↑

Properties

Label 10-1911e5-1.1-c1e5-0-2
Degree $10$
Conductor $2.549\times 10^{16}$
Sign $1$
Analytic cond. $827349.$
Root an. cond. $3.90632$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 5·3-s + 3·5-s + 10·6-s − 2·8-s + 15·9-s + 6·10-s + 6·11-s + 5·13-s + 15·15-s − 3·16-s − 10·17-s + 30·18-s + 5·19-s + 12·22-s + 23-s − 10·24-s + 10·26-s + 35·27-s + 17·29-s + 30·30-s + 31-s − 4·32-s + 30·33-s − 20·34-s + 4·37-s + 10·38-s + ⋯
L(s)  = 1  + 1.41·2-s + 2.88·3-s + 1.34·5-s + 4.08·6-s − 0.707·8-s + 5·9-s + 1.89·10-s + 1.80·11-s + 1.38·13-s + 3.87·15-s − 3/4·16-s − 2.42·17-s + 7.07·18-s + 1.14·19-s + 2.55·22-s + 0.208·23-s − 2.04·24-s + 1.96·26-s + 6.73·27-s + 3.15·29-s + 5.47·30-s + 0.179·31-s − 0.707·32-s + 5.22·33-s − 3.42·34-s + 0.657·37-s + 1.62·38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{5} \cdot 7^{10} \cdot 13^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{5} \cdot 7^{10} \cdot 13^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(3^{5} \cdot 7^{10} \cdot 13^{5}\)
Sign: $1$
Analytic conductor: \(827349.\)
Root analytic conductor: \(3.90632\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((10,\ 3^{5} \cdot 7^{10} \cdot 13^{5} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(88.89242096\)
\(L(\frac12)\) \(\approx\) \(88.89242096\)
\(L(\frac{3}{2})\) 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)$
bad3$C_1$ \( ( 1 - T )^{5} \)
7 \( 1 \)
13$C_1$ \( ( 1 - T )^{5} \)
good2$C_2 \wr S_5$ \( 1 - p T + p^{2} T^{2} - 3 p T^{3} + 11 T^{4} - p^{4} T^{5} + 11 p T^{6} - 3 p^{3} T^{7} + p^{5} T^{8} - p^{5} T^{9} + p^{5} T^{10} \)
5$C_2 \wr S_5$ \( 1 - 3 T + 9 T^{2} - 4 p T^{3} + 6 p T^{4} - 54 T^{5} + 6 p^{2} T^{6} - 4 p^{3} T^{7} + 9 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} \)
11$C_2 \wr S_5$ \( 1 - 6 T + 39 T^{2} - 128 T^{3} + 502 T^{4} - 1372 T^{5} + 502 p T^{6} - 128 p^{2} T^{7} + 39 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} \)
17$C_2 \wr S_5$ \( 1 + 10 T + 53 T^{2} + 232 T^{3} + 1274 T^{4} + 5980 T^{5} + 1274 p T^{6} + 232 p^{2} T^{7} + 53 p^{3} T^{8} + 10 p^{4} T^{9} + p^{5} T^{10} \)
19$C_2 \wr S_5$ \( 1 - 5 T + 39 T^{2} + 36 T^{3} - 286 T^{4} + 5042 T^{5} - 286 p T^{6} + 36 p^{2} T^{7} + 39 p^{3} T^{8} - 5 p^{4} T^{9} + p^{5} T^{10} \)
23$C_2 \wr S_5$ \( 1 - T + 55 T^{2} - 116 T^{3} + 1694 T^{4} - 4694 T^{5} + 1694 p T^{6} - 116 p^{2} T^{7} + 55 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} \)
29$C_2 \wr S_5$ \( 1 - 17 T + 177 T^{2} - 1228 T^{3} + 242 p T^{4} - 36566 T^{5} + 242 p^{2} T^{6} - 1228 p^{2} T^{7} + 177 p^{3} T^{8} - 17 p^{4} T^{9} + p^{5} T^{10} \)
31$C_2 \wr S_5$ \( 1 - T + 107 T^{2} - 92 T^{3} + 5402 T^{4} - 3846 T^{5} + 5402 p T^{6} - 92 p^{2} T^{7} + 107 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} \)
37$C_2 \wr S_5$ \( 1 - 4 T + 97 T^{2} - 144 T^{3} + 3490 T^{4} + 360 T^{5} + 3490 p T^{6} - 144 p^{2} T^{7} + 97 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
41$C_2 \wr S_5$ \( 1 - 14 T + 173 T^{2} - 1648 T^{3} + 13862 T^{4} - 90956 T^{5} + 13862 p T^{6} - 1648 p^{2} T^{7} + 173 p^{3} T^{8} - 14 p^{4} T^{9} + p^{5} T^{10} \)
43$C_2 \wr S_5$ \( 1 + T + 143 T^{2} + 76 T^{3} + 10098 T^{4} + 4630 T^{5} + 10098 p T^{6} + 76 p^{2} T^{7} + 143 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} \)
47$C_2 \wr S_5$ \( 1 + 3 T + 63 T^{2} - 164 T^{3} + 5162 T^{4} + 8886 T^{5} + 5162 p T^{6} - 164 p^{2} T^{7} + 63 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} \)
