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Properties

Label 10-89e5-1.1-c1e5-0-0
Degree $10$
Conductor $5584059449$
Sign $1$
Analytic cond. $0.181274$
Root an. cond. $0.843011$
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-s − 3·3-s + 4-s − 5-s + 3·6-s + 8·7-s + 8-s − 2·9-s + 10-s + 6·11-s − 3·12-s − 8·14-s + 3·15-s − 4·16-s − 13·17-s + 2·18-s + 13·19-s − 20-s − 24·21-s − 6·22-s + 23-s − 3·24-s − 10·25-s + 19·27-s + 8·28-s + 2·29-s − 3·30-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.73·3-s + 1/2·4-s − 0.447·5-s + 1.22·6-s + 3.02·7-s + 0.353·8-s − 2/3·9-s + 0.316·10-s + 1.80·11-s − 0.866·12-s − 2.13·14-s + 0.774·15-s − 16-s − 3.15·17-s + 0.471·18-s + 2.98·19-s − 0.223·20-s − 5.23·21-s − 1.27·22-s + 0.208·23-s − 0.612·24-s − 2·25-s + 3.65·27-s + 1.51·28-s + 0.371·29-s − 0.547·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(89^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(89^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(89^{5}\)
Sign: $1$
Analytic conductor: \(0.181274\)
Root analytic conductor: \(0.843011\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((10,\ 89^{5} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.4165612504\)
\(L(\frac12)\) \(\approx\) \(0.4165612504\)
\(L(\frac{3}{2})\) 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)$
bad89$C_1$ \( ( 1 - T )^{5} \)
good2$C_2 \wr S_5$ \( 1 + T - p T^{3} + T^{4} + T^{5} + p T^{6} - p^{3} T^{7} + p^{4} T^{9} + p^{5} T^{10} \)
3$C_2 \wr S_5$ \( 1 + p T + 11 T^{2} + 20 T^{3} + 5 p^{2} T^{4} + 65 T^{5} + 5 p^{3} T^{6} + 20 p^{2} T^{7} + 11 p^{3} T^{8} + p^{5} T^{9} + p^{5} T^{10} \)
5$C_2 \wr S_5$ \( 1 + T + 11 T^{2} + 6 T^{3} + 69 T^{4} + 23 T^{5} + 69 p T^{6} + 6 p^{2} T^{7} + 11 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} \)
7$C_2 \wr S_5$ \( 1 - 8 T + 45 T^{2} - 188 T^{3} + 632 T^{4} - 260 p T^{5} + 632 p T^{6} - 188 p^{2} T^{7} + 45 p^{3} T^{8} - 8 p^{4} T^{9} + p^{5} T^{10} \)
11$C_2 \wr S_5$ \( 1 - 6 T + 35 T^{2} - 152 T^{3} + 630 T^{4} - 2004 T^{5} + 630 p T^{6} - 152 p^{2} T^{7} + 35 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} \)
13$C_2 \wr S_5$ \( 1 + 37 T^{2} - 56 T^{3} + 46 p T^{4} - 1440 T^{5} + 46 p^{2} T^{6} - 56 p^{2} T^{7} + 37 p^{3} T^{8} + p^{5} T^{10} \)
17$C_2 \wr S_5$ \( 1 + 13 T + 7 p T^{2} + 730 T^{3} + 3833 T^{4} + 16423 T^{5} + 3833 p T^{6} + 730 p^{2} T^{7} + 7 p^{4} T^{8} + 13 p^{4} T^{9} + p^{5} T^{10} \)
19$C_2 \wr S_5$ \( 1 - 13 T + 137 T^{2} - 946 T^{3} + 5707 T^{4} - 26363 T^{5} + 5707 p T^{6} - 946 p^{2} T^{7} + 137 p^{3} T^{8} - 13 p^{4} T^{9} + p^{5} T^{10} \)
23$C_2 \wr S_5$ \( 1 - T + 53 T^{2} + 58 T^{3} + 1643 T^{4} + 2069 T^{5} + 1643 p T^{6} + 58 p^{2} T^{7} + 53 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} \)
29$C_2 \wr S_5$ \( 1 - 2 T + 73 T^{2} + 80 T^{3} + 2098 T^{4} + 7220 T^{5} + 2098 p T^{6} + 80 p^{2} T^{7} + 73 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \)
31$C_2 \wr S_5$ \( 1 - 19 T + 257 T^{2} - 2470 T^{3} + 19109 T^{4} - 116615 T^{5} + 19109 p T^{6} - 2470 p^{2} T^{7} + 257 p^{3} T^{8} - 19 p^{4} T^{9} + p^{5} T^{10} \)
37$C_2 \wr S_5$ \( 1 + 14 T + 193 T^{2} + 1736 T^{3} + 14658 T^{4} + 91252 T^{5} + 14658 p T^{6} + 1736 p^{2} T^{7} + 193 p^{3} T^{8} + 14 p^{4} T^{9} + p^{5} T^{10} \)
41$C_2 \wr S_5$ \( 1 + 2 T + 145 T^{2} + 304 T^{3} + 10230 T^{4} + 17132 T^{5} + 10230 p T^{6} + 304 p^{2} T^{7} + 145 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
43$C_2 \wr S_5$ \( 1 - T + 147 T^{2} - 228 T^{3} + 10595 T^{4} - 14337 T^{5} + 10595 p T^{6} - 228 p^{2} T^{7} + 147 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} \)
47$C_2 \wr S_5$ \( 1 + 4 T + 191 T^{2} + 784 T^{3} + 15998 T^{4} + 56008 T^{5} + 15998 p T^{6} + 784 p^{2} T^{7} + 191 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \)
53$C_2 \wr S_5$ \( 1 + 11 T + 259 T^{2} + 1990 T^{3} + 26589 T^{4} + 150461 T^{5} + 26589 p T^{6} + 1990 p^{2} T^{7} + 259 p^{3} T^{8} + 11 p^{4} T^{9} + p^{5} T^{10} \)
59$C_2 \wr S_5$ \( 1 + 3 p T^{2} + 784 T^{3} + 12024 T^{4} + 94092 T^{5} + 12024 p T^{6} + 784 p^{2} T^{7} + 3 p^{4} T^{8} + p^{5} T^{10} \)
61$C_2 \wr S_5$ \( 1 - 4 T + 297 T^{2} - 952 T^{3} + 35762 T^{4} - 86392 T^{5} + 35762 p T^{6} - 952 p^{2} T^{7} + 297 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
67$C_2 \wr S_5$ \( 1 - 4 T + 199 T^{2} - 832 T^{3} + 22354 T^{4} - 73576 T^{5} + 22354 p T^{6} - 832 p^{2} T^{7} + 199 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
71$C_2 \wr S_5$ \( 1 + 2 T + 75 T^{2} - 56 T^{3} + 10050 T^{4} + 18892 T^{5} + 10050 p T^{6} - 56 p^{2} T^{7} + 75 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
73$C_2 \wr S_5$ \( 1 + 25 T + 551 T^{2} + 7534 T^{3} + 92429 T^{4} + 830039 T^{5} + 92429 p T^{6} + 7534 p^{2} T^{7} + 551 p^{3} T^{8} + 25 p^{4} T^{9} + p^{5} T^{10} \)
79$C_2 \wr S_5$ \( 1 - 54 T + 1491 T^{2} - 27416 T^{3} + 367554 T^{4} - 3732164 T^{5} + 367554 p T^{6} - 27416 p^{2} T^{7} + 1491 p^{3} T^{8} - 54 p^{4} T^{9} + p^{5} T^{10} \)
83$C_2 \wr S_5$ \( 1 + 20 T + 493 T^{2} + 6396 T^{3} + 88140 T^{4} + 786372 T^{5} + 88140 p T^{6} + 6396 p^{2} T^{7} + 493 p^{3} T^{8} + 20 p^{4} T^{9} + p^{5} T^{10} \)
97$C_2 \wr S_5$ \( 1 - 13 T + 355 T^{2} - 2294 T^{3} + 42401 T^{4} - 178803 T^{5} + 42401 p T^{6} - 2294 p^{2} T^{7} + 355 p^{3} T^{8} - 13 p^{4} T^{9} + p^{5} T^{10} \)
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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

