Properties

Degree 12
Conductor $ 17^{6} \cdot 43^{6} $
Sign $1$
Motivic weight 1
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 − 3·3-s − 3·4-s + 3·5-s + 3·6-s − 7·7-s + 8-s − 6·9-s − 3·10-s + 4·11-s + 9·12-s − 10·13-s + 7·14-s − 9·15-s + 4·16-s − 6·17-s + 6·18-s − 20·19-s − 9·20-s + 21·21-s − 4·22-s − 3·23-s − 3·24-s − 14·25-s + 10·26-s + 28·27-s + 21·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.73·3-s − 3/2·4-s + 1.34·5-s + 1.22·6-s − 2.64·7-s + 0.353·8-s − 2·9-s − 0.948·10-s + 1.20·11-s + 2.59·12-s − 2.77·13-s + 1.87·14-s − 2.32·15-s + 16-s − 1.45·17-s + 1.41·18-s − 4.58·19-s − 2.01·20-s + 4.58·21-s − 0.852·22-s − 0.625·23-s − 0.612·24-s − 2.79·25-s + 1.96·26-s + 5.38·27-s + 3.96·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(17^{6} \cdot 43^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(17^{6} \cdot 43^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(12\)
\( N \)  =  \(17^{6} \cdot 43^{6}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{731} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  6
Selberg data  =  $(12,\ 17^{6} \cdot 43^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{17,\;43\}$,\(F_p(T)\) is a polynomial of degree 12. If $p \in \{17,\;43\}$, then $F_p(T)$ is a polynomial of degree at most 11.
$p$$F_p(T)$
bad17 \( ( 1 + T )^{6} \)
43 \( ( 1 + T )^{6} \)
good2 \( 1 + T + p^{2} T^{2} + 3 p T^{3} + 13 T^{4} + 15 T^{5} + 35 T^{6} + 15 p T^{7} + 13 p^{2} T^{8} + 3 p^{4} T^{9} + p^{6} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
3 \( 1 + p T + 5 p T^{2} + 35 T^{3} + 100 T^{4} + 184 T^{5} + 383 T^{6} + 184 p T^{7} + 100 p^{2} T^{8} + 35 p^{3} T^{9} + 5 p^{5} T^{10} + p^{6} T^{11} + p^{6} T^{12} \)
5 \( 1 - 3 T + 23 T^{2} - 13 p T^{3} + 49 p T^{4} - 607 T^{5} + 1551 T^{6} - 607 p T^{7} + 49 p^{3} T^{8} - 13 p^{4} T^{9} + 23 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
7 \( 1 + p T + 44 T^{2} + 172 T^{3} + 2 p^{3} T^{4} + 2077 T^{5} + 6303 T^{6} + 2077 p T^{7} + 2 p^{5} T^{8} + 172 p^{3} T^{9} + 44 p^{4} T^{10} + p^{6} T^{11} + p^{6} T^{12} \)
11 \( 1 - 4 T + 65 T^{2} - 206 T^{3} + 1767 T^{4} - 4386 T^{5} + 25807 T^{6} - 4386 p T^{7} + 1767 p^{2} T^{8} - 206 p^{3} T^{9} + 65 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 + 10 T + 80 T^{2} + 36 p T^{3} + 2427 T^{4} + 10503 T^{5} + 41451 T^{6} + 10503 p T^{7} + 2427 p^{2} T^{8} + 36 p^{4} T^{9} + 80 p^{4} T^{10} + 10 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 + 20 T + 236 T^{2} + 1934 T^{3} + 12368 T^{4} + 65720 T^{5} + 304071 T^{6} + 65720 p T^{7} + 12368 p^{2} T^{8} + 1934 p^{3} T^{9} + 236 p^{4} T^{10} + 20 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 3 T + 104 T^{2} + 263 T^{3} + 5078 T^{4} + 10753 T^{5} + 147609 T^{6} + 10753 p T^{7} + 5078 p^{2} T^{8} + 263 p^{3} T^{9} + 104 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + 15 T + 186 T^{2} + 1492 T^{3} + 10238 T^{4} + 59357 T^{5} + 325051 T^{6} + 59357 p T^{7} + 10238 p^{2} T^{8} + 1492 p^{3} T^{9} + 186 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 - 12 T + 159 T^{2} - 1316 T^{3} + 11142 T^{4} - 70444 T^{5} + 447555 T^{6} - 70444 p T^{7} + 11142 p^{2} T^{8} - 1316 p^{3} T^{9} + 159 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 + 14 T + 125 T^{2} + 1205 T^{3} + 10985 T^{4} + 74918 T^{5} + 440181 T^{6} + 74918 p T^{7} + 10985 p^{2} T^{8} + 1205 p^{3} T^{9} + 125 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 + 2 T + 33 T^{3} + 3835 T^{4} + 4315 T^{5} + 10197 T^{6} + 4315 p T^{7} + 3835 p^{2} T^{8} + 33 p^{3} T^{9} + 2 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 + 11 T + 223 T^{2} + 1961 T^{3} + 22621 T^{4} + 155625 T^{5} + 1348507 T^{6} + 155625 p T^{7} + 22621 p^{2} T^{8} + 1961 p^{3} T^{9} + 223 p^{4} T^{10} + 11 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 - 3 T + 226 T^{2} - 