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Properties

Label 10-161e5-1.1-c1e5-0-0
Degree $10$
Conductor $108175616801$
Sign $1$
Analytic cond. $3.51167$
Root an. cond. $1.13383$
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  + 2·2-s + 3·4-s − 4·5-s + 5·7-s + 3·8-s − 2·9-s − 8·10-s − 4·11-s − 6·13-s + 10·14-s − 16-s − 12·17-s − 4·18-s + 6·19-s − 12·20-s − 8·22-s − 5·23-s + 5·25-s − 12·26-s + 15·28-s − 4·29-s + 30·31-s − 5·32-s − 24·34-s − 20·35-s − 6·36-s + 4·37-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s − 1.78·5-s + 1.88·7-s + 1.06·8-s − 2/3·9-s − 2.52·10-s − 1.20·11-s − 1.66·13-s + 2.67·14-s − 1/4·16-s − 2.91·17-s − 0.942·18-s + 1.37·19-s − 2.68·20-s − 1.70·22-s − 1.04·23-s + 25-s − 2.35·26-s + 2.83·28-s − 0.742·29-s + 5.38·31-s − 0.883·32-s − 4.11·34-s − 3.38·35-s − 36-s + 0.657·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{5} \cdot 23^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{5} \cdot 23^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(7^{5} \cdot 23^{5}\)
Sign: $1$
Analytic conductor: \(3.51167\)
Root analytic conductor: \(1.13383\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((10,\ 7^{5} \cdot 23^{5} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.899406470\)
\(L(\frac12)\) \(\approx\) \(1.899406470\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{5} \)
23$C_1$ \( ( 1 + T )^{5} \)
good2$C_2 \wr S_5$ \( 1 - p T + T^{2} + T^{3} + p T^{4} - 7 T^{5} + p^{2} T^{6} + p^{2} T^{7} + p^{3} T^{8} - p^{5} T^{9} + p^{5} T^{10} \)
3$C_2 \wr S_5$ \( 1 + 2 T^{2} + 11 T^{4} + 10 T^{5} + 11 p T^{6} + 2 p^{3} T^{8} + p^{5} T^{10} \)
5$C_2 \wr S_5$ \( 1 + 4 T + 11 T^{2} + 26 T^{3} + 92 T^{4} + 228 T^{5} + 92 p T^{6} + 26 p^{2} T^{7} + 11 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \)
11$C_2 \wr S_5$ \( 1 + 4 T + 27 T^{2} + 28 T^{3} + 126 T^{4} - 400 T^{5} + 126 p T^{6} + 28 p^{2} T^{7} + 27 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \)
13$C_2 \wr S_5$ \( 1 + 6 T + 56 T^{2} + 266 T^{3} + 1351 T^{4} + 4944 T^{5} + 1351 p T^{6} + 266 p^{2} T^{7} + 56 p^{3} T^{8} + 6 p^{4} T^{9} + p^{5} T^{10} \)
17$C_2 \wr S_5$ \( 1 + 12 T + 91 T^{2} + 430 T^{3} + 1692 T^{4} + 6148 T^{5} + 1692 p T^{6} + 430 p^{2} T^{7} + 91 p^{3} T^{8} + 12 p^{4} T^{9} + p^{5} T^{10} \)
19$C_2 \wr S_5$ \( 1 - 6 T + 67 T^{2} - 360 T^{3} + 2334 T^{4} - 9220 T^{5} + 2334 p T^{6} - 360 p^{2} T^{7} + 67 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} \)
29$C_2 \wr S_5$ \( 1 + 4 T + 34 T^{2} + 214 T^{3} + 1217 T^{4} + 4232 T^{5} + 1217 p T^{6} + 214 p^{2} T^{7} + 34 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \)
31$C_2 \wr S_5$ \( 1 - 30 T + 502 T^{2} - 5646 T^{3} + 46995 T^{4} - 297598 T^{5} + 46995 p T^{6} - 5646 p^{2} T^{7} + 502 p^{3} T^{8} - 30 p^{4} T^{9} + p^{5} T^{10} \)
37$C_2 \wr S_5$ \( 1 - 4 T + 109 T^{2} - 216 T^{3} + 5126 T^{4} - 5064 T^{5} + 5126 p T^{6} - 216 p^{2} T^{7} + 109 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
41$C_2 \wr S_5$ \( 1 - 6 T + 176 T^{2} - 838 T^{3} + 13295 T^{4} - 49000 T^{5} + 13295 p T^{6} - 838 p^{2} T^{7} + 176 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} \)
43$C_2 \wr S_5$ \( 1 + 12 T + 191 T^{2} + 1696 T^{3} + 16226 T^{4} + 101736 T^{5} + 16226 p T^{6} + 1696 p^{2} T^{7} + 191 p^{3} T^{8} + 12 p^{4} T^{9} + p^{5} T^{10} \)
47$C_2 \wr S_5$ \( 1 - 10 T + 110 T^{2} + 38 T^{3} - 3609 T^{4} + 58894 T^{5} - 3609 p T^{6} + 38 p^{2} T^{7} + 110 p^{3} T^{8} - 10 p^{4} T^{9} + p^{5} T^{10} \)
53$C_2 \wr S_5$ \( 1 - 16 T + 317 T^{2} - 3240 T^{3} + 35942 T^{4} - 254032 T^{5} + 35942 p T^{6} - 3240 p^{2} T^{7} + 317 p^{3} T^{8} - 16 p^{4} T^{9} + p^{5} T^{10} \)
59$C_2 \wr S_5$ \( 1 - 22 T + 7 p T^{2} - 5194 T^{3} + 54600 T^{4} - 458288 T^{5} + 54600 p T^{6} - 5194 p^{2} T^{7} + 7 p^{4} T^{8} - 22 p^{4} T^{9} + p^{5} T^{10} \)
61$C_2 \wr S_5$ \( 1 + 18 T + 339 T^{2} + 3954 T^{3} + 43744 T^{4} + 348488 T^{5} + 43744 p T^{6} + 3954 p^{2} T^{7} + 339 p^{3} T^{8} + 18 p^{4} T^{9} + p^{5} T^{10} \)
67$C_2 \wr S_5$ \( 1 + 2 T + 35 T^{2} - 204 T^{3} + 6790 T^{4} + 16644 T^{5} + 6790 p T^{6} - 204 p^{2} T^{7} + 35 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
71$C_2 \wr S_5$ \( 1 - 4 T + 254 T^{2} - 860 T^{3} + 30833 T^{4} - 86976 T^{5} + 30833 p T^{6} - 860 p^{2} T^{7} + 254 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
73$C_2 \wr S_5$ \( 1 + 2 T + 168 T^{2} + 946 T^{3} + 18175 T^{4} + 89144 T^{5} + 18175 p T^{6} + 946 p^{2} T^{7} + 168 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
79$C_2 \wr S_5$ \( 1 - 30 T + 703 T^{2} - 10676 T^{3} + 136374 T^{4} - 1310412 T^{5} + 136374 p T^{6} - 10676 p^{2} T^{7} + 703 p^{3} T^{8} - 30 p^{4} T^{9} + p^{5} T^{10} \)
83$C_2 \wr S_5$ \( 1 - 8 T + 359 T^{2} - 2224 T^{3} + 55778 T^{4} - 264336 T^{5} + 55778 p T^{6} - 2224 p^{2} T^{7} + 359 p^{3} T^{8} - 8 p^{4} T^{9} + p^{5} T^{10} \)
89$C_2 \wr S_5$ \( 1 + 20 T + 511 T^{2} + 6422 T^{3} + 92672 T^{4} + 821572 T^{5} + 92672 p T^{6} + 6422 p^{2} T^{7} + 511 p^{3} T^{8} + 20 p^{4} T^{9} + p^{5} T^{10} \)
97$C_2 \wr S_5$ \( 1 + 12 T + 371 T^{2} + 3294 T^{3} + 61432 T^{4} + 417340 T^{5} + 61432 p T^{6} + 3294 p^{2} T^{7} + 371 p^{3} T^{8} + 12 p^{4} T^{9} + p^{5} T^{10} \)
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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

