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Properties

Label 10-4235e5-1.1-c1e5-0-6
Degree $10$
Conductor $1.362\times 10^{18}$
Sign $-1$
Analytic cond. $4.42234\times 10^{7}$
Root an. cond. $5.81520$
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  − 2·2-s + 2·3-s − 4-s + 5·5-s − 4·6-s − 5·7-s + 6·8-s − 6·9-s − 10·10-s − 2·12-s − 8·13-s + 10·14-s + 10·15-s − 5·16-s − 6·17-s + 12·18-s − 7·19-s − 5·20-s − 10·21-s + 3·23-s + 12·24-s + 15·25-s + 16·26-s − 15·27-s + 5·28-s − 17·29-s − 20·30-s + ⋯
L(s)  = 1  − 1.41·2-s + 1.15·3-s − 1/2·4-s + 2.23·5-s − 1.63·6-s − 1.88·7-s + 2.12·8-s − 2·9-s − 3.16·10-s − 0.577·12-s − 2.21·13-s + 2.67·14-s + 2.58·15-s − 5/4·16-s − 1.45·17-s + 2.82·18-s − 1.60·19-s − 1.11·20-s − 2.18·21-s + 0.625·23-s + 2.44·24-s + 3·25-s + 3.13·26-s − 2.88·27-s + 0.944·28-s − 3.15·29-s − 3.65·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{5} \cdot 7^{5} \cdot 11^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{5} \cdot 7^{5} \cdot 11^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(5^{5} \cdot 7^{5} \cdot 11^{10}\)
Sign: $-1$
Analytic conductor: \(4.42234\times 10^{7}\)
Root analytic conductor: \(5.81520\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(5\)
Selberg data: \((10,\ 5^{5} \cdot 7^{5} \cdot 11^{10} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ -1 )\)

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)$
bad5$C_1$ \( ( 1 - T )^{5} \)
7$C_1$ \( ( 1 + T )^{5} \)
11 \( 1 \)
good2$C_2 \wr S_5$ \( 1 + p T + 5 T^{2} + 3 p T^{3} + 5 p T^{4} + 9 T^{5} + 5 p^{2} T^{6} + 3 p^{3} T^{7} + 5 p^{3} T^{8} + p^{5} T^{9} + p^{5} T^{10} \)
3$C_2 \wr S_5$ \( 1 - 2 T + 10 T^{2} - 17 T^{3} + 52 T^{4} - 23 p T^{5} + 52 p T^{6} - 17 p^{2} T^{7} + 10 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \)
13$C_2 \wr S_5$ \( 1 + 8 T + 69 T^{2} + 371 T^{3} + 1886 T^{4} + 6933 T^{5} + 1886 p T^{6} + 371 p^{2} T^{7} + 69 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} \)
17$C_2 \wr S_5$ \( 1 + 6 T + 23 T^{2} + 97 T^{3} + 740 T^{4} + 3295 T^{5} + 740 p T^{6} + 97 p^{2} T^{7} + 23 p^{3} T^{8} + 6 p^{4} T^{9} + p^{5} T^{10} \)
19$C_2 \wr S_5$ \( 1 + 7 T + 4 p T^{2} + 408 T^{3} + 2706 T^{4} + 10737 T^{5} + 2706 p T^{6} + 408 p^{2} T^{7} + 4 p^{4} T^{8} + 7 p^{4} T^{9} + p^{5} T^{10} \)
23$C_2 \wr S_5$ \( 1 - 3 T + 67 T^{2} - 377 T^{3} + 1951 T^{4} - 14137 T^{5} + 1951 p T^{6} - 377 p^{2} T^{7} + 67 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} \)
29$C_2 \wr S_5$ \( 1 + 17 T + 237 T^{2} + 2133 T^{3} + 16431 T^{4} + 95031 T^{5} + 16431 p T^{6} + 2133 p^{2} T^{7} + 237 p^{3} T^{8} + 17 p^{4} T^{9} + p^{5} T^{10} \)
31$C_2 \wr S_5$ \( 1 - 12 T + 136 T^{2} - 1027 T^{3} + 7372 T^{4} - 42433 T^{5} + 7372 p T^{6} - 1027 p^{2} T^{7} + 136 p^{3} T^{8} - 12 p^{4} T^{9} + p^{5} T^{10} \)
37$C_2 \wr S_5$ \( 1 + 2 T + 112 T^{2} + 453 T^{3} + 5742 T^{4} + 27635 T^{5} + 5742 p T^{6} + 453 p^{2} T^{7} + 112 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
41$C_2 \wr S_5$ \( 1 + 5 T + 135 T^{2} + 729 T^{3} + 9549 T^{4} + 41163 T^{5} + 9549 p T^{6} + 729 p^{2} T^{7} + 135 p^{3} T^{8} + 5 p^{4} T^{9} + p^{5} T^{10} \)
43$C_2 \wr S_5$ \( 1 + 4 T + 138 T^{2} + 851 T^{3} + 8512 T^{4} + 58393 T^{5} + 8512 p T^{6} + 851 p^{2} T^{7} + 138 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \)
47$C_2 \wr S_5$ \( 1 + 144 T^{2} + 40 T^{3} + 10803 T^{4} + 1187 T^{5} + 10803 p T^{6} + 40 p^{2} T^{7} + 144 p^{3} T^{8} + p^{5} T^{10} \)
53$C_2 \wr S_5$ \( 1 - 8 T + 176 T^{2} - 479 T^{3} + 9276 T^{4} + 59 T^{5} + 9276 p T^{6} - 479 p^{2} T^{7} + 176 p^{3} T^{8} - 8 p^{4} T^{9} + p^{5} T^{10} \)
59$C_2 \wr S_5$ \( 1 + 16 T + 309 T^{2} + 3319 T^{3} + 36450 T^{4} + 282083 T^{5} + 36450 p T^{6} + 3319 p^{2} T^{7} + 309 p^{3} T^{8} + 16 p^{4} T^{9} + p^{5} T^{10} \)
61$C_2 \wr S_5$ \( 1 + 24 T + 440 T^{2} + 5524 T^{3} + 59319 T^{4} + 497795 T^{5} + 59319 p T^{6} + 5524 p^{2} T^{7} + 440 p^{3} T^{8} + 24 p^{4} T^{9} + p^{5} T^{10} \)
67$C_2 \wr S_5$ \( 1 + 9 T + 280 T^{2} + 1866 T^{3} + 33498 T^{4} + 2527 p T^{5} + 33498 p T^{6} + 1866 p^{2} T^{7} + 280 p^{3} T^{8} + 9 p^{4} T^{9} + p^{5} T^{10} \)
71$C_2 \wr S_5$ \( 1 + 10 T + 100 T^{2} + 927 T^{3} + 10862 T^{4} + 132847 T^{5} + 10862 p T^{6} + 927 p^{2} T^{7} + 100 p^{3} T^{8} + 10 p^{4} T^{9} + p^{5} T^{10} \)
73$C_2 \wr S_5$ \( 1 + 11 T + 318 T^{2} + 2504 T^{3} + 40988 T^{4} + 246675 T^{5} + 40988 p T^{6} + 2504 p^{2} T^{7} + 318 p^{3} T^{8} + 11 p^{4} T^{9} + p^{5} T^{10} \)
79$C_2 \wr S_5$ \( 1 + 9 T + 248 T^{2} + 2620 T^{3} + 27880 T^{4} + 301773 T^{5} + 27880 p T^{6} + 2620 p^{2} T^{7} + 248 p^{3} T^{8} + 9 p^{4} T^{9} + p^{5} T^{10} \)
83$C_2 \wr S_5$ \( 1 + 10 T + 63 T^{2} + 505 T^{3} - 1036 T^{4} - 743 p T^{5} - 1036 p T^{6} + 505 p^{2} T^{7} + 63 p^{3} T^{8} + 10 p^{4} T^{9} + p^{5} T^{10} \)
89$C_2 \wr S_5$ \( 1 + 9 T + 376 T^{2} + 2187 T^{3} + 57289 T^{4} + 243527 T^{5} + 57289 p T^{6} + 2187 p^{2} T^{7} + 376 p^{3} T^{8} + 9 p^{4} T^{9} + p^{5} T^{10} \)
97$C_2 \wr S_5$ \( 1 - 11 T + 258 T^{2} - 2775 T^{3} + 36459 T^{4} - 379185 T^{5} + 36459 p T^{6} - 2775 p^{2} T^{7} + 258 p^{3} T^{8} - 11 p^{4} T^{9} + p^{5} T^{10} \)
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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

