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Properties

Label 10-1334e5-1.1-c1e5-0-1
Degree $10$
Conductor $4.225\times 10^{15}$
Sign $-1$
Analytic cond. $137140.$
Root an. cond. $3.26374$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $5$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 5·2-s + 3-s + 15·4-s − 5·5-s − 5·6-s − 2·7-s − 35·8-s − 6·9-s + 25·10-s + 11-s + 15·12-s − 11·13-s + 10·14-s − 5·15-s + 70·16-s + 4·17-s + 30·18-s − 12·19-s − 75·20-s − 2·21-s − 5·22-s + 5·23-s − 35·24-s − 2·25-s + 55·26-s − 9·27-s − 30·28-s + ⋯
L(s)  = 1  − 3.53·2-s + 0.577·3-s + 15/2·4-s − 2.23·5-s − 2.04·6-s − 0.755·7-s − 12.3·8-s − 2·9-s + 7.90·10-s + 0.301·11-s + 4.33·12-s − 3.05·13-s + 2.67·14-s − 1.29·15-s + 35/2·16-s + 0.970·17-s + 7.07·18-s − 2.75·19-s − 16.7·20-s − 0.436·21-s − 1.06·22-s + 1.04·23-s − 7.14·24-s − 2/5·25-s + 10.7·26-s − 1.73·27-s − 5.66·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{5} \cdot 23^{5} \cdot 29^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{5} \cdot 23^{5} \cdot 29^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(2^{5} \cdot 23^{5} \cdot 29^{5}\)
Sign: $-1$
Analytic conductor: \(137140.\)
Root analytic conductor: \(3.26374\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(5\)
Selberg data: \((10,\ 2^{5} \cdot 23^{5} \cdot 29^{5} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ -1 )\)

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)$
bad2$C_1$ \( ( 1 + T )^{5} \)
23$C_1$ \( ( 1 - T )^{5} \)
29$C_1$ \( ( 1 - T )^{5} \)
good3$C_2 \wr S_5$ \( 1 - T + 7 T^{2} - 4 T^{3} + 29 T^{4} - 13 T^{5} + 29 p T^{6} - 4 p^{2} T^{7} + 7 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} \)
5$C_2 \wr S_5$ \( 1 + p T + 27 T^{2} + 18 p T^{3} + 277 T^{4} + 651 T^{5} + 277 p T^{6} + 18 p^{3} T^{7} + 27 p^{3} T^{8} + p^{5} T^{9} + p^{5} T^{10} \)
7$C_2 \wr S_5$ \( 1 + 2 T + 27 T^{2} + 6 p T^{3} + 338 T^{4} + 414 T^{5} + 338 p T^{6} + 6 p^{3} T^{7} + 27 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
11$C_2 \wr S_5$ \( 1 - T + 21 T^{2} + 26 T^{3} + 151 T^{4} + 797 T^{5} + 151 p T^{6} + 26 p^{2} T^{7} + 21 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} \)
13$C_2 \wr S_5$ \( 1 + 11 T + 81 T^{2} + 438 T^{3} + 2111 T^{4} + 8253 T^{5} + 2111 p T^{6} + 438 p^{2} T^{7} + 81 p^{3} T^{8} + 11 p^{4} T^{9} + p^{5} T^{10} \)
17$C_2 \wr S_5$ \( 1 - 4 T + 29 T^{2} - 82 T^{3} + 40 p T^{4} - 1794 T^{5} + 40 p^{2} T^{6} - 82 p^{2} T^{7} + 29 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
19$C_2 \wr S_5$ \( 1 + 12 T + 121 T^{2} + 872 T^{3} + 5088 T^{4} + 24476 T^{5} + 5088 p T^{6} + 872 p^{2} T^{7} + 121 p^{3} T^{8} + 12 p^{4} T^{9} + p^{5} T^{10} \)
31$C_2 \wr S_5$ \( 1 + 5 T + 65 T^{2} - 38 T^{3} + 311 T^{4} - 12309 T^{5} + 311 p T^{6} - 38 p^{2} T^{7} + 65 p^{3} T^{8} + 5 p^{4} T^{9} + p^{5} T^{10} \)
37$C_2 \wr S_5$ \( 1 + 115 T^{2} + 82 T^{3} + 6540 T^{4} + 6034 T^{5} + 6540 p T^{6} + 82 p^{2} T^{7} + 115 p^{3} T^{8} + p^{5} T^{10} \)
41$C_2 \wr S_5$ \( 1 + 6 T + 67 T^{2} + 46 T^{3} + 716 T^{4} - 12954 T^{5} + 716 p T^{6} + 46 p^{2} T^{7} + 67 p^{3} T^{8} + 6 p^{4} T^{9} + p^{5} T^{10} \)
43$C_2 \wr S_5$ \( 1 + 7 T + 109 T^{2} + 830 T^{3} + 7979 T^{4} + 43257 T^{5} + 7979 p T^{6} + 830 p^{2} T^{7} + 109 p^{3} T^{8} + 7 p^{4} T^{9} + p^{5} T^{10} \)
47$C_2 \wr S_5$ \( 1 - 5 T + 195 T^{2} - 924 T^{3} + 16547 T^{4} - 64717 T^{5} + 16547 p T^{6} - 924 p^{2} T^{7} + 195 p^{3} T^{8} - 5 p^{4} T^{9} + p^{5} T^{10} \)
53$C_2 \wr S_5$ \( 1 + T + 91 T^{2} + 402 T^{3} + 6397 T^{4} + 20591 T^{5} + 6397 p T^{6} + 402 p^{2} T^{7} + 91 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} \)
59$C_2 \wr S_5$ \( 1 + 12 T + 91 T^{2} + 656 T^{3} + 3014 T^{4} - 3312 T^{5} + 3014 p T^{6} + 656 p^{2} T^{7} + 91 p^{3} T^{8} + 12 p^{4} T^{9} + p^{5} T^{10} \)
61$C_2 \wr S_5$ \( 1 + 20 T + 215 T^{2} + 1318 T^{3} + 7436 T^{4} + 43314 T^{5} + 7436 p T^{6} + 1318 p^{2} T^{7} + 215 p^{3} T^{8} + 20 p^{4} T^{9} + p^{5} T^{10} \)
67$C_2 \wr S_5$ \( 1 + 4 T + 189 T^{2} + 1372 T^{3} + 16862 T^{4} + 146098 T^{5} + 16862 p T^{6} + 1372 p^{2} T^{7} + 189 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \)
71$C_2 \wr S_5$ \( 1 + 4 T + 131 T^{2} + 208 T^{3} + 10794 T^{4} - 488 T^{5} + 10794 p T^{6} + 208 p^{2} T^{7} + 131 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \)
73$C_2 \wr S_5$ \( 1 - 4 T + 87 T^{2} + 244 T^{3} + 2844 T^{4} + 64110 T^{5} + 2844 p T^{6} + 244 p^{2} T^{7} + 87 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
79$C_2 \wr S_5$ \( 1 + T + 143 T^{2} + 824 T^{3} + 14257 T^{4} + 80637 T^{5} + 14257 p T^{6} + 824 p^{2} T^{7} + 143 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} \)
83$C_2 \wr S_5$ \( 1 - 22 T + 471 T^{2} - 5452 T^{3} + 66994 T^{4} - 564900 T^{5} + 66994 p T^{6} - 5452 p^{2} T^{7} + 471 p^{3} T^{8} - 22 p^{4} T^{9} + p^{5} T^{10} \)
89$C_2 \wr S_5$ \( 1 - 8 T + 289 T^{2} - 2208 T^{3} + 37212 T^{4} - 266790 T^{5} + 37212 p T^{6} - 2208 p^{2} T^{7} + 289 p^{3} T^{8} - 8 p^{4} T^{9} + p^{5} T^{10} \)
97$C_2 \wr S_5$ \( 1 + 46 T + 1301 T^{2} + 24812 T^{3} + 360074 T^{4} + 3992628 T^{5} + 360074 p T^{6} + 24812 p^{2} T^{7} + 1301 p^{3} T^{8} + 46 p^{4} T^{9} + p^{5} T^{10} \)
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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