−8.035771243439483362509867940648, −7.907471033152448265907195428918, −7.61489126818776635600391953912, −7.53526661867581573397086025621, −7.28290748898792270077344866140, −6.85984234081130562306068305763, −6.75646667751723041342704203637, −6.48632362644247565916978827873, −6.24688083086619781452494916208, −6.02759012649510552902070519615, −5.54680757206596033225223810660, −5.14530960437138326241383388487, −5.08445505576253656187737109081, −5.04521824085128416881283244476, −4.62782847782411448078646020745, −4.33270854012757008946164679016, −4.24611476617390687411547707736, −3.92277744644971369831156098576, −3.85986265226988419907850523999, −2.86259767353401268034234081739, −2.77720480966674936482102278629, −2.63621495415498626468537384881, −2.29996003692481769146752978162, −1.92414527917823031586076958206, −0.69557337749308828478254653264, 0.69557337749308828478254653264, 1.92414527917823031586076958206, 2.29996003692481769146752978162, 2.63621495415498626468537384881, 2.77720480966674936482102278629, 2.86259767353401268034234081739, 3.85986265226988419907850523999, 3.92277744644971369831156098576, 4.24611476617390687411547707736, 4.33270854012757008946164679016, 4.62782847782411448078646020745, 5.04521824085128416881283244476, 5.08445505576253656187737109081, 5.14530960437138326241383388487, 5.54680757206596033225223810660, 6.02759012649510552902070519615, 6.24688083086619781452494916208, 6.48632362644247565916978827873, 6.75646667751723041342704203637, 6.85984234081130562306068305763, 7.28290748898792270077344866140, 7.53526661867581573397086025621, 7.61489126818776635600391953912, 7.907471033152448265907195428918, 8.035771243439483362509867940648

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