−5.42049005418412094937971998221, −5.25147119328122329303314963506, −4.99015394011700638046303149733, −4.92883563048022245586215875056, −4.87333078898600046084320193431, −4.38794917561730855980850855869, −4.30564949353422620068452773722, −4.30116019537131075817535330012, −4.22538423603848286221148617914, −4.06404762524259747440787156153, −3.50164222817974160671474442466, −3.39907739636554988906903799327, −3.07899729358329068650898053192, −3.06999622439817827632076581668, −3.02638818875737486681386195744, −2.89059041816905023459557071049, −2.61641217107731837924224236907, −2.40351610099818246152966978000, −2.21652041786518890032812559174, −2.11251482446648819272670600525, −2.04451814696164241182238491453, −1.57513401715968866560066632796, −1.55224747906821533473191267333, −1.07980545084374918005882586896, −1.01123106651860485255262457096, 0, 0, 0, 0, 0, 1.01123106651860485255262457096, 1.07980545084374918005882586896, 1.55224747906821533473191267333, 1.57513401715968866560066632796, 2.04451814696164241182238491453, 2.11251482446648819272670600525, 2.21652041786518890032812559174, 2.40351610099818246152966978000, 2.61641217107731837924224236907, 2.89059041816905023459557071049, 3.02638818875737486681386195744, 3.06999622439817827632076581668, 3.07899729358329068650898053192, 3.39907739636554988906903799327, 3.50164222817974160671474442466, 4.06404762524259747440787156153, 4.22538423603848286221148617914, 4.30116019537131075817535330012, 4.30564949353422620068452773722, 4.38794917561730855980850855869, 4.87333078898600046084320193431, 4.92883563048022245586215875056, 4.99015394011700638046303149733, 5.25147119328122329303314963506, 5.42049005418412094937971998221

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