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Properties

Label 10-1071e5-1.1-c1e5-0-0
Degree $10$
Conductor $1.409\times 10^{15}$
Sign $1$
Analytic cond. $45743.8$
Root an. cond. $2.92437$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 2·4-s − 5·7-s − 2·8-s + 2·11-s + 2·13-s + 10·14-s − 2·16-s − 5·17-s + 6·19-s − 4·22-s + 10·23-s − 2·25-s − 4·26-s − 10·28-s + 8·29-s + 7·32-s + 10·34-s + 8·37-s − 12·38-s − 18·41-s + 8·43-s + 4·44-s − 20·46-s + 10·47-s + 15·49-s + 4·50-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s − 1.88·7-s − 0.707·8-s + 0.603·11-s + 0.554·13-s + 2.67·14-s − 1/2·16-s − 1.21·17-s + 1.37·19-s − 0.852·22-s + 2.08·23-s − 2/5·25-s − 0.784·26-s − 1.88·28-s + 1.48·29-s + 1.23·32-s + 1.71·34-s + 1.31·37-s − 1.94·38-s − 2.81·41-s + 1.21·43-s + 0.603·44-s − 2.94·46-s + 1.45·47-s + 15/7·49-s + 0.565·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{10} \cdot 7^{5} \cdot 17^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{10} \cdot 7^{5} \cdot 17^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(3^{10} \cdot 7^{5} \cdot 17^{5}\)
Sign: $1$
Analytic conductor: \(45743.8\)
Root analytic conductor: \(2.92437\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((10,\ 3^{10} \cdot 7^{5} \cdot 17^{5} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.471600926\)
\(L(\frac12)\) \(\approx\) \(1.471600926\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_1$ \( ( 1 + T )^{5} \)
17$C_1$ \( ( 1 + T )^{5} \)
good2$C_2 \wr S_5$ \( 1 + p T + p T^{2} + p T^{3} + 3 p T^{4} + 9 T^{5} + 3 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} + p^{5} T^{9} + p^{5} T^{10} \)
5$C_2 \wr S_5$ \( 1 + 2 T^{2} - 18 T^{3} + 36 T^{4} - 2 T^{5} + 36 p T^{6} - 18 p^{2} T^{7} + 2 p^{3} T^{8} + p^{5} T^{10} \)
11$C_2 \wr S_5$ \( 1 - 2 T + p T^{2} - 48 T^{3} + 254 T^{4} - 380 T^{5} + 254 p T^{6} - 48 p^{2} T^{7} + p^{4} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \)
13$C_2 \wr S_5$ \( 1 - 2 T + 25 T^{2} - 48 T^{3} + 482 T^{4} - 1116 T^{5} + 482 p T^{6} - 48 p^{2} T^{7} + 25 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \)
19$C_2 \wr S_5$ \( 1 - 6 T + 83 T^{2} - 400 T^{3} + 2974 T^{4} - 10932 T^{5} + 2974 p T^{6} - 400 p^{2} T^{7} + 83 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} \)
23$C_2 \wr S_5$ \( 1 - 10 T + 107 T^{2} - 776 T^{3} + 5010 T^{4} - 24988 T^{5} + 5010 p T^{6} - 776 p^{2} T^{7} + 107 p^{3} T^{8} - 10 p^{4} T^{9} + p^{5} T^{10} \)
29$C_2 \wr S_5$ \( 1 - 8 T + 73 T^{2} - 16 p T^{3} + 3362 T^{4} - 16048 T^{5} + 3362 p T^{6} - 16 p^{3} T^{7} + 73 p^{3} T^{8} - 8 p^{4} T^{9} + p^{5} T^{10} \)
31$C_2 \wr S_5$ \( 1 + 122 T^{2} - 94 T^{3} + 6464 T^{4} - 5844 T^{5} + 6464 p T^{6} - 94 p^{2} T^{7} + 122 p^{3} T^{8} + p^{5} T^{10} \)
37$C_2 \wr S_5$ \( 1 - 8 T + 81 T^{2} - 752 T^{3} + 5730 T^{4} - 29360 T^{5} + 5730 p T^{6} - 752 p^{2} T^{7} + 81 p^{3} T^{8} - 8 p^{4} T^{9} + p^{5} T^{10} \)
41$C_2 \wr S_5$ \( 1 + 18 T + 284 T^{2} + 3016 T^{3} + 26390 T^{4} + 186634 T^{5} + 26390 p T^{6} + 3016 p^{2} T^{7} + 284 p^{3} T^{8} + 18 p^{4} T^{9} + p^{5} T^{10} \)
43$C_2 \wr S_5$ \( 1 - 8 T + 184 T^{2} - 1160 T^{3} + 14648 T^{4} - 71228 T^{5} + 14648 p T^{6} - 1160 p^{2} T^{7} + 184 p^{3} T^{8} - 8 p^{4} T^{9} + p^{5} T^{10} \)
47$C_2 \wr S_5$ \( 1 - 10 T + 187 T^{2} - 1064 T^{3} + 12618 T^{4} - 53532 T^{5} + 12618 p T^{6} - 1064 p^{2} T^{7} + 187 p^{3} T^{8} - 10 p^{4} T^{9} + p^{5} T^{10} \)
