↑

Properties

Label 10-6003e5-1.1-c1e5-0-0
Degree $10$
Conductor $7.795\times 10^{18}$
Sign $1$
Analytic cond. $2.53062\times 10^{8}$
Root an. cond. $6.92345$
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 + 4-s + 3·5-s − 5·7-s − 8-s + 6·10-s + 8·11-s + 5·13-s − 10·14-s − 8·16-s − 2·17-s − 9·19-s + 3·20-s + 16·22-s − 5·23-s − 7·25-s + 10·26-s − 5·28-s − 5·29-s − 6·31-s − 13·32-s − 4·34-s − 15·35-s + 10·37-s − 18·38-s − 3·40-s + 11·41-s + ⋯
L(s)  = 1  + 1.41·2-s + 1/2·4-s + 1.34·5-s − 1.88·7-s − 0.353·8-s + 1.89·10-s + 2.41·11-s + 1.38·13-s − 2.67·14-s − 2·16-s − 0.485·17-s − 2.06·19-s + 0.670·20-s + 3.41·22-s − 1.04·23-s − 7/5·25-s + 1.96·26-s − 0.944·28-s − 0.928·29-s − 1.07·31-s − 2.29·32-s − 0.685·34-s − 2.53·35-s + 1.64·37-s − 2.91·38-s − 0.474·40-s + 1.71·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{10} \cdot 23^{5} \cdot 29^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{10} \cdot 23^{5} \cdot 29^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(3^{10} \cdot 23^{5} \cdot 29^{5}\)
Sign: $1$
Analytic conductor: \(2.53062\times 10^{8}\)
Root analytic conductor: \(6.92345\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((10,\ 3^{10} \cdot 23^{5} \cdot 29^{5} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(5.731548731\)
\(L(\frac12)\) \(\approx\) \(5.731548731\)
\(L(\frac{3}{2})\) 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 \( 1 \)
23$C_1$ \( ( 1 + T )^{5} \)
29$C_1$ \( ( 1 + T )^{5} \)
good2$C_2 \wr S_5$ \( 1 - p T + 3 T^{2} - 3 T^{3} + 9 T^{4} - 15 T^{5} + 9 p T^{6} - 3 p^{2} T^{7} + 3 p^{3} T^{8} - p^{5} T^{9} + p^{5} T^{10} \)
5$C_2 \wr S_5$ \( 1 - 3 T + 16 T^{2} - 32 T^{3} + 132 T^{4} - 227 T^{5} + 132 p T^{6} - 32 p^{2} T^{7} + 16 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} \)
7$C_2 \wr S_5$ \( 1 + 5 T + 33 T^{2} + 115 T^{3} + 447 T^{4} + 1151 T^{5} + 447 p T^{6} + 115 p^{2} T^{7} + 33 p^{3} T^{8} + 5 p^{4} T^{9} + p^{5} T^{10} \)
11$C_2 \wr S_5$ \( 1 - 8 T + 71 T^{2} - 359 T^{3} + 1734 T^{4} - 5961 T^{5} + 1734 p T^{6} - 359 p^{2} T^{7} + 71 p^{3} T^{8} - 8 p^{4} T^{9} + p^{5} T^{10} \)
13$C_2 \wr S_5$ \( 1 - 5 T + 61 T^{2} - 237 T^{3} + 1563 T^{4} - 4463 T^{5} + 1563 p T^{6} - 237 p^{2} T^{7} + 61 p^{3} T^{8} - 5 p^{4} T^{9} + p^{5} T^{10} \)
17$C_2 \wr S_5$ \( 1 + 2 T + 80 T^{2} + 125 T^{3} + 2634 T^{4} + 3097 T^{5} + 2634 p T^{6} + 125 p^{2} T^{7} + 80 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
19$C_2 \wr S_5$ \( 1 + 9 T + 89 T^{2} + 438 T^{3} + 2569 T^{4} + 9583 T^{5} + 2569 p T^{6} + 438 p^{2} T^{7} + 89 p^{3} T^{8} + 9 p^{4} T^{9} + p^{5} T^{10} \)
31$C_2 \wr S_5$ \( 1 + 6 T + 115 T^{2} + 685 T^{3} + 6096 T^{4} + 31077 T^{5} + 6096 p T^{6} + 685 p^{2} T^{7} + 115 p^{3} T^{8} + 6 p^{4} T^{9} + p^{5} T^{10} \)
37$C_2 \wr S_5$ \( 1 - 10 T + 161 T^{2} - 1277 T^{3} + 11230 T^{4} - 67939 T^{5} + 11230 p T^{6} - 1277 p^{2} T^{7} + 161 p^{3} T^{8} - 10 p^{4} T^{9} + p^{5} T^{10} \)
41$C_2 \wr S_5$ \( 1 - 11 T + 184 T^{2} - 1340 T^{3} + 12920 T^{4} - 71977 T^{5} + 12920 p T^{6} - 1340 p^{2} T^{7} + 184 p^{3} T^{8} - 11 p^{4} T^{9} + p^{5} T^{10} \)
43$C_2 \wr S_5$ \( 1 + 9 T + 140 T^{2} + 922 T^{3} + 10096 T^{4} + 53457 T^{5} + 10096 p T^{6} + 922 p^{2} T^{7} + 140 p^{3} T^{8} + 9 p^{4} T^{9} + p^{5} T^{10} \)
47$C_2 \wr S_5$ \( 1 - 13 T + 90 T^{2} - 567 T^{3} + 5553 T^{4} - 45003 T^{5} + 5553 p T^{6} - 567 p^{2} T^{7} + 90 p^{3} T^{8} - 13 p^{4} T^{9} + p^{5} T^{10} \)
53$C_2 \wr S_5$ \( 1 + T + 216 T^{2} + 192 T^{3} + 20850 T^{4} + 14373 T^{5} + 20850 p T^{6} + 192 p^{2} T^{7} + 216 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} \)
59$C_2 \wr S_5$ \( 1 + 6 T + 91 T^{2} + 8 T^{3} - 4194 T^{4} - 42652 T^{5} - 4194 p T^{6} + 8 p^{2} T^{7} + 91 p^{3} T^{8} + 6 p^{4} T^{9} + p^{5} T^{10} \)
61$C_2 \wr S_5$ \( 1 - 23 T + 389 T^{2} - 4749 T^{3} + 48365 T^{4} - 408493 T^{5} + 48365 p T^{6} - 4749 p^{2} T^{7} + 389 p^{3} T^{8} - 23 p^{4} T^{9} + p^{5} T^{10} \)
67$C_2 \wr S_5$ \( 1 + 10 T + 202 T^{2} + 1197 T^{3} + 16424 T^{4} + 70105 T^{5} + 16424 p T^{6} + 1197 p^{2} T^{7} + 202 p^{3} T^{8} + 10 p^{4} T^{9} + p^{5} T^{10} \)
71$C_2 \wr S_5$ \( 1 - 11 T + 357 T^{2} - 3005 T^{3} + 50921 T^{4} - 315865 T^{5} + 50921 p T^{6} - 3005 p^{2} T^{7} + 357 p^{3} T^{8} - 11 p^{4} T^{9} + p^{5} T^{10} \)
73$C_2 \wr S_5$ \( 1 - 31 T + 648 T^{2} - 9721 T^{3} + 114923 T^{4} - 1087551 T^{5} + 114923 p T^{6} - 9721 p^{2} T^{7} + 648 p^{3} T^{8} - 31 p^{4} T^{9} + p^{5} T^{10} \)
79$C_2 \wr S_5$ \( 1 - 8 T + 333 T^{2} - 2069 T^{3} + 48048 T^{4} - 229967 T^{5} + 48048 p T^{6} - 2069 p^{2} T^{7} + 333 p^{3} T^{8} - 8 p^{4} T^{9} + p^{5} T^{10} \)
83$C_2 \wr S_5$ \( 1 + 7 T + 130 T^{2} + 1342 T^{3} + 15890 T^{4} + 85583 T^{5} + 15890 p T^{6} + 1342 p^{2} T^{7} + 130 p^{3} T^{8} + 7 p^{4} T^{9} + p^{5} T^{10} \)
89$C_2 \wr S_5$ \( 1 + 3 T + 295 T^{2} + 1825 T^{3} + 37881 T^{4} + 278011 T^{5} + 37881 p T^{6} + 1825 p^{2} T^{7} + 295 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} \)
97$C_2 \wr S_5$ \( 1 - 3 T + 217 T^{2} + 949 T^{3} + 10747 T^{4} + 244897 T^{5} + 10747 p T^{6} + 949 p^{2} T^{7} + 217 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} \)
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

