Properties

Degree 12
Conductor $ 29^{6} $
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  − 2·2-s − 5·3-s + 2·4-s + 5-s + 10·6-s + 7-s − 7·8-s + 17·9-s − 2·10-s − 11·11-s − 10·12-s − 5·13-s − 2·14-s − 5·15-s + 14·16-s + 8·17-s − 34·18-s + 19-s + 2·20-s − 5·21-s + 22·22-s − 7·23-s + 35·24-s − 9·25-s + 10·26-s − 49·27-s + 2·28-s + ⋯
L(s)  = 1  − 1.41·2-s − 2.88·3-s + 4-s + 0.447·5-s + 4.08·6-s + 0.377·7-s − 2.47·8-s + 17/3·9-s − 0.632·10-s − 3.31·11-s − 2.88·12-s − 1.38·13-s − 0.534·14-s − 1.29·15-s + 7/2·16-s + 1.94·17-s − 8.01·18-s + 0.229·19-s + 0.447·20-s − 1.09·21-s + 4.69·22-s − 1.45·23-s + 7.14·24-s − 9/5·25-s + 1.96·26-s − 9.43·27-s + 0.377·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut &\left(29^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr =\mathstrut & \,\Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut &\left(29^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(12\)
\( N \)  =  \(29^{6}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{29} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(12,\ 29^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )$
$L(1)$  $\approx$  $0.0319149$
$L(\frac12)$  $\approx$  $0.0319149$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 29$, \(F_p\) is a polynomial of degree 12. If $p = 29$, then $F_p$ is a polynomial of degree at most 11.
$p$$F_p$
bad29 \( 1 - 6 T - 13 T^{2} + 316 T^{3} - 13 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
good2 \( 1 + p T + p T^{2} + 7 T^{3} + 5 p T^{4} + 13 T^{5} + 27 T^{6} + 13 p T^{7} + 5 p^{3} T^{8} + 7 p^{3} T^{9} + p^{5} T^{10} + p^{6} T^{11} + p^{6} T^{12} \)
3 \( 1 + 5 T + 8 T^{2} + 4 T^{3} + p T^{4} + p T^{5} - 8 T^{6} + p^{2} T^{7} + p^{3} T^{8} + 4 p^{3} T^{9} + 8 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} \)
5 \( 1 - T + 2 p T^{2} - 12 T^{3} + 81 T^{4} - 91 T^{5} + 456 T^{6} - 91 p T^{7} + 81 p^{2} T^{8} - 12 p^{3} T^{9} + 2 p^{5} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
7 \( 1 - T + 8 T^{2} - 22 T^{3} + 127 T^{4} - 169 T^{5} + 848 T^{6} - 169 p T^{7} + 127 p^{2} T^{8} - 22 p^{3} T^{9} + 8 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 + p T + 68 T^{2} + 354 T^{3} + 153 p T^{4} + 6723 T^{5} + 23296 T^{6} + 6723 p T^{7} + 153 p^{3} T^{8} + 354 p^{3} T^{9} + 68 p^{4} T^{10} + p^{6} T^{11} + p^{6} T^{12} \)
13 \( 1 + 5 T + 12 T^{2} - 40 T^{3} - 111 T^{4} + 875 T^{5} + 6308 T^{6} + 875 p T^{7} - 111 p^{2} T^{8} - 40 p^{3} T^{9} + 12 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} \)
17 \( ( 1 - 4 T + 47 T^{2} - 128 T^{3} + 47 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
19 \( 1 - T - 18 T^{2} + 156 T^{3} + 67 T^{4} - 1127 T^{5} + 9612 T^{6} - 1127 p T^{7} + 67 p^{2} T^{8} + 156 p^{3} T^{9} - 18 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 7 T + 26 T^{2} + 84 T^{3} + 431 T^{4} - 175 T^{5} - 12020 T^{6} - 175 p T^{7} + 431 p^{2} T^{8} + 84 p^{3} T^{9} + 26 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 - 5 T - 34 T^{2} + 388 T^{3} - 1649 T^{4} - 7283 T^{5} + 109220 T^{6} - 7283 p T^{7} - 1649 p^{2} T^{8} + 388 p^{3} T^{9} - 34 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 - 11 T + 42 T^{2} - 328 T^{3} + 4049 T^{4} - 22869 T^{5} + 100360 T^{6} - 22869 p T^{7} + 4049 p^{2} T^{8} - 328 p^{3} T^{9} + 42 p^{4} T^{10} - 11 p^{5} T^{11} + p^{6} T^{12} \)
41 \( ( 1 - 10 T + 147 T^{2} - 828 T^{3} + 147 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
43 \( 1 - 13 T + 42 T^{2} - 50 T^{3} + 251 T^{4} + 17703 T^{5} - 219260 T^{6} + 17703 p T^{7} + 251 p^{2} T^{8} - 50 p^{3} T^{9} + 42 p^{4} T^{10} - 13 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 - 11 T + 18 T^{2} + 410 T^{3} - 2409 T^{4} - 12679 T^{5} + 224300 T^{6} - 12679 p T^{7} - 2409 p^{2} T^{8} + 410 p^{3} T^{9} + 18 p^{4} T^{10} - 11 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 - 3 T - 58 T^{2} + 648 T^{3} - 431 T^{4} - 23629 T^{5} + 179088 T^{6} - 23629 p T^{7} - 431 p^{2} T^{8} + 648 p^{3} T^{9} - 58 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
59 \( ( 1 + 28 T + 429 T^{2} + 4032 T^{3} + 429 p T^{4} + 28 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
61 \( 1 - 3 T - 24 T^{2} - 74 T^{3} + 4633 T^{4} - 24883 T^{5} - 67236 T^{6} - 24883 p T^{7} + 4633 p^{2} T^{8} - 74 p^{3} T^{9} - 24 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 19 T + 168 T^{2} - 904 T^{3} + 1419 T^{4} + 3983 T^{5} + 31592 T^{6} + 3983 p T^{7} + 1419 p^{2} T^{8} - 904 p^{3} T^{9} + 168 p^{4} T^{10} - 19 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 - 21 T + 118 T^{2} + 546 T^{3} - 4409 T^{4} - 107373 T^{5} + 1690660 T^{6} - 107373 p T^{7} - 4409 p^{2} T^{8} + 546 p^{3} T^{9} + 118 p^{4} T^{10} - 21 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 + 25 T + 300 T^{2} + 3400 T^{3} + 39069 T^{4} + 348943 T^{5} + 2806700 T^{6} + 348943 p T^{7} + 39069 p^{2} T^{8} + 3400 p^{3} T^{9} + 300 p^{4} T^{10} + 25 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 9 T + 2 T^{2} + 210 T^{3} - 977 T^{4} - 72339 T^{5} - 812260 T^{6} - 72339 p T^{7} - 977 p^{2} T^{8} + 210 p^{3} T^{9} + 2 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 - 17 T + 136 T^{2} - 1776 T^{3} + 27843 T^{4} - 251219 T^{5} + 1969848 T^{6} - 251219 p T^{7} + 27843 p^{2} T^{8} - 1776 p^{3} T^{9} + 136 p^{4} T^{10} - 17 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 - 7 T + 30 T^{2} + 532 T^{3} + 7389 T^{4} - 23261 T^{5} + 663944 T^{6} - 23261 p T^{7} + 7389 p^{2} T^{8} + 532 p^{3} T^{9} + 30 p^{4} T^{10} - 7 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - T + 114 T^{2} + 382 T^{3} + 20669 T^{4} + 10611 T^{5} + 2196928 T^{6} + 10611 p T^{7} + 20669 p^{2} T^{8} + 382 p^{3} T^{9} + 114 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
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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

