Properties

Degree 14
Conductor $ 2^{21} \cdot 3^{7} \cdot 5^{7} \cdot 67^{7} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 7·3-s − 7·5-s + 10·7-s + 28·9-s − 13-s − 49·15-s − 2·17-s + 9·19-s + 70·21-s + 2·23-s + 28·25-s + 84·27-s − 29-s + 9·31-s − 70·35-s + 23·37-s − 7·39-s + 5·41-s − 3·43-s − 196·45-s + 11·47-s + 32·49-s − 14·51-s + 13·53-s + 63·57-s + 59-s + 4·61-s + ⋯
L(s)  = 1  + 4.04·3-s − 3.13·5-s + 3.77·7-s + 28/3·9-s − 0.277·13-s − 12.6·15-s − 0.485·17-s + 2.06·19-s + 15.2·21-s + 0.417·23-s + 28/5·25-s + 16.1·27-s − 0.185·29-s + 1.61·31-s − 11.8·35-s + 3.78·37-s − 1.12·39-s + 0.780·41-s − 0.457·43-s − 29.2·45-s + 1.60·47-s + 32/7·49-s − 1.96·51-s + 1.78·53-s + 8.34·57-s + 0.130·59-s + 0.512·61-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{21} \cdot 3^{7} \cdot 5^{7} \cdot 67^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr =\mathstrut & \,\Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{21} \cdot 3^{7} \cdot 5^{7} \cdot 67^{7}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{7} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(14\)
\( N \)  =  \(2^{21} \cdot 3^{7} \cdot 5^{7} \cdot 67^{7}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{8040} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(14,\ 2^{21} \cdot 3^{7} \cdot 5^{7} \cdot 67^{7} ,\ ( \ : [1/2]^{7} ),\ 1 )$
$L(1)$  $\approx$  $429.2284222$
$L(\frac12)$  $\approx$  $429.2284222$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;5,\;67\}$, \(F_p\) is a polynomial of degree 14. If $p \in \{2,\;3,\;5,\;67\}$, then $F_p$ is a polynomial of degree at most 13.
$p$$F_p$
bad2 \( 1 \)
3 \( ( 1 - T )^{7} \)
5 \( ( 1 + T )^{7} \)
67 \( ( 1 - T )^{7} \)
good7 \( 1 - 10 T + 68 T^{2} - 346 T^{3} + 1471 T^{4} - 5261 T^{5} + 16684 T^{6} - 46598 T^{7} + 16684 p T^{8} - 5261 p^{2} T^{9} + 1471 p^{3} T^{10} - 346 p^{4} T^{11} + 68 p^{5} T^{12} - 10 p^{6} T^{13} + p^{7} T^{14} \)
11 \( 1 + 36 T^{2} + 18 T^{3} + 811 T^{4} + 415 T^{5} + 12300 T^{6} + 6304 T^{7} + 12300 p T^{8} + 415 p^{2} T^{9} + 811 p^{3} T^{10} + 18 p^{4} T^{11} + 36 p^{5} T^{12} + p^{7} T^{14} \)
13 \( 1 + T + 36 T^{2} + 9 T^{3} + 945 T^{4} + 441 T^{5} + 16234 T^{6} + 2666 T^{7} + 16234 p T^{8} + 441 p^{2} T^{9} + 945 p^{3} T^{10} + 9 p^{4} T^{11} + 36 p^{5} T^{12} + p^{6} T^{13} + p^{7} T^{14} \)
17 \( 1 + 2 T + 43 T^{2} + 15 p T^{3} + 1262 T^{4} + 7372 T^{5} + 38452 T^{6} + 7426 p T^{7} + 38452 p T^{8} + 7372 p^{2} T^{9} + 1262 p^{3} T^{10} + 15 p^{5} T^{11} + 43 p^{5} T^{12} + 2 p^{6} T^{13} + p^{7} T^{14} \)
19 \( 1 - 9 T + 102 T^{2} - 579 T^{3} + 3771 T^{4} - 15441 T^{5} + 80646 T^{6} - 292046 T^{7} + 80646 p T^{8} - 15441 p^{2} T^{9} + 3771 p^{3} T^{10} - 579 p^{4} T^{11} + 102 p^{5} T^{12} - 9 p^{6} T^{13} + p^{7} T^{14} \)
