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Properties

Label 10-4600e5-1.1-c1e5-0-2
Degree $10$
Conductor $2.060\times 10^{18}$
Sign $1$
Analytic cond. $6.68612\times 10^{7}$
Root an. cond. $6.06062$
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  + 4·7-s − 6·9-s + 4·13-s + 6·17-s − 5·23-s + 3·27-s + 12·29-s − 18·31-s + 10·37-s − 6·41-s + 10·43-s + 22·47-s − 2·49-s + 10·53-s − 59-s + 10·61-s − 24·63-s + 8·67-s + 8·71-s + 6·73-s + 9·81-s − 2·83-s + 14·89-s + 16·91-s + 6·97-s − 3·101-s + 2·103-s + ⋯
L(s)  = 1  + 1.51·7-s − 2·9-s + 1.10·13-s + 1.45·17-s − 1.04·23-s + 0.577·27-s + 2.22·29-s − 3.23·31-s + 1.64·37-s − 0.937·41-s + 1.52·43-s + 3.20·47-s − 2/7·49-s + 1.37·53-s − 0.130·59-s + 1.28·61-s − 3.02·63-s + 0.977·67-s + 0.949·71-s + 0.702·73-s + 81-s − 0.219·83-s + 1.48·89-s + 1.67·91-s + 0.609·97-s − 0.298·101-s + 0.197·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{15} \cdot 5^{10} \cdot 23^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{15} \cdot 5^{10} \cdot 23^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(2^{15} \cdot 5^{10} \cdot 23^{5}\)
Sign: $1$
Analytic conductor: \(6.68612\times 10^{7}\)
Root analytic conductor: \(6.06062\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((10,\ 2^{15} \cdot 5^{10} \cdot 23^{5} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(14.78660752\)
\(L(\frac12)\) \(\approx\) \(14.78660752\)
\(L(\frac{3}{2})\) 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)$
bad2 \( 1 \)
5 \( 1 \)
23$C_1$ \( ( 1 + T )^{5} \)
good3$C_2 \wr S_5$ \( 1 + 2 p T^{2} - p T^{3} + p^{3} T^{4} - 2 p T^{5} + p^{4} T^{6} - p^{3} T^{7} + 2 p^{4} T^{8} + p^{5} T^{10} \)
7$C_2 \wr S_5$ \( 1 - 4 T + 18 T^{2} - 36 T^{3} + 153 T^{4} - 312 T^{5} + 153 p T^{6} - 36 p^{2} T^{7} + 18 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
11$C_2 \wr S_5$ \( 1 + 40 T^{2} - 6 T^{3} + 763 T^{4} - 108 T^{5} + 763 p T^{6} - 6 p^{2} T^{7} + 40 p^{3} T^{8} + p^{5} T^{10} \)
13$C_2 \wr S_5$ \( 1 - 4 T + 33 T^{2} - 9 p T^{3} + 729 T^{4} - 2037 T^{5} + 729 p T^{6} - 9 p^{3} T^{7} + 33 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
17$C_2 \wr S_5$ \( 1 - 6 T + 37 T^{2} - 48 T^{3} - 230 T^{4} + 2220 T^{5} - 230 p T^{6} - 48 p^{2} T^{7} + 37 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} \)
19$C_2 \wr S_5$ \( 1 + 80 T^{2} - 6 T^{3} + 2803 T^{4} - 204 T^{5} + 2803 p T^{6} - 6 p^{2} T^{7} + 80 p^{3} T^{8} + p^{5} T^{10} \)
29$C_2 \wr S_5$ \( 1 - 12 T + 5 p T^{2} - 1053 T^{3} + 8137 T^{4} - 43677 T^{5} + 8137 p T^{6} - 1053 p^{2} T^{7} + 5 p^{4} T^{8} - 12 p^{4} T^{9} + p^{5} T^{10} \)
31$C_2 \wr S_5$ \( 1 + 18 T + 8 p T^{2} + 2277 T^{3} + 17743 T^{4} + 106350 T^{5} + 17743 p T^{6} + 2277 p^{2} T^{7} + 8 p^{4} T^{8} + 18 p^{4} T^{9} + p^{5} T^{10} \)
37$C_2 \wr S_5$ \( 1 - 10 T + 153 T^{2} - 984 T^{3} + 9210 T^{4} - 46044 T^{5} + 9210 p T^{6} - 984 p^{2} T^{7} + 153 p^{3} T^{8} - 10 p^{4} T^{9} + p^{5} T^{10} \)
41$C_2 \wr S_5$ \( 1 + 6 T + 139 T^{2} + 747 T^{3} + 10003 T^{4} + 41535 T^{5} + 10003 p T^{6} + 747 p^{2} T^{7} + 139 p^{3} T^{8} + 6 p^{4} T^{9} + p^{5} T^{10} \)
43$C_2 \wr S_5$ \( 1 - 10 T + 192 T^{2} - 1368 T^{3} + 15411 T^{4} - 82140 T^{5} + 15411 p T^{6} - 1368 p^{2} T^{7} + 192 p^{3} T^{8} - 10 p^{4} T^{9} + p^{5} T^{10} \)
47$C_2 \wr S_5$ \( 1 - 22 T + 332 T^{2} - 3835 T^{3} + 35433 T^{4} - 263290 T^{5} + 35433 p T^{6} - 3835 p^{2} T^{7} + 332 p^{3} T^{8} - 22 p^{4} T^{9} + p^{5} T^{10} \)
53$C_2 \wr S_5$ \( 1 - 10 T + 173 T^{2} - 1720 T^{3} + 14166 T^{4} - 125884 T^{5} + 14166 p T^{6} - 1720 p^{2} T^{7} + 173 p^{3} T^{8} - 10 p^{4} T^{9} + p^{5} T^{10} \)
59$C_2 \wr S_5$ \( 1 + T + 212 T^{2} + 343 T^{3} + 21441 T^{4} + 30508 T^{5} + 21441 p T^{6} + 343 p^{2} T^{7} + 212 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} \)
61$C_2 \wr S_5$ \( 1 - 10 T + 105 T^{2} - 1368 T^{3} + 13410 T^{4} - 84828 T^{5} + 13410 p T^{6} - 1368 p^{2} T^{7} + 105 p^{3} T^{8} - 10 p^{4} T^{9} + p^{5} T^{10} \)
67$C_2 \wr S_5$ \( 1 - 8 T + 147 T^{2} + 264 T^{3} - 1602 T^{4} + 116928 T^{5} - 1602 p T^{6} + 264 p^{2} T^{7} + 147 p^{3} T^{8} - 8 p^{4} T^{9} + p^{5} T^{10} \)
71$C_2 \wr S_5$ \( 1 - 8 T + 176 T^{2} - 1583 T^{3} + 18243 T^{4} - 136046 T^{5} + 18243 p T^{6} - 1583 p^{2} T^{7} + 176 p^{3} T^{8} - 8 p^{4} T^{9} + p^{5} T^{10} \)
73$C_2 \wr S_5$ \( 1 - 6 T + 143 T^{2} - 1713 T^{3} + 15079 T^{4} - 157581 T^{5} + 15079 p T^{6} - 1713 p^{2} T^{7} + 143 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} \)
79$C_2 \wr S_5$ \( 1 + 242 T^{2} - 36 T^{3} + 31333 T^{4} - 7416 T^{5} + 31333 p T^{6} - 36 p^{2} T^{7} + 242 p^{3} T^{8} + p^{5} T^{10} \)
83$C_2 \wr S_5$ \( 1 + 2 T + 224 T^{2} - 448 T^{3} + 21855 T^{4} - 99532 T^{5} + 21855 p T^{6} - 448 p^{2} T^{7} + 224 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
89$C_2 \wr S_5$ \( 1 - 14 T + 401 T^{2} - 3704 T^{3} + 62598 T^{4} - 434420 T^{5} + 62598 p T^{6} - 3704 p^{2} T^{7} + 401 p^{3} T^{8} - 14 p^{4} T^{9} + p^{5} T^{10} \)
97$C_2 \wr S_5$ \( 1 - 6 T + 341 T^{2} - 1512 T^{3} + 52762 T^{4} - 180324 T^{5} + 52762 p T^{6} - 1512 p^{2} T^{7} + 341 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} \)
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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

