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Properties

Label 10-91e10-1.1-c1e5-0-1
Degree $10$
Conductor $3.894\times 10^{19}$
Sign $-1$
Analytic cond. $1.26415\times 10^{9}$
Root an. cond. $8.13167$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $5$

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 + 3·16-s − 5·17-s + 24·18-s − 9·19-s − 14·20-s + 44·22-s + 10·23-s − 6·25-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 − 77·44-s + 12·45-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 + 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 − 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 − 11.6·44-s + 1.78·45-s + ⋯

Functional equation

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

Invariants

Degree: \(10\)
Conductor: \(7^{10} \cdot 13^{10}\)
Sign: $-1$
Analytic conductor: \(1.26415\times 10^{9}\)
Root analytic conductor: \(8.13167\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(5\)
Selberg data: \((10,\ 7^{10} \cdot 13^{10} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ -1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
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

−4.78886692787068515945565070587, −4.71252340065477595832335883337, −4.71177483625010844389279259328, −4.69396017967954693293597437543, −4.45092302787211012556151664671, −4.30388395410394458368668280784, −3.86423039121182627644882553609, −3.86207666121627903671376796732, −3.70576584173178193877144449576, −3.42963065462163923243534397810, −3.30368040025380187539949936712, −3.22845578793350396747517473944, −2.80458193265823700825759844304, −2.75726408968915537855891981205, −2.66962406394006875189031468462, −2.61321866803953122754878859460, −2.51816864270507658915657682372, −2.13586241878404068434996373492, −1.88663092786401848700713472869, −1.78892686518619849664152797520, −1.78820638119764745327803755263, −1.42959639566292443230172912160, −1.00669315699895309555803057258, −0.73532680424079142632935069130, −0.71204636625999536028728804519, 0, 0, 0, 0, 0, 0.71204636625999536028728804519, 0.73532680424079142632935069130, 1.00669315699895309555803057258, 1.42959639566292443230172912160, 1.78820638119764745327803755263, 1.78892686518619849664152797520, 1.88663092786401848700713472869, 2.13586241878404068434996373492, 2.51816864270507658915657682372, 2.61321866803953122754878859460, 2.66962406394006875189031468462, 2.75726408968915537855891981205, 2.80458193265823700825759844304, 3.22845578793350396747517473944, 3.30368040025380187539949936712, 3.42963065462163923243534397810, 3.70576584173178193877144449576, 3.86207666121627903671376796732, 3.86423039121182627644882553609, 4.30388395410394458368668280784, 4.45092302787211012556151664671, 4.69396017967954693293597437543, 4.71177483625010844389279259328, 4.71252340065477595832335883337, 4.78886692787068515945565070587

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