Properties

Degree 14
Conductor $ 3^{14} \cdot 5^{7} \cdot 89^{7} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s + 5·4-s − 7·5-s − 16·7-s − 28·10-s + 10·11-s − 7·13-s − 64·14-s − 5·16-s + 13·17-s − 7·19-s − 35·20-s + 40·22-s + 13·23-s + 28·25-s − 28·26-s − 80·28-s + 4·29-s + 31-s − 7·32-s + 52·34-s + 112·35-s − 5·37-s − 28·38-s − 5·41-s − 31·43-s + 50·44-s + ⋯
L(s)  = 1  + 2.82·2-s + 5/2·4-s − 3.13·5-s − 6.04·7-s − 8.85·10-s + 3.01·11-s − 1.94·13-s − 17.1·14-s − 5/4·16-s + 3.15·17-s − 1.60·19-s − 7.82·20-s + 8.52·22-s + 2.71·23-s + 28/5·25-s − 5.49·26-s − 15.1·28-s + 0.742·29-s + 0.179·31-s − 1.23·32-s + 8.91·34-s + 18.9·35-s − 0.821·37-s − 4.54·38-s − 0.780·41-s − 4.72·43-s + 7.53·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{14} \cdot 5^{7} \cdot 89^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{14} \cdot 5^{7} \cdot 89^{7}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(14\)
\( N \)  =  \(3^{14} \cdot 5^{7} \cdot 89^{7}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{4005} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(14,\ 3^{14} \cdot 5^{7} \cdot 89^{7} ,\ ( \ : [1/2]^{7} ),\ 1 )$
$L(1)$  $\approx$  $3.535920734$
$L(\frac12)$  $\approx$  $3.535920734$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;89\}$,\(F_p(T)\) is a polynomial of degree 14. If $p \in \{3,\;5,\;89\}$, then $F_p(T)$ is a polynomial of degree at most 13.
$p$$F_p(T)$
bad3 \( 1 \)
5 \( ( 1 + T )^{7} \)
89 \( ( 1 + T )^{7} \)
good2 \( 1 - p^{2} T + 11 T^{2} - 3 p^{3} T^{3} + 23 p T^{4} - 77 T^{5} + 59 p T^{6} - 171 T^{7} + 59 p^{2} T^{8} - 77 p^{2} T^{9} + 23 p^{4} T^{10} - 3 p^{7} T^{11} + 11 p^{5} T^{12} - p^{8} T^{13} + p^{7} T^{14} \)
7 \( 1 + 16 T + 143 T^{2} + 908 T^{3} + 92 p^{2} T^{4} + 18272 T^{5} + 61958 T^{6} + 177804 T^{7} + 61958 p T^{8} + 18272 p^{2} T^{9} + 92 p^{5} T^{10} + 908 p^{4} T^{11} + 143 p^{5} T^{12} + 16 p^{6} T^{13} + p^{7} T^{14} \)
11 \( 1 - 10 T + 91 T^{2} - 511 T^{3} + 2672 T^{4} - 966 p T^{5} + 41830 T^{6} - 136602 T^{7} + 41830 p T^{8} - 966 p^{3} T^{9} + 2672 p^{3} T^{10} - 511 p^{4} T^{11} + 91 p^{5} T^{12} - 10 p^{6} T^{13} + p^{7} T^{14} \)
13 \( 1 + 7 T + 47 T^{2} + 168 T^{3} + 622 T^{4} + 2167 T^{5} + 9118 T^{6} + 36132 T^{7} + 9118 p T^{8} + 2167 p^{2} T^{9} + 622 p^{3} T^{10} + 168 p^{4} T^{11} + 47 p^{5} T^{12} + 7 p^{6} T^{13} + p^{7} T^{14} \)
17 \( 1 - 13 T + 108 T^{2} - 591 T^{3} + 3017 T^{4} - 14845 T^{5} + 78122 T^{6} - 344702 T^{7} + 78122 p T^{8} - 14845 p^{2} T^{9} + 3017 p^{3} T^{10} - 591 p^{4} T^{11} + 108 p^{5} T^{12} - 13 p^{6} T^{13} + p^{7} T^{14} \)
19 \( 1 + 7 T + 81 T^{2} + 366 T^{3} + 2784 T^{4} + 9257 T^{5} + 61132 T^{6} + 176376 T^{7} + 61132 p T^{8} + 9257 p^{2} T^{9} + 2784 p^{3} T^{10} + 366 p^{4} T^{11} + 81 p^{5} T^{12} + 7 p^{6} T^{13} + p^{7} T^{14} \)