−8.867916251429708211538924688531, −8.689788781707687792554591580430, −8.508260957313873413365094327982, −8.501901491841015067494698068757, −8.050516926738481523314434039074, −7.88611926396401112330469801069, −7.61779099427155181338745850111, −7.46851669293819603467494867547, −6.88395964427172721733780234031, −6.59052672984160179889265750991, −6.58266311964925475644416378412, −6.43568761622454221303857860629, −5.94986829848227808964312766630, −5.53222976586399877342488961564, −5.40863567161974978265036147088, −4.95283656456386803345642958433, −4.71415627639495115020629138393, −4.67021185486536516232972203166, −4.41029326705218714972973232598, −3.91926158936289874458722067095, −3.22594983474203489706154103308, −2.96274501696815526029123212601, −1.98034343477875218669591159961, −1.91341662104729503875907175725, −1.08802207448069004225823918985, 1.08802207448069004225823918985, 1.91341662104729503875907175725, 1.98034343477875218669591159961, 2.96274501696815526029123212601, 3.22594983474203489706154103308, 3.91926158936289874458722067095, 4.41029326705218714972973232598, 4.67021185486536516232972203166, 4.71415627639495115020629138393, 4.95283656456386803345642958433, 5.40863567161974978265036147088, 5.53222976586399877342488961564, 5.94986829848227808964312766630, 6.43568761622454221303857860629, 6.58266311964925475644416378412, 6.59052672984160179889265750991, 6.88395964427172721733780234031, 7.46851669293819603467494867547, 7.61779099427155181338745850111, 7.88611926396401112330469801069, 8.050516926738481523314434039074, 8.501901491841015067494698068757, 8.508260957313873413365094327982, 8.689788781707687792554591580430, 8.867916251429708211538924688531

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