−10.45055346583592329462401095693, −9.930267503431574766805015088400, −9.850540473922797701672075212363, −9.596757290493155621182311957349, −9.389528532047503245348854437306, −9.316771376289671385980121419354, −9.136575573695839191221125418177, −8.115387743568871256004602444772, −7.965079999948813456966409270612, −7.83506387503764677641644822333, −7.75962509752452597228196744040, −7.74090712307790665573723381081, −7.03706179140409979728308016066, −7.02558343554993958193798692433, −6.16587374479562321953016828178, −5.99289058272781368313125466695, −5.96165855796953113502760425775, −5.66325346560503472403248174041, −5.63591874046606001267864523225, −5.08866170452463378308470265617, −4.67898505343392185417733475953, −4.45939968424702078647269062605, −3.76461178074972908672241363844, −2.78194491136502306249726834988, −2.46259427346092974642579421634, 2.46259427346092974642579421634, 2.78194491136502306249726834988, 3.76461178074972908672241363844, 4.45939968424702078647269062605, 4.67898505343392185417733475953, 5.08866170452463378308470265617, 5.63591874046606001267864523225, 5.66325346560503472403248174041, 5.96165855796953113502760425775, 5.99289058272781368313125466695, 6.16587374479562321953016828178, 7.02558343554993958193798692433, 7.03706179140409979728308016066, 7.74090712307790665573723381081, 7.75962509752452597228196744040, 7.83506387503764677641644822333, 7.965079999948813456966409270612, 8.115387743568871256004602444772, 9.136575573695839191221125418177, 9.316771376289671385980121419354, 9.389528532047503245348854437306, 9.596757290493155621182311957349, 9.850540473922797701672075212363, 9.930267503431574766805015088400, 10.45055346583592329462401095693

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

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