23 \( 1 - 13 T + 144 T^{2} - 1105 T^{3} + 7943 T^{4} - 47855 T^{5} + 274740 T^{6} - 1359354 T^{7} + 274740 p T^{8} - 47855 p^{2} T^{9} + 7943 p^{3} T^{10} - 1105 p^{4} T^{11} + 144 p^{5} T^{12} - 13 p^{6} T^{13} + p^{7} T^{14} \)
29 \( 1 - 4 T + 113 T^{2} - 625 T^{3} + 6740 T^{4} - 40902 T^{5} + 266814 T^{6} - 1538922 T^{7} + 266814 p T^{8} - 40902 p^{2} T^{9} + 6740 p^{3} T^{10} - 625 p^{4} T^{11} + 113 p^{5} T^{12} - 4 p^{6} T^{13} + p^{7} T^{14} \)
31 \( 1 - T + 77 T^{2} - 34 T^{3} + 3570 T^{4} - 2367 T^{5} + 134894 T^{6} - 138904 T^{7} + 134894 p T^{8} - 2367 p^{2} T^{9} + 3570 p^{3} T^{10} - 34 p^{4} T^{11} + 77 p^{5} T^{12} - p^{6} T^{13} + p^{7} T^{14} \)
37 \( 1 + 5 T + 168 T^{2} + 803 T^{3} + 14161 T^{4} + 59989 T^{5} + 771494 T^{6} + 2748646 T^{7} + 771494 p T^{8} + 59989 p^{2} T^{9} + 14161 p^{3} T^{10} + 803 p^{4} T^{11} + 168 p^{5} T^{12} + 5 p^{6} T^{13} + p^{7} T^{14} \)
41 \( 1 + 5 T + 92 T^{2} + 673 T^{3} + 7237 T^{4} + 46413 T^{5} + 387726 T^{6} + 2290138 T^{7} + 387726 p T^{8} + 46413 p^{2} T^{9} + 7237 p^{3} T^{10} + 673 p^{4} T^{11} + 92 p^{5} T^{12} + 5 p^{6} T^{13} + p^{7} T^{14} \)
43 \( 1 + 31 T + 687 T^{2} + 10436 T^{3} + 129854 T^{4} + 1291529 T^{5} + 10962460 T^{6} + 77409324 T^{7} + 10962460 p T^{8} + 1291529 p^{2} T^{9} + 129854 p^{3} T^{10} + 10436 p^{4} T^{11} + 687 p^{5} T^{12} + 31 p^{6} T^{13} + p^{7} T^{14} \)
47 \( 1 - 14 T + 271 T^{2} - 2621 T^{3} + 31508 T^{4} - 240678 T^{5} + 2207362 T^{6} - 13822254 T^{7} + 2207362 p T^{8} - 240678 p^{2} T^{9} + 31508 p^{3} T^{10} - 2621 p^{4} T^{11} + 271 p^{5} T^{12} - 14 p^{6} T^{13} + p^{7} T^{14} \)
53 \( 1 - 13 T + 269 T^{2} - 2568 T^{3} + 34060 T^{4} - 265121 T^{5} + 2673394 T^{6} - 17174660 T^{7} + 2673394 p T^{8} - 265121 p^{2} T^{9} + 34060 p^{3} T^{10} - 2568 p^{4} T^{11} + 269 p^{5} T^{12} - 13 p^{6} T^{13} + p^{7} T^{14} \)
59 \( 1 - 14 T + 253 T^{2} - 2446 T^{3} + 31472 T^{4} - 260310 T^{5} + 2563464 T^{6} - 17624256 T^{7} + 2563464 p T^{8} - 260310 p^{2} T^{9} + 31472 p^{3} T^{10} - 2446 p^{4} T^{11} + 253 p^{5} T^{12} - 14 p^{6} T^{13} + p^{7} T^{14} \)
61 \( 1 - 3 T + 222 T^{2} - 759 T^{3} + 29403 T^{4} - 83651 T^{5} + 2524030 T^{6} - 6527094 T^{7} + 2524030 p T^{8} - 83651 p^{2} T^{9} + 29403 p^{3} T^{10} - 759 p^{4} T^{11} + 222 p^{5} T^{12} - 3 p^{6} T^{13} + p^{7} T^{14} \)
67 \( 1 - T + 328 T^{2} - 279 T^{3} + 53091 T^{4} - 39287 T^{5} + 5339384 T^{6} - 3265170 T^{7} + 5339384 p T^{8} - 39287 p^{2} T^{9} + 53091 p^{3} T^{10} - 279 p^{4} T^{11} + 328 p^{5} T^{12} - p^{6} T^{13} + p^{7} T^{14} \)
71 \( 1 - 8 T + 187 T^{2} - 1328 T^{3} + 22776 T^{4} - 168048 T^{5} + 2047314 T^{6} - 13028752 T^{7} + 2047314 p T^{8} - 168048 p^{2} T^{9} + 22776 p^{3} T^{10} - 1328 p^{4} T^{11} + 187 p^{5} T^{12} - 8 p^{6} T^{13} + p^{7} T^{14} \)
