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Properties

Label 10-637e5-1.1-c1e5-0-1
Degree $10$
Conductor $1.049\times 10^{14}$
Sign $1$
Analytic cond. $3404.73$
Root an. cond. $2.25532$
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  + 4·2-s + 7·4-s + 2·5-s + 7·8-s − 6·9-s + 8·10-s + 11·11-s + 5·13-s + 3·16-s − 5·17-s − 24·18-s + 9·19-s + 14·20-s + 44·22-s + 10·23-s − 6·25-s + 20·26-s − 3·29-s − 6·31-s − 3·32-s − 20·34-s − 42·36-s + 4·37-s + 36·38-s + 14·40-s + 14·41-s + 2·43-s + ⋯
L(s)  = 1  + 2.82·2-s + 7/2·4-s + 0.894·5-s + 2.47·8-s − 2·9-s + 2.52·10-s + 3.31·11-s + 1.38·13-s + 3/4·16-s − 1.21·17-s − 5.65·18-s + 2.06·19-s + 3.13·20-s + 9.38·22-s + 2.08·23-s − 6/5·25-s + 3.92·26-s − 0.557·29-s − 1.07·31-s − 0.530·32-s − 3.42·34-s − 7·36-s + 0.657·37-s + 5.83·38-s + 2.21·40-s + 2.18·41-s + 0.304·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{10} \cdot 13^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{10} \cdot 13^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(7^{10} \cdot 13^{5}\)
Sign: $1$
Analytic conductor: \(3404.73\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((10,\ 7^{10} \cdot 13^{5} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(26.18214034\)
\(L(\frac12)\) \(\approx\) \(26.18214034\)
\(L(\frac{3}{2})\) 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)$
bad7 \( 1 \)
13$C_1$ \( ( 1 - T )^{5} \)
good2$C_2 \wr S_5$ \( 1 - p^{2} T + 9 T^{2} - 15 T^{3} + 11 p T^{4} - 31 T^{5} + 11 p^{2} T^{6} - 15 p^{2} T^{7} + 9 p^{3} T^{8} - p^{6} T^{9} + p^{5} T^{10} \)
3$C_2 \wr S_5$ \( 1 + 2 p T^{2} + 25 T^{4} + 4 T^{5} + 25 p T^{6} + 2 p^{4} T^{8} + p^{5} T^{10} \)
5$C_2 \wr S_5$ \( 1 - 2 T + 2 p T^{2} - 4 p T^{3} + 73 T^{4} - 148 T^{5} + 73 p T^{6} - 4 p^{3} T^{7} + 2 p^{4} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \)
11$C_2 \wr S_5$ \( 1 - p T + 91 T^{2} - 46 p T^{3} + 2353 T^{4} - 767 p T^{5} + 2353 p T^{6} - 46 p^{3} T^{7} + 91 p^{3} T^{8} - p^{5} T^{9} + p^{5} T^{10} \)
17$C_2 \wr S_5$ \( 1 + 5 T + 63 T^{2} + 234 T^{3} + 1861 T^{4} + 5495 T^{5} + 1861 p T^{6} + 234 p^{2} T^{7} + 63 p^{3} T^{8} + 5 p^{4} T^{9} + p^{5} T^{10} \)
19$C_2 \wr S_5$ \( 1 - 9 T + 81 T^{2} - 508 T^{3} + 2985 T^{4} - 13029 T^{5} + 2985 p T^{6} - 508 p^{2} T^{7} + 81 p^{3} T^{8} - 9 p^{4} T^{9} + p^{5} T^{10} \)
23$C_2 \wr S_5$ \( 1 - 10 T + 146 T^{2} - 946 T^{3} + 7417 T^{4} - 32924 T^{5} + 7417 p T^{6} - 946 p^{2} T^{7} + 146 p^{3} T^{8} - 10 p^{4} T^{9} + p^{5} T^{10} \)
29$C_2 \wr S_5$ \( 1 + 3 T + 120 T^{2} + 329 T^{3} + 6379 T^{4} + 13928 T^{5} + 6379 p T^{6} + 329 p^{2} T^{7} + 120 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} \)
31$C_2 \wr S_5$ \( 1 + 6 T + 94 T^{2} + 642 T^{3} + 4445 T^{4} + 27916 T^{5} + 4445 p T^{6} + 642 p^{2} T^{7} + 94 p^{3} T^{8} + 6 p^{4} T^{9} + p^{5} T^{10} \)
37$C_2 \wr S_5$ \( 1 - 4 T + 2 p T^{2} + 86 T^{3} + 2029 T^{4} + 10280 T^{5} + 2029 p T^{6} + 86 p^{2} T^{7} + 2 p^{4} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
41$C_2 \wr S_5$ \( 1 - 14 T + 177 T^{2} - 1356 T^{3} + 10822 T^{4} - 65708 T^{5} + 10822 p T^{6} - 1356 p^{2} T^{7} + 177 p^{3} T^{8} - 14 p^{4} T^{9} + p^{5} T^{10} \)
43$C_2 \wr S_5$ \( 1 - 2 T + 143 T^{2} - 36 T^{3} + 8914 T^{4} + 4364 T^{5} + 8914 p T^{6} - 36 p^{2} T^{7} + 143 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \)
47$C_2 \wr S_5$ \( 1 - T + 111 T^{2} - 162 T^{3} + 7417 T^{4} - 15979 T^{5} + 7417 p T^{6} - 162 p^{2} T^{7} + 111 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} \)
53$C_2 \wr S_5$ \( 1 - 17 T + 191 T^{2} - 1178 T^{3} + 3565 T^{4} - 9403 T^{5} + 3565 p T^{6} - 1178 p^{2} T^{7} + 191 p^{3} T^{8} - 17 p^{4} T^{9} + p^{5} T^{10} \)
59$C_2 \wr S_5$ \( 1 - 11 T + 331 T^{2} - 2618 T^{3} + 41137 T^{4} - 232309 T^{5} + 41137 p T^{6} - 2618 p^{2} T^{7} + 331 p^{3} T^{8} - 11 p^{4} T^{9} + p^{5} T^{10} \)
61$C_2 \wr S_5$ \( 1 + 11 T + 3 p T^{2} + 1918 T^{3} + 20765 T^{4} + 143673 T^{5} + 20765 p T^{6} + 1918 p^{2} T^{7} + 3 p^{4} T^{8} + 11 p^{4} T^{9} + p^{5} T^{10} \)
67$C_2 \wr S_5$ \( 1 - 13 T + 173 T^{2} - 1324 T^{3} + 11737 T^{4} - 83401 T^{5} + 11737 p T^{6} - 1324 p^{2} T^{7} + 173 p^{3} T^{8} - 13 p^{4} T^{9} + p^{5} T^{10} \)
71$C_2 \wr S_5$ \( 1 - 15 T + 330 T^{2} - 3407 T^{3} + 44629 T^{4} - 338900 T^{5} + 44629 p T^{6} - 3407 p^{2} T^{7} + 330 p^{3} T^{8} - 15 p^{4} T^{9} + p^{5} T^{10} \)
73$C_2 \wr S_5$ \( 1 + 290 T^{2} + 42 T^{3} + 37565 T^{4} + 5420 T^{5} + 37565 p T^{6} + 42 p^{2} T^{7} + 290 p^{3} T^{8} + p^{5} T^{10} \)
79$C_2 \wr S_5$ \( 1 - 2 T + 258 T^{2} - 822 T^{3} + 31441 T^{4} - 105912 T^{5} + 31441 p T^{6} - 822 p^{2} T^{7} + 258 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \)
83$C_2 \wr S_5$ \( 1 - 6 T + 291 T^{2} - 1684 T^{3} + 40702 T^{4} - 204364 T^{5} + 40702 p T^{6} - 1684 p^{2} T^{7} + 291 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} \)
89$C_2 \wr S_5$ \( 1 + 4 T + 290 T^{2} + 730 T^{3} + 39973 T^{4} + 74264 T^{5} + 39973 p T^{6} + 730 p^{2} T^{7} + 290 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \)
97$C_2 \wr S_5$ \( 1 + 12 T + 469 T^{2} + 4044 T^{3} + 87194 T^{4} + 556336 T^{5} + 87194 p T^{6} + 4044 p^{2} T^{7} + 469 p^{3} T^{8} + 12 p^{4} T^{9} + p^{5} T^{10} \)
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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

