↑

Properties

Label 10-381e5-1.1-c1e5-0-0
Degree $10$
Conductor $8.028\times 10^{12}$
Sign $1$
Analytic cond. $260.621$
Root an. cond. $1.74421$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 5·3-s + 5-s − 10·6-s − 2·8-s + 15·9-s + 2·10-s + 14·11-s − 5·13-s − 5·15-s − 16-s + 4·17-s + 30·18-s + 4·19-s + 28·22-s + 15·23-s + 10·24-s − 15·25-s − 10·26-s − 35·27-s + 9·29-s − 10·30-s + 3·31-s − 70·33-s + 8·34-s − 5·37-s + 8·38-s + ⋯
L(s)  = 1  + 1.41·2-s − 2.88·3-s + 0.447·5-s − 4.08·6-s − 0.707·8-s + 5·9-s + 0.632·10-s + 4.22·11-s − 1.38·13-s − 1.29·15-s − 1/4·16-s + 0.970·17-s + 7.07·18-s + 0.917·19-s + 5.96·22-s + 3.12·23-s + 2.04·24-s − 3·25-s − 1.96·26-s − 6.73·27-s + 1.67·29-s − 1.82·30-s + 0.538·31-s − 12.1·33-s + 1.37·34-s − 0.821·37-s + 1.29·38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{5} \cdot 127^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{5} \cdot 127^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(3^{5} \cdot 127^{5}\)
Sign: $1$
Analytic conductor: \(260.621\)
Root analytic conductor: \(1.74421\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((10,\ 3^{5} \cdot 127^{5} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.699984237\)
\(L(\frac12)\) \(\approx\) \(2.699984237\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad3$C_1$ \( ( 1 + T )^{5} \)
127$C_1$ \( ( 1 - T )^{5} \)
good2$C_2 \wr S_5$ \( 1 - p T + p^{2} T^{2} - 3 p T^{3} + 9 T^{4} - 3 p^{2} T^{5} + 9 p T^{6} - 3 p^{3} T^{7} + p^{5} T^{8} - p^{5} T^{9} + p^{5} T^{10} \)
5$C_2 \wr S_5$ \( 1 - T + 16 T^{2} - 9 T^{3} + 122 T^{4} - 41 T^{5} + 122 p T^{6} - 9 p^{2} T^{7} + 16 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} \)
7$C_2 \wr S_5$ \( 1 + 22 T^{2} - 4 T^{3} + 250 T^{4} - 54 T^{5} + 250 p T^{6} - 4 p^{2} T^{7} + 22 p^{3} T^{8} + p^{5} T^{10} \)
11$C_2 \wr S_5$ \( 1 - 14 T + 118 T^{2} - 712 T^{3} + 302 p T^{4} - 12268 T^{5} + 302 p^{2} T^{6} - 712 p^{2} T^{7} + 118 p^{3} T^{8} - 14 p^{4} T^{9} + p^{5} T^{10} \)
13$C_2 \wr S_5$ \( 1 + 5 T + 62 T^{2} + 227 T^{3} + 1564 T^{4} + 4231 T^{5} + 1564 p T^{6} + 227 p^{2} T^{7} + 62 p^{3} T^{8} + 5 p^{4} T^{9} + p^{5} T^{10} \)
17$C_2 \wr S_5$ \( 1 - 4 T + 53 T^{2} - 208 T^{3} + 1562 T^{4} - 4696 T^{5} + 1562 p T^{6} - 208 p^{2} T^{7} + 53 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
19$C_2 \wr S_5$ \( 1 - 4 T + 55 T^{2} - 240 T^{3} + 1522 T^{4} - 6488 T^{5} + 1522 p T^{6} - 240 p^{2} T^{7} + 55 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
23$C_2 \wr S_5$ \( 1 - 15 T + 179 T^{2} - 1384 T^{3} + 9258 T^{4} - 47202 T^{5} + 9258 p T^{6} - 1384 p^{2} T^{7} + 179 p^{3} T^{8} - 15 p^{4} T^{9} + p^{5} T^{10} \)
29$C_2 \wr S_5$ \( 1 - 9 T + 114 T^{2} - 697 T^{3} + 198 p T^{4} - 27567 T^{5} + 198 p^{2} T^{6} - 697 p^{2} T^{7} + 114 p^{3} T^{8} - 9 p^{4} T^{9} + p^{5} T^{10} \)
31$C_2 \wr S_5$ \( 1 - 3 T + 55 T^{2} - 376 T^{3} + 1734 T^{4} - 15706 T^{5} + 1734 p T^{6} - 376 p^{2} T^{7} + 55 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} \)
37$C_2 \wr S_5$ \( 1 + 5 T + 94 T^{2} + 87 T^{3} + 2920 T^{4} - 6345 T^{5} + 2920 p T^{6} + 87 p^{2} T^{7} + 94 p^{3} T^{8} + 5 p^{4} T^{9} + p^{5} T^{10} \)
41$C_2 \wr S_5$ \( 1 - 4 T + 89 T^{2} - 336 T^{3} + 3966 T^{4} - 16472 T^{5} + 3966 p T^{6} - 336 p^{2} T^{7} + 89 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
43$C_2 \wr S_5$ \( 1 - 10 T + 126 T^{2} - 974 T^{3} + 8670 T^{4} - 58056 T^{5} + 8670 p T^{6} - 974 p^{2} T^{7} + 126 p^{3} T^{8} - 10 p^{4} T^{9} + p^{5} T^{10} \)
47$C_2 \wr S_5$ \( 1 + 4 T + 186 T^{2} + 638 T^{3} + 15762 T^{4} + 42452 T^{5} + 15762 p T^{6} + 638 p^{2} T^{7} + 186 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \)