53$C_2 \wr S_5$ \( 1 + 4 T + 232 T^{2} + 772 T^{3} + 23144 T^{4} + 59222 T^{5} + 23144 p T^{6} + 772 p^{2} T^{7} + 232 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \)
59$C_2 \wr S_5$ \( 1 + 8 T + 215 T^{2} + 1248 T^{3} + 20906 T^{4} + 94640 T^{5} + 20906 p T^{6} + 1248 p^{2} T^{7} + 215 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} \)
61$C_2 \wr S_5$ \( 1 - 22 T + 448 T^{2} - 5408 T^{3} + 61002 T^{4} - 490510 T^{5} + 61002 p T^{6} - 5408 p^{2} T^{7} + 448 p^{3} T^{8} - 22 p^{4} T^{9} + p^{5} T^{10} \)
67$C_2 \wr S_5$ \( 1 - 16 T + 384 T^{2} - 3984 T^{3} + 52992 T^{4} - 388340 T^{5} + 52992 p T^{6} - 3984 p^{2} T^{7} + 384 p^{3} T^{8} - 16 p^{4} T^{9} + p^{5} T^{10} \)
71$C_2 \wr S_5$ \( 1 - 2 T + 119 T^{2} + 304 T^{3} + 7614 T^{4} + 49636 T^{5} + 7614 p T^{6} + 304 p^{2} T^{7} + 119 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \)
73$C_2 \wr S_5$ \( 1 - 10 T + 188 T^{2} - 708 T^{3} + 10310 T^{4} - 7906 T^{5} + 10310 p T^{6} - 708 p^{2} T^{7} + 188 p^{3} T^{8} - 10 p^{4} T^{9} + p^{5} T^{10} \)
79$C_2 \wr S_5$ \( 1 - 18 T + 435 T^{2} - 5144 T^{3} + 69218 T^{4} - 585004 T^{5} + 69218 p T^{6} - 5144 p^{2} T^{7} + 435 p^{3} T^{8} - 18 p^{4} T^{9} + p^{5} T^{10} \)
83$C_2 \wr S_5$ \( 1 - 12 T + 351 T^{2} - 3032 T^{3} + 51082 T^{4} - 339960 T^{5} + 51082 p T^{6} - 3032 p^{2} T^{7} + 351 p^{3} T^{8} - 12 p^{4} T^{9} + p^{5} T^{10} \)
89$C_2 \wr S_5$ \( 1 + 20 T + 345 T^{2} + 3568 T^{3} + 38318 T^{4} + 310808 T^{5} + 38318 p T^{6} + 3568 p^{2} T^{7} + 345 p^{3} T^{8} + 20 p^{4} T^{9} + p^{5} T^{10} \)
97$C_2 \wr S_5$ \( 1 - 12 T + 246 T^{2} - 1890 T^{3} + 26704 T^{4} - 140626 T^{5} + 26704 p T^{6} - 1890 p^{2} T^{7} + 246 p^{3} T^{8} - 12 p^{4} T^{9} + p^{5} T^{10} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{10} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−6.12277928181555638927119791763, −5.91043727314138732023415863911, −5.56787873343245879746524978389, −5.56445135232467930104627945510, −5.34318490317540622857010272722, −5.05340582085850481872070777486, −4.79019524407272346299057270507, −4.66780330693902575057879829698, −4.47534413424811872047347786377, −4.38309781529412704613397285556, −3.79133324137922745833337569838, −3.71069896447623736204659261089, −3.70338840549253129316472031337, −3.42733643909814904550420526065, −3.09195625690659682642723465326, −2.96579243777451587405165608742, −2.73885673412750403632093243225, −2.51799675826562187779957782681, −2.13671444526211712241896816675, −2.07755020929834001197210001220, −1.75105383020301437125779422172, −1.06263679834800434929898506156, −1.01277777960950756433345751810, −0.64226962698804774321457072623, −0.45303037374101693397675516248, 0.45303037374101693397675516248, 0.64226962698804774321457072623, 1.01277777960950756433345751810, 1.06263679834800434929898506156, 1.75105383020301437125779422172, 2.07755020929834001197210001220, 2.13671444526211712241896816675, 2.51799675826562187779957782681, 2.73885673412750403632093243225, 2.96579243777451587405165608742, 3.09195625690659682642723465326, 3.42733643909814904550420526065, 3.70338840549253129316472031337, 3.71069896447623736204659261089, 3.79133324137922745833337569838, 4.38309781529412704613397285556, 4.47534413424811872047347786377, 4.66780330693902575057879829698, 4.79019524407272346299057270507, 5.05340582085850481872070777486, 5.34318490317540622857010272722, 5.56445135232467930104627945510, 5.56787873343245879746524978389, 5.91043727314138732023415863911, 6.12277928181555638927119791763

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