23 \( 1 - 2 T + 85 T^{2} - 118 T^{3} + 3724 T^{4} - 4338 T^{5} + 113348 T^{6} - 116524 T^{7} + 113348 p T^{8} - 4338 p^{2} T^{9} + 3724 p^{3} T^{10} - 118 p^{4} T^{11} + 85 p^{5} T^{12} - 2 p^{6} T^{13} + p^{7} T^{14} \)
29 \( 1 + T + 170 T^{2} + 211 T^{3} + 13017 T^{4} + 16747 T^{5} + 588216 T^{6} + 665298 T^{7} + 588216 p T^{8} + 16747 p^{2} T^{9} + 13017 p^{3} T^{10} + 211 p^{4} T^{11} + 170 p^{5} T^{12} + p^{6} T^{13} + p^{7} T^{14} \)
31 \( 1 - 9 T + 213 T^{2} - 1418 T^{3} + 18840 T^{4} - 98361 T^{5} + 940870 T^{6} - 3912856 T^{7} + 940870 p T^{8} - 98361 p^{2} T^{9} + 18840 p^{3} T^{10} - 1418 p^{4} T^{11} + 213 p^{5} T^{12} - 9 p^{6} T^{13} + p^{7} T^{14} \)
37 \( 1 - 23 T + 349 T^{2} - 3810 T^{3} + 35828 T^{4} - 288872 T^{5} + 2113606 T^{6} - 13534932 T^{7} + 2113606 p T^{8} - 288872 p^{2} T^{9} + 35828 p^{3} T^{10} - 3810 p^{4} T^{11} + 349 p^{5} T^{12} - 23 p^{6} T^{13} + p^{7} T^{14} \)
41 \( 1 - 5 T + 259 T^{2} - 1186 T^{3} + 728 p T^{4} - 118499 T^{5} + 1976994 T^{6} - 6418792 T^{7} + 1976994 p T^{8} - 118499 p^{2} T^{9} + 728 p^{4} T^{10} - 1186 p^{4} T^{11} + 259 p^{5} T^{12} - 5 p^{6} T^{13} + p^{7} T^{14} \)
43 \( 1 + 3 T + 181 T^{2} + 260 T^{3} + 14294 T^{4} + 123 T^{5} + 723462 T^{6} - 486012 T^{7} + 723462 p T^{8} + 123 p^{2} T^{9} + 14294 p^{3} T^{10} + 260 p^{4} T^{11} + 181 p^{5} T^{12} + 3 p^{6} T^{13} + p^{7} T^{14} \)
47 \( 1 - 11 T + 258 T^{2} - 2359 T^{3} + 31469 T^{4} - 238279 T^{5} + 2303160 T^{6} - 14200638 T^{7} + 2303160 p T^{8} - 238279 p^{2} T^{9} + 31469 p^{3} T^{10} - 2359 p^{4} T^{11} + 258 p^{5} T^{12} - 11 p^{6} T^{13} + p^{7} T^{14} \)
53 \( 1 - 13 T + 242 T^{2} - 1941 T^{3} + 19547 T^{4} - 100915 T^{5} + 795342 T^{6} - 3675094 T^{7} + 795342 p T^{8} - 100915 p^{2} T^{9} + 19547 p^{3} T^{10} - 1941 p^{4} T^{11} + 242 p^{5} T^{12} - 13 p^{6} T^{13} + p^{7} T^{14} \)
59 \( 1 - T + 300 T^{2} - 621 T^{3} + 40407 T^{4} - 114341 T^{5} + 3366752 T^{6} - 9579194 T^{7} + 3366752 p T^{8} - 114341 p^{2} T^{9} + 40407 p^{3} T^{10} - 621 p^{4} T^{11} + 300 p^{5} T^{12} - p^{6} T^{13} + p^{7} T^{14} \)
61 \( 1 - 4 T + 271 T^{2} - 378 T^{3} + 527 p T^{4} + 19309 T^{5} + 2451547 T^{6} + 3397408 T^{7} + 2451547 p T^{8} + 19309 p^{2} T^{9} + 527 p^{4} T^{10} - 378 p^{4} T^{11} + 271 p^{5} T^{12} - 4 p^{6} T^{13} + p^{7} T^{14} \)
71 \( 1 - T + 295 T^{2} - 381 T^{3} + 40643 T^{4} - 59882 T^{5} + 3670009 T^{6} - 5435836 T^{7} + 3670009 p T^{8} - 59882 p^{2} T^{9} + 40643 p^{3} T^{10} - 381 p^{4} T^{11} + 295 p^{5} T^{12} - p^{6} T^{13} + p^{7} T^{14} \)
73 \( 1 - 14 T + 397 T^{2} - 4457 T^{3} + 70926 T^{4} - 647590 T^{5} + 7660356 T^{6} - 57927654 T^{7} + 7660356 p T^{8} - 647590 p^{2} T^{9} + 70926 p^{3} T^{10} - 4457 p^{4} T^{11} + 397 p^{5} T^{12} - 14 p^{6} T^{13} + p^{7} T^{14} \)