53$C_2 \wr S_5$ \( 1 - 17 T + 281 T^{2} - 2988 T^{3} + 29466 T^{4} - 225334 T^{5} + 29466 p T^{6} - 2988 p^{2} T^{7} + 281 p^{3} T^{8} - 17 p^{4} T^{9} + p^{5} T^{10} \)
59$C_2 \wr S_5$ \( 1 + 8 T + 111 T^{2} + 736 T^{3} + 10894 T^{4} + 65984 T^{5} + 10894 p T^{6} + 736 p^{2} T^{7} + 111 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} \)
61$C_2 \wr S_5$ \( 1 + 18 T + 313 T^{2} + 3448 T^{3} + 37090 T^{4} + 295308 T^{5} + 37090 p T^{6} + 3448 p^{2} T^{7} + 313 p^{3} T^{8} + 18 p^{4} T^{9} + p^{5} T^{10} \)
67$C_2 \wr S_5$ \( 1 - 16 T + 319 T^{2} - 3456 T^{3} + 41290 T^{4} - 330720 T^{5} + 41290 p T^{6} - 3456 p^{2} T^{7} + 319 p^{3} T^{8} - 16 p^{4} T^{9} + p^{5} T^{10} \)
71$C_2 \wr S_5$ \( 1 + 16 T + 259 T^{2} + 2408 T^{3} + 24806 T^{4} + 179216 T^{5} + 24806 p T^{6} + 2408 p^{2} T^{7} + 259 p^{3} T^{8} + 16 p^{4} T^{9} + p^{5} T^{10} \)
73$C_2 \wr S_5$ \( 1 - 23 T + 525 T^{2} - 6964 T^{3} + 87738 T^{4} - 771786 T^{5} + 87738 p T^{6} - 6964 p^{2} T^{7} + 525 p^{3} T^{8} - 23 p^{4} T^{9} + p^{5} T^{10} \)
79$C_2 \wr S_5$ \( 1 - 3 T + 163 T^{2} - 20 p T^{3} + 14482 T^{4} - 186258 T^{5} + 14482 p T^{6} - 20 p^{3} T^{7} + 163 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} \)
83$C_2 \wr S_5$ \( 1 + 11 T + 331 T^{2} + 3612 T^{3} + 48066 T^{4} + 448086 T^{5} + 48066 p T^{6} + 3612 p^{2} T^{7} + 331 p^{3} T^{8} + 11 p^{4} T^{9} + p^{5} T^{10} \)
89$C_2 \wr S_5$ \( 1 + 35 T + 789 T^{2} + 12884 T^{3} + 167254 T^{4} + 1734086 T^{5} + 167254 p T^{6} + 12884 p^{2} T^{7} + 789 p^{3} T^{8} + 35 p^{4} T^{9} + p^{5} T^{10} \)
97$C_2 \wr S_5$ \( 1 - 27 T + 453 T^{2} - 5204 T^{3} + 49498 T^{4} - 446498 T^{5} + 49498 p T^{6} - 5204 p^{2} T^{7} + 453 p^{3} T^{8} - 27 p^{4} T^{9} + p^{5} T^{10} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{10} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−5.26105842826723202578343688175, −5.15032953101238092916470604939, −5.14513393771494117181060597308, −4.92276192356313005487559334605, −4.72563522752497256673146541647, −4.44162931248109931797296372852, −4.25297511335136044373640567742, −4.15474987903849962362905157752, −4.14645332058451061249425017480, −4.06100846370850389237241836001, −3.75978012763046835514091070946, −3.58404412583352911081190564222, −3.28469524352250821264678836790, −3.17818147481539020834179722043, −2.81127989708204524205399997932, −2.63586365205553203957023559549, −2.59875855707702307679318252279, −2.55747761969898080442141468259, −2.14493178733141205477688851685, −1.80191683249567918048936962061, −1.63283213968166080994241378839, −1.54256875830053036575373601403, −1.19624813318876455203423792625, −0.74355486434998916531227558738, −0.73984690079900993542128137302, 0.73984690079900993542128137302, 0.74355486434998916531227558738, 1.19624813318876455203423792625, 1.54256875830053036575373601403, 1.63283213968166080994241378839, 1.80191683249567918048936962061, 2.14493178733141205477688851685, 2.55747761969898080442141468259, 2.59875855707702307679318252279, 2.63586365205553203957023559549, 2.81127989708204524205399997932, 3.17818147481539020834179722043, 3.28469524352250821264678836790, 3.58404412583352911081190564222, 3.75978012763046835514091070946, 4.06100846370850389237241836001, 4.14645332058451061249425017480, 4.15474987903849962362905157752, 4.25297511335136044373640567742, 4.44162931248109931797296372852, 4.72563522752497256673146541647, 4.92276192356313005487559334605, 5.14513393771494117181060597308, 5.15032953101238092916470604939, 5.26105842826723202578343688175

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