73 \( 1 - 9 T + 286 T^{2} - 1613 T^{3} + 37567 T^{4} - 166989 T^{5} + 3689378 T^{6} - 14466938 T^{7} + 3689378 p T^{8} - 166989 p^{2} T^{9} + 37567 p^{3} T^{10} - 1613 p^{4} T^{11} + 286 p^{5} T^{12} - 9 p^{6} T^{13} + p^{7} T^{14} \)
79 \( 1 - 9 T + 430 T^{2} - 2867 T^{3} + 81347 T^{4} - 413191 T^{5} + 9305818 T^{6} - 38366618 T^{7} + 9305818 p T^{8} - 413191 p^{2} T^{9} + 81347 p^{3} T^{10} - 2867 p^{4} T^{11} + 430 p^{5} T^{12} - 9 p^{6} T^{13} + p^{7} T^{14} \)
83 \( 1 - 42 T + 1008 T^{2} - 16390 T^{3} + 2485 p T^{4} - 2129142 T^{5} + 19880600 T^{6} - 179268272 T^{7} + 19880600 p T^{8} - 2129142 p^{2} T^{9} + 2485 p^{4} T^{10} - 16390 p^{4} T^{11} + 1008 p^{5} T^{12} - 42 p^{6} T^{13} + p^{7} T^{14} \)
97 \( 1 + 7 T + 609 T^{2} + 3724 T^{3} + 164856 T^{4} + 856227 T^{5} + 25710150 T^{6} + 108804964 T^{7} + 25710150 p T^{8} + 856227 p^{2} T^{9} + 164856 p^{3} T^{10} + 3724 p^{4} T^{11} + 609 p^{5} T^{12} + 7 p^{6} T^{13} + p^{7} T^{14} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{14} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−3.79212293051712988607555376318, −3.74683221530736935911329436194, −3.67030964179931905986957882141, −3.64642158549463268920430725952, −3.48102645695819032440271692892, −3.28075390742662929475083161135, −3.22515363721044854592135700249, −3.22152493562563395571105655414, −3.13308294132510319229821892928, −3.01389020421385741416430281131, −2.78843777062835554305536232888, −2.55705563754935317431589997304, −2.53960951102450828506834178394, −2.33994961611655180784545503647, −2.21669833011689861232081657837, −1.92367402177552971373473021125, −1.70020202082051440673147507162, −1.45617055868804096323483676381, −1.31586027371826108649607455120, −0.904603493956570931623112412073, −0.861202382816179963795175573148, −0.71471803262329141204066186509, −0.50813911943900697291309316977, −0.31640057740750242824242205185, −0.24563271759140740798653463895, 0.24563271759140740798653463895, 0.31640057740750242824242205185, 0.50813911943900697291309316977, 0.71471803262329141204066186509, 0.861202382816179963795175573148, 0.904603493956570931623112412073, 1.31586027371826108649607455120, 1.45617055868804096323483676381, 1.70020202082051440673147507162, 1.92367402177552971373473021125, 2.21669833011689861232081657837, 2.33994961611655180784545503647, 2.53960951102450828506834178394, 2.55705563754935317431589997304, 2.78843777062835554305536232888, 3.01389020421385741416430281131, 3.13308294132510319229821892928, 3.22152493562563395571105655414, 3.22515363721044854592135700249, 3.28075390742662929475083161135, 3.48102645695819032440271692892, 3.64642158549463268920430725952, 3.67030964179931905986957882141, 3.74683221530736935911329436194, 3.79212293051712988607555376318

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

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