−6.19977804894643360592538824872, −6.12837648925918115314368737128, −6.10104838859536239741567484487, −6.02073922220545041077559287206, −5.91847855967045210766437037775, −5.17527565028895333180432040569, −5.09942579779263774265585368354, −5.04356918860612027975406636387, −5.03972504405646154719767628873, −4.57877451447239730844354447556, −4.37902096165342156751143929137, −4.04418269448573426303692638671, −3.73969192555920505482568885472, −3.61049994991494313934052127874, −3.59421489314926515280920663276, −3.27369955456161043831708398565, −3.00783367308752479142225379246, −2.70644420108074226899315811588, −2.67249902453927827794535144265, −2.50285312409096883899484087150, −2.29770890664454914422425095925, −1.95902348021154892525180146900, −1.68022475861967904016597614803, −1.30921993052179508683228468690, −1.21077866954330363826115610317, 0, 0, 0, 0, 0, 1.21077866954330363826115610317, 1.30921993052179508683228468690, 1.68022475861967904016597614803, 1.95902348021154892525180146900, 2.29770890664454914422425095925, 2.50285312409096883899484087150, 2.67249902453927827794535144265, 2.70644420108074226899315811588, 3.00783367308752479142225379246, 3.27369955456161043831708398565, 3.59421489314926515280920663276, 3.61049994991494313934052127874, 3.73969192555920505482568885472, 4.04418269448573426303692638671, 4.37902096165342156751143929137, 4.57877451447239730844354447556, 5.03972504405646154719767628873, 5.04356918860612027975406636387, 5.09942579779263774265585368354, 5.17527565028895333180432040569, 5.91847855967045210766437037775, 6.02073922220545041077559287206, 6.10104838859536239741567484487, 6.12837648925918115314368737128, 6.19977804894643360592538824872

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