−4.82745333768920034223328865853, −4.80853747565466031015092006648, −4.79071292494401068504155742439, −4.45867154156994732650905069020, −4.22688832558102923691103565392, −3.97460475904813273782382496982, −3.88883917580553656383597929590, −3.81288269839144253535458725592, −3.71792069853896557984936132081, −3.48593461368859395516351503268, −3.16981120625340703592729183770, −3.03225650938068322147102048799, −2.90910733755853297359107939447, −2.79863612313931827073232021774, −2.54364560648639716520477317487, −2.19036557682398692104585216422, −2.10503048406690081825881614053, −2.05576486004353758940028588503, −1.69513452499762571237276808780, −1.69355216813757696917366459326, −1.18993679354799318998732468260, −0.848774523513021746028498612602, −0.821528322779340646549064741760, −0.54656808089251075008767709253, −0.51249814676406134814082655197, 0.51249814676406134814082655197, 0.54656808089251075008767709253, 0.821528322779340646549064741760, 0.848774523513021746028498612602, 1.18993679354799318998732468260, 1.69355216813757696917366459326, 1.69513452499762571237276808780, 2.05576486004353758940028588503, 2.10503048406690081825881614053, 2.19036557682398692104585216422, 2.54364560648639716520477317487, 2.79863612313931827073232021774, 2.90910733755853297359107939447, 3.03225650938068322147102048799, 3.16981120625340703592729183770, 3.48593461368859395516351503268, 3.71792069853896557984936132081, 3.81288269839144253535458725592, 3.88883917580553656383597929590, 3.97460475904813273782382496982, 4.22688832558102923691103565392, 4.45867154156994732650905069020, 4.79071292494401068504155742439, 4.80853747565466031015092006648, 4.82745333768920034223328865853

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