−4.43744899661310374654275864577, −4.43123955556116798054274042362, −4.41592624948539382872632009721, −4.38333084921669531294534803832, −4.16379948005214299826485468050, −3.84153613724456801363398713425, −3.76576008429150312664728388752, −3.74462250829620736519922825864, −3.66285829373687060448852572224, −3.57786336273429890426853870673, −3.22124816691401522485198495773, −2.93358738684096659864053347802, −2.91203876691062904076612220578, −2.61828340321222722327521360351, −2.25452938494520594431259026717, −2.22351418273919309448779457440, −2.15438108284872180995596219527, −2.04465523259547825795701381960, −1.77806326085170616051252296932, −1.55640174135404040268195725499, −1.27184636647894386110002681967, −1.12109256493675249475466339954, −0.71157951596913940535758396694, −0.36345353685894041128122060480, −0.24655011630279649231392011666, 0.24655011630279649231392011666, 0.36345353685894041128122060480, 0.71157951596913940535758396694, 1.12109256493675249475466339954, 1.27184636647894386110002681967, 1.55640174135404040268195725499, 1.77806326085170616051252296932, 2.04465523259547825795701381960, 2.15438108284872180995596219527, 2.22351418273919309448779457440, 2.25452938494520594431259026717, 2.61828340321222722327521360351, 2.91203876691062904076612220578, 2.93358738684096659864053347802, 3.22124816691401522485198495773, 3.57786336273429890426853870673, 3.66285829373687060448852572224, 3.74462250829620736519922825864, 3.76576008429150312664728388752, 3.84153613724456801363398713425, 4.16379948005214299826485468050, 4.38333084921669531294534803832, 4.41592624948539382872632009721, 4.43123955556116798054274042362, 4.43744899661310374654275864577

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