53$C_2 \wr S_5$ \( 1 - 3 T + 210 T^{2} - 331 T^{3} + 18870 T^{4} - 18021 T^{5} + 18870 p T^{6} - 331 p^{2} T^{7} + 210 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} \)
59$C_2 \wr S_5$ \( 1 - 23 T + 423 T^{2} - 5144 T^{3} + 54970 T^{4} - 448386 T^{5} + 54970 p T^{6} - 5144 p^{2} T^{7} + 423 p^{3} T^{8} - 23 p^{4} T^{9} + p^{5} T^{10} \)
61$C_2 \wr S_5$ \( 1 + 15 T + 114 T^{2} + 769 T^{3} + 9652 T^{4} + 101401 T^{5} + 9652 p T^{6} + 769 p^{2} T^{7} + 114 p^{3} T^{8} + 15 p^{4} T^{9} + p^{5} T^{10} \)
67$C_2 \wr S_5$ \( 1 - 18 T + 198 T^{2} - 842 T^{3} - 1242 T^{4} + 58612 T^{5} - 1242 p T^{6} - 842 p^{2} T^{7} + 198 p^{3} T^{8} - 18 p^{4} T^{9} + p^{5} T^{10} \)
71$C_2 \wr S_5$ \( 1 - 12 T + 374 T^{2} - 3302 T^{3} + 54322 T^{4} - 348006 T^{5} + 54322 p T^{6} - 3302 p^{2} T^{7} + 374 p^{3} T^{8} - 12 p^{4} T^{9} + p^{5} T^{10} \)
73$C_2 \wr S_5$ \( 1 + 43 T + 1030 T^{2} + 16873 T^{3} + 208660 T^{4} + 2006533 T^{5} + 208660 p T^{6} + 16873 p^{2} T^{7} + 1030 p^{3} T^{8} + 43 p^{4} T^{9} + p^{5} T^{10} \)
79$C_2 \wr S_5$ \( 1 - 16 T + 459 T^{2} - 4992 T^{3} + 77114 T^{4} - 589280 T^{5} + 77114 p T^{6} - 4992 p^{2} T^{7} + 459 p^{3} T^{8} - 16 p^{4} T^{9} + p^{5} T^{10} \)
83$C_2 \wr S_5$ \( 1 - 11 T + 239 T^{2} - 908 T^{3} + 14090 T^{4} + 13950 T^{5} + 14090 p T^{6} - 908 p^{2} T^{7} + 239 p^{3} T^{8} - 11 p^{4} T^{9} + p^{5} T^{10} \)
89$C_2 \wr S_5$ \( 1 - 9 T + 292 T^{2} - 2533 T^{3} + 46058 T^{4} - 295137 T^{5} + 46058 p T^{6} - 2533 p^{2} T^{7} + 292 p^{3} T^{8} - 9 p^{4} T^{9} + p^{5} T^{10} \)
97$C_2 \wr S_5$ \( 1 + 20 T + 513 T^{2} + 6480 T^{3} + 95726 T^{4} + 875640 T^{5} + 95726 p T^{6} + 6480 p^{2} T^{7} + 513 p^{3} T^{8} + 20 p^{4} T^{9} + p^{5} T^{10} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{10} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−6.89654338734222994969848541946, −6.63705863446918261982184080774, −6.44716678812268283764297419773, −6.28498033832619238285531494314, −6.13316256483590013361967559290, −6.11051504396111775663755261799, −5.51783641871794438086446835130, −5.42426411114909420688219141878, −5.33363774119557692493095933649, −5.25851696795298904772811143161, −4.76912157529016353728906159049, −4.60860164942559480964908538502, −4.46578228984830177267078167336, −4.33908683931741630421550683143, −4.24171092207917730202305869727, −3.78151993291473791399775376071, −3.44276225568264629494022461382, −3.36519238243805848197391976289, −3.25508213597345487875338324018, −2.50149868095198710052026801612, −2.07814505629374111329003509797, −1.64157834670226032765876295070, −1.31344207674110783166934010484, −1.00221365864898740273951875206, −0.69802288257484510620986445215, 0.69802288257484510620986445215, 1.00221365864898740273951875206, 1.31344207674110783166934010484, 1.64157834670226032765876295070, 2.07814505629374111329003509797, 2.50149868095198710052026801612, 3.25508213597345487875338324018, 3.36519238243805848197391976289, 3.44276225568264629494022461382, 3.78151993291473791399775376071, 4.24171092207917730202305869727, 4.33908683931741630421550683143, 4.46578228984830177267078167336, 4.60860164942559480964908538502, 4.76912157529016353728906159049, 5.25851696795298904772811143161, 5.33363774119557692493095933649, 5.42426411114909420688219141878, 5.51783641871794438086446835130, 6.11051504396111775663755261799, 6.13316256483590013361967559290, 6.28498033832619238285531494314, 6.44716678812268283764297419773, 6.63705863446918261982184080774, 6.89654338734222994969848541946

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