 p^{6} T^{9} + 6203 p^{8} T^{10} + 102 p^{10} T^{11} + p^{12} T^{12} \)
31 \( 1 - 3421 T^{2} + 5823134 T^{4} - 6536013745 T^{6} + 5823134 p^{4} T^{8} - 3421 p^{8} T^{10} + p^{12} T^{12} \)
37 \( 1 - 3501 T^{2} + 9576054 T^{4} - 14575249001 T^{6} + 9576054 p^{4} T^{8} - 3501 p^{8} T^{10} + p^{12} T^{12} \)
41 \( 1 - 96 T + 8867 T^{2} - 556320 T^{3} + 34233206 T^{4} - 1610757504 T^{5} + 73428522359 T^{6} - 1610757504 p^{2} T^{7} + 34233206 p^{4} T^{8} - 556320 p^{6} T^{9} + 8867 p^{8} T^{10} - 96 p^{10} T^{11} + p^{12} T^{12} \)
43 \( 1 + 107 T + 2315 T^{2} + 109092 T^{3} + 20090029 T^{4} + 716895265 T^{5} + 7701663934 T^{6} + 716895265 p^{2} T^{7} + 20090029 p^{4} T^{8} + 109092 p^{6} T^{9} + 2315 p^{8} T^{10} + 107 p^{10} T^{11} + p^{12} T^{12} \)
47 \( 1 - 50 T - 4651 T^{2} + 79434 T^{3} + 24415642 T^{4} - 226764442 T^{5} - 55863211187 T^{6} - 226764442 p^{2} T^{7} + 24415642 p^{4} T^{8} + 79434 p^{6} T^{9} - 4651 p^{8} T^{10} - 50 p^{10} T^{11} + p^{12} T^{12} \)
53 \( 1 + 90 T + 123 p T^{2} + 343710 T^{3} + 14892630 T^{4} + 694668222 T^{5} + 20439623059 T^{6} + 694668222 p^{2} T^{7} + 14892630 p^{4} T^{8} + 343710 p^{6} T^{9} + 123 p^{9} T^{10} + 90 p^{10} T^{11} + p^{12} T^{12} \)
59 \( 1 + 9395 T^{2} + 55562030 T^{4} - 82187460 T^{5} + 222918474911 T^{6} - 82187460 p^{2} T^{7} + 55562030 p^{4} T^{8} + 9395 p^{8} T^{10} + p^{12} T^{12} \)
61 \( 1 - 27 T - 8277 T^{2} + 88964 T^{3} + 43113009 T^{4} - 59789481 T^{5} - 181278951354 T^{6} - 59789481 p^{2} T^{7} + 43113009 p^{4} T^{8} + 88964 p^{6} T^{9} - 8277 p^{8} T^{10} - 27 p^{10} T^{11} + p^{12} T^{12} \)
67 \( 1 - 39 T + 7859 T^{2} - 286728 T^{3} + 23155985 T^{4} - 405814017 T^{5} + 60263661254 T^{6} - 405814017 p^{2} T^{7} + 23155985 p^{4} T^{8} - 286728 p^{6} T^{9} + 7859 p^{8} T^{10} - 39 p^{10} T^{11} + p^{12} T^{12} \)
71 \( 1 + 84 T + 10371 T^{2} + 673596 T^{3} + 28816422 T^{4} - 720991884 T^{5} + 18547655047 T^{6} - 720991884 p^{2} T^{7} + 28816422 p^{4} T^{8} + 673596 p^{6} T^{9} + 10371 p^{8} T^{10} + 84 p^{10} T^{11} + p^{12} T^{12} \)
73 \( 1 + 77 T - 1517 T^{2} + 250636 T^{3} + 3339001 T^{4} - 37295617 p T^{5} - 135125227066 T^{6} - 37295617 p^{3} T^{7} + 3339001 p^{4} T^{8} + 250636 p^{6} T^{9} - 1517 p^{8} T^{10} + 77 p^{10} T^{11} + p^{12} T^{12} \)
79 \( 1 + 9 T + 11711 T^{2} + 105156 T^{3} + 64875497 T^{4} + 4632616851 T^{5} + 425401555742 T^{6} + 4632616851 p^{2} T^{7} + 64875497 p^{4} T^{8} + 105156 p^{6} T^{9} + 11711 p^{8} T^{10} + 9 p^{10} T^{11} + p^{12} T^{12} \)
83 \( ( 1 - 174 T + 16527 T^{2} - 1223300 T^{3} + 16527 p^{2} T^{4} - 174 p^{4} T^{5} + p^{6} T^{6} )^{2} \)
89 \( 1 + 72 T + 12639 T^{2} + 785592 T^{3} + 28024818 T^{4} - 6608294460 T^{5} - 147146320001 T^{6} - 6608294460 p^{2} T^{7} + 28024818 p^{4} T^{8} + 785592 p^{6} T^{9} + 12639 p^{8} T^{10} + 72 p^{10} T^{11} + p^{12} T^{12} \)
97 \( 1 + 228 T + 49763 T^{2} + 7395180 T^{3} + 1074017366 T^{4} + 119443328292 T^{5} + 12875825313911 T^{6} + 119443328292 p^{2} T^{7} + 1074017366 p^{4} T^{8} + 7395180 p^{6} T^{9} + 49763 p^{8} T^{10} + 228 p^{10} T^{11} + p^{12} T^{12} \)
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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

−5.05544512933246073031641130515, −4.90332605524740025645185182422, −4.51890201106753202623596182500, −4.50740215561729105721878391394, −4.49126151420481762746888035930, −4.45977674064496134637637520605, −4.08754300956466918703298741470, −3.68592835071177982053350655042, −3.62529796242779277247770814157, −3.56182730401913329809326028371, −3.55810158920937104382442766508, −3.47706200104227061997900524911, −2.89805559422437532682659405679, −2.88512413899760675762133594860, −2.77376167965285578524015071376, −2.19640976377351877891036936244, −2.03102092744390129974719414426, −2.00586518643531680793137913780, −1.79487472944977619141785993823, −1.77726468133607707783078230001, −1.37250589518619758881907597632, −1.28766385106219709133154168023, −1.22801405318415385105632395657, −0.890632445643735087952025284014, −0.30133813315200498856104728754, 0.30133813315200498856104728754, 0.890632445643735087952025284014, 1.22801405318415385105632395657, 1.28766385106219709133154168023, 1.37250589518619758881907597632, 1.77726468133607707783078230001, 1.79487472944977619141785993823, 2.00586518643531680793137913780, 2.03102092744390129974719414426, 2.19640976377351877891036936244, 2.77376167965285578524015071376, 2.88512413899760675762133594860, 2.89805559422437532682659405679, 3.47706200104227061997900524911, 3.55810158920937104382442766508, 3.56182730401913329809326028371, 3.62529796242779277247770814157, 3.68592835071177982053350655042, 4.08754300956466918703298741470, 4.45977674064496134637637520605, 4.49126151420481762746888035930, 4.50740215561729105721878391394, 4.51890201106753202623596182500, 4.90332605524740025645185182422, 5.05544512933246073031641130515

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

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