783 T^{3} + 24895 T^{4} - 77814 T^{5} + 1667071 T^{6} - 77814 p T^{7} + 24895 p^{2} T^{8} - 783 p^{3} T^{9} + 226 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 - 2 T + 211 T^{2} - 300 T^{3} + 23930 T^{4} - 24433 T^{5} + 1701233 T^{6} - 24433 p T^{7} + 23930 p^{2} T^{8} - 300 p^{3} T^{9} + 211 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 20 T + 444 T^{2} + 5838 T^{3} + 73861 T^{4} + 698027 T^{5} + 6160733 T^{6} + 698027 p T^{7} + 73861 p^{2} T^{8} + 5838 p^{3} T^{9} + 444 p^{4} T^{10} + 20 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 2 T + 138 T^{2} + 230 T^{3} + 18283 T^{4} + 23305 T^{5} + 1253885 T^{6} + 23305 p T^{7} + 18283 p^{2} T^{8} + 230 p^{3} T^{9} + 138 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 - T + 245 T^{2} - 535 T^{3} + 32231 T^{4} - 67267 T^{5} + 2837077 T^{6} - 67267 p T^{7} + 32231 p^{2} T^{8} - 535 p^{3} T^{9} + 245 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 - 13 T + 305 T^{2} - 3567 T^{3} + 41863 T^{4} - 443657 T^{5} + 3637205 T^{6} - 443657 p T^{7} + 41863 p^{2} T^{8} - 3567 p^{3} T^{9} + 305 p^{4} T^{10} - 13 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 26 T + 543 T^{2} + 6874 T^{3} + 79324 T^{4} + 693295 T^{5} + 6626843 T^{6} + 693295 p T^{7} + 79324 p^{2} T^{8} + 6874 p^{3} T^{9} + 543 p^{4} T^{10} + 26 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 - 10 T + 350 T^{2} - 3350 T^{3} + 60903 T^{4} - 476772 T^{5} + 6438532 T^{6} - 476772 p T^{7} + 60903 p^{2} T^{8} - 3350 p^{3} T^{9} + 350 p^{4} T^{10} - 10 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 + 15 T + 414 T^{2} + 4690 T^{3} + 79231 T^{4} + 720074 T^{5} + 9003067 T^{6} + 720074 p T^{7} + 79231 p^{2} T^{8} + 4690 p^{3} T^{9} + 414 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 + 22 T + 472 T^{2} + 4530 T^{3} + 44838 T^{4} + 114330 T^{5} + 1499567 T^{6} + 114330 p T^{7} + 44838 p^{2} T^{8} + 4530 p^{3} T^{9} + 472 p^{4} T^{10} + 22 p^{5} T^{11} + p^{6} T^{12} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−6.14386475681703177786822373728, −6.08689915276856061145258482038, −5.66088088436561406179123231277, −5.56621458477830555065521841142, −5.50115242423230845791444184146, −5.33597653404152324270618841383, −5.26586651723312163148572646294, −4.75710408721870182009411095392, −4.70970347629156050612747991983, −4.56610985216660071580919635244, −4.55853515132601337156290026062, −4.17760811363505970694803797222, −3.99543869560287126613495169358, −3.88745321227365620147515252028, −3.51167641547133325165950826536, −3.50560523527321925559032479873, −3.30481716018561055051528022488, −2.89263121935266338762566114469, −2.65902646729624576157635952031, −2.58519782493905735782650059119, −2.46931529185023526413853953681, −2.04753551522320719886945198100, −1.89555902249478005299047405181, −1.87901819194857645507979719686, −1.36727566051240818908008943907, 0, 0, 0, 0, 0, 0, 1.36727566051240818908008943907, 1.87901819194857645507979719686, 1.89555902249478005299047405181, 2.04753551522320719886945198100, 2.46931529185023526413853953681, 2.58519782493905735782650059119, 2.65902646729624576157635952031, 2.89263121935266338762566114469, 3.30481716018561055051528022488, 3.50560523527321925559032479873, 3.51167641547133325165950826536, 3.88745321227365620147515252028, 3.99543869560287126613495169358, 4.17760811363505970694803797222, 4.55853515132601337156290026062, 4.56610985216660071580919635244, 4.70970347629156050612747991983, 4.75710408721870182009411095392, 5.26586651723312163148572646294, 5.33597653404152324270618841383, 5.50115242423230845791444184146, 5.56621458477830555065521841142, 5.66088088436561406179123231277, 6.08689915276856061145258482038, 6.14386475681703177786822373728

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

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