−6.44117932276499983603005377882, −6.04606178243473389147000976931, −6.02883304303346113051358692948, −5.83501730523927416146340520122, −5.65871437213022781620982719046, −5.22509066933486439733252162204, −5.22160992529478884201996075507, −5.21836943092787905449818572111, −5.15561657819510436261130487641, −4.49434860961139003451501846553, −4.27510560068539662712211202578, −4.12836263864436883298067059119, −4.07052869479064403728805460215, −3.94629995237744310438685607732, −3.57686451437127174993477717016, −3.46232434429809137797318091278, −3.25133560467073639152613808583, −2.96031699784824758921161016485, −2.63643294720511834793319575028, −2.46106698038607966999029540644, −2.06131719640071757778923039939, −1.91451706145432172953967256008, −1.19271414644531081978401165037, −1.17932930021618709396735525087, −0.75742506270143291825187926934, 0.75742506270143291825187926934, 1.17932930021618709396735525087, 1.19271414644531081978401165037, 1.91451706145432172953967256008, 2.06131719640071757778923039939, 2.46106698038607966999029540644, 2.63643294720511834793319575028, 2.96031699784824758921161016485, 3.25133560467073639152613808583, 3.46232434429809137797318091278, 3.57686451437127174993477717016, 3.94629995237744310438685607732, 4.07052869479064403728805460215, 4.12836263864436883298067059119, 4.27510560068539662712211202578, 4.49434860961139003451501846553, 5.15561657819510436261130487641, 5.21836943092787905449818572111, 5.22160992529478884201996075507, 5.22509066933486439733252162204, 5.65871437213022781620982719046, 5.83501730523927416146340520122, 6.02883304303346113051358692948, 6.04606178243473389147000976931, 6.44117932276499983603005377882

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