79 \( 1 - 25 T + 631 T^{2} - 10230 T^{3} + 154556 T^{4} - 1855255 T^{5} + 20468128 T^{6} - 189895608 T^{7} + 20468128 p T^{8} - 1855255 p^{2} T^{9} + 154556 p^{3} T^{10} - 10230 p^{4} T^{11} + 631 p^{5} T^{12} - 25 p^{6} T^{13} + p^{7} T^{14} \)
83 \( 1 + 29 T + 773 T^{2} + 13433 T^{3} + 212237 T^{4} + 2656366 T^{5} + 30355947 T^{6} + 289170114 T^{7} + 30355947 p T^{8} + 2656366 p^{2} T^{9} + 212237 p^{3} T^{10} + 13433 p^{4} T^{11} + 773 p^{5} T^{12} + 29 p^{6} T^{13} + p^{7} T^{14} \)
89 \( 1 - 7 T + 319 T^{2} - 387 T^{3} + 44043 T^{4} + 46200 T^{5} + 5508333 T^{6} + 2398220 T^{7} + 5508333 p T^{8} + 46200 p^{2} T^{9} + 44043 p^{3} T^{10} - 387 p^{4} T^{11} + 319 p^{5} T^{12} - 7 p^{6} T^{13} + p^{7} T^{14} \)
97 \( 1 - 38 T + 893 T^{2} - 15098 T^{3} + 216505 T^{4} - 2773777 T^{5} + 336877 p T^{6} - 341607190 T^{7} + 336877 p^{2} T^{8} - 2773777 p^{2} T^{9} + 216505 p^{3} T^{10} - 15098 p^{4} T^{11} + 893 p^{5} T^{12} - 38 p^{6} T^{13} + p^{7} T^{14} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{14} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−3.46834827105846367734832791169, −3.44995415818261639306327940672, −3.44435859095274915156808371700, −3.39056787515570647044632171492, −2.89761550669903835452931355758, −2.75916564878224707979032789946, −2.73900783598996625954076472781, −2.68900605842701647950875201038, −2.68072120223795021521353047163, −2.66551215691499248762232404322, −2.49355200325312525564979424923, −2.00665248696187530211995344546, −2.00053063930800597301056121365, −1.98144757096651863868258708814, −1.80310609603500606193762052574, −1.71135429017082406503312239783, −1.70394726455677187179456561056, −1.57131317624822094289778061884, −0.983596852258812971532847292287, −0.960880405052537624384991842643, −0.945971938827641472057716976106, −0.924196896347061758655866552655, −0.59250365428848048760976506317, −0.58280721254064627998106802362, −0.53318340533996517484602872159, 0.53318340533996517484602872159, 0.58280721254064627998106802362, 0.59250365428848048760976506317, 0.924196896347061758655866552655, 0.945971938827641472057716976106, 0.960880405052537624384991842643, 0.983596852258812971532847292287, 1.57131317624822094289778061884, 1.70394726455677187179456561056, 1.71135429017082406503312239783, 1.80310609603500606193762052574, 1.98144757096651863868258708814, 2.00053063930800597301056121365, 2.00665248696187530211995344546, 2.49355200325312525564979424923, 2.66551215691499248762232404322, 2.68072120223795021521353047163, 2.68900605842701647950875201038, 2.73900783598996625954076472781, 2.75916564878224707979032789946, 2.89761550669903835452931355758, 3.39056787515570647044632171492, 3.44435859095274915156808371700, 3.44995415818261639306327940672, 3.46834827105846367734832791169

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

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