↑

Properties

Label 10-9576e5-1.1-c1e5-0-1
Degree $10$
Conductor $8.052\times 10^{19}$
Sign $1$
Analytic cond. $2.61400\times 10^{9}$
Root an. cond. $8.74441$
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  − 2·5-s + 5·7-s − 2·11-s + 8·13-s + 2·17-s + 5·19-s − 2·23-s − 25-s − 4·29-s − 4·31-s − 10·35-s + 10·37-s + 6·41-s + 2·47-s + 15·49-s − 16·53-s + 4·55-s + 12·59-s + 30·61-s − 16·65-s + 18·67-s + 10·71-s + 14·73-s − 10·77-s − 2·79-s + 6·83-s − 4·85-s + ⋯
L(s)  = 1  − 0.894·5-s + 1.88·7-s − 0.603·11-s + 2.21·13-s + 0.485·17-s + 1.14·19-s − 0.417·23-s − 1/5·25-s − 0.742·29-s − 0.718·31-s − 1.69·35-s + 1.64·37-s + 0.937·41-s + 0.291·47-s + 15/7·49-s − 2.19·53-s + 0.539·55-s + 1.56·59-s + 3.84·61-s − 1.98·65-s + 2.19·67-s + 1.18·71-s + 1.63·73-s − 1.13·77-s − 0.225·79-s + 0.658·83-s − 0.433·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{15} \cdot 3^{10} \cdot 7^{5} \cdot 19^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{15} \cdot 3^{10} \cdot 7^{5} \cdot 19^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(2^{15} \cdot 3^{10} \cdot 7^{5} \cdot 19^{5}\)
Sign: $1$
Analytic conductor: \(2.61400\times 10^{9}\)
Root analytic conductor: \(8.74441\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((10,\ 2^{15} \cdot 3^{10} \cdot 7^{5} \cdot 19^{5} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(16.57703599\)
\(L(\frac12)\) \(\approx\) \(16.57703599\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3 \( 1 \)
7$C_1$ \( ( 1 - T )^{5} \)
19$C_1$ \( ( 1 - T )^{5} \)
good5$C_2 \wr S_5$ \( 1 + 2 T + p T^{2} + 8 T^{3} + 14 T^{4} + 44 T^{5} + 14 p T^{6} + 8 p^{2} T^{7} + p^{4} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
11$C_2 \wr S_5$ \( 1 + 2 T + 27 T^{2} + 48 T^{3} + 446 T^{4} + 700 T^{5} + 446 p T^{6} + 48 p^{2} T^{7} + 27 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
13$C_2 \wr S_5$ \( 1 - 8 T + 53 T^{2} - 16 p T^{3} + 870 T^{4} - 2768 T^{5} + 870 p T^{6} - 16 p^{3} T^{7} + 53 p^{3} T^{8} - 8 p^{4} T^{9} + p^{5} T^{10} \)
17$C_2 \wr S_5$ \( 1 - 2 T - 7 T^{2} + 56 T^{3} + 182 T^{4} - 2444 T^{5} + 182 p T^{6} + 56 p^{2} T^{7} - 7 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \)
23$C_2 \wr S_5$ \( 1 + 2 T + 79 T^{2} + 208 T^{3} + 2966 T^{4} + 7324 T^{5} + 2966 p T^{6} + 208 p^{2} T^{7} + 79 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
29$C_2 \wr S_5$ \( 1 + 4 T + 93 T^{2} + 200 T^{3} + 4014 T^{4} + 6248 T^{5} + 4014 p T^{6} + 200 p^{2} T^{7} + 93 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \)
31$C_2 \wr S_5$ \( 1 + 4 T + 75 T^{2} + 288 T^{3} + 3578 T^{4} + 9912 T^{5} + 3578 p T^{6} + 288 p^{2} T^{7} + 75 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \)
37$C_2 \wr S_5$ \( 1 - 10 T + 177 T^{2} - 1256 T^{3} + 12498 T^{4} - 65916 T^{5} + 12498 p T^{6} - 1256 p^{2} T^{7} + 177 p^{3} T^{8} - 10 p^{4} T^{9} + p^{5} T^{10} \)
41$C_2 \wr S_5$ \( 1 - 6 T + 133 T^{2} - 744 T^{3} + 8994 T^{4} - 41188 T^{5} + 8994 p T^{6} - 744 p^{2} T^{7} + 133 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} \)
43$C_2$ \( ( 1 + p T^{2} )^{5} \)
47$C_2 \wr S_5$ \( 1 - 2 T + 55 T^{2} - 456 T^{3} + 3334 T^{4} - 16300 T^{5} + 3334 p T^{6} - 456 p^{2} T^{7} + 55 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \)
53$C_2 \wr S_5$ \( 1 + 16 T + 309 T^{2} + 3288 T^{3} + 662 p T^{4} + 258672 T^{5} + 662 p^{2} T^{6} + 3288 p^{2} T^{7} + 309 p^{3} T^{8} + 16 p^{4} T^{9} + p^{5} T^{10} \)
59$C_2 \wr S_5$ \( 1 - 12 T + 39 T^{2} + 112 T^{3} + 506 T^{4} - 8712 T^{5} + 506 p T^{6} + 112 p^{2} T^{7} + 39 p^{3} T^{8} - 12 p^{4} T^{9} + p^{5} T^{10} \)
61$C_2$ \( ( 1 - 6 T + p T^{2} )^{5} \)
67$C_2 \wr S_5$ \( 1 - 18 T + 227 T^{2} - 1840 T^{3} + 12398 T^{4} - 74332 T^{5} + 12398 p T^{6} - 1840 p^{2} T^{7} + 227 p^{3} T^{8} - 18 p^{4} T^{9} + p^{5} T^{10} \)
71$C_2 \wr S_5$ \( 1 - 10 T + 183 T^{2} - 1000 T^{3} + 16654 T^{4} - 85468 T^{5} + 16654 p T^{6} - 1000 p^{2} T^{7} + 183 p^{3} T^{8} - 10 p^{4} T^{9} + p^{5} T^{10} \)
73$C_2 \wr S_5$ \( 1 - 14 T + 261 T^{2} - 3336 T^{3} + 36290 T^{4} - 328852 T^{5} + 36290 p T^{6} - 3336 p^{2} T^{7} + 261 p^{3} T^{8} - 14 p^{4} T^{9} + p^{5} T^{10} \)
79$C_2 \wr S_5$ \( 1 + 2 T + 239 T^{2} + 1136 T^{3} + 25310 T^{4} + 154012 T^{5} + 25310 p T^{6} + 1136 p^{2} T^{7} + 239 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
83$C_2 \wr S_5$ \( 1 - 6 T + 323 T^{2} - 1544 T^{3} + 47518 T^{4} - 177732 T^{5} + 47518 p T^{6} - 1544 p^{2} T^{7} + 323 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} \)
89$C_2 \wr S_5$ \( 1 + 10 T + 277 T^{2} + 3096 T^{3} + 36802 T^{4} + 398972 T^{5} + 36802 p T^{6} + 3096 p^{2} T^{7} + 277 p^{3} T^{8} + 10 p^{4} T^{9} + p^{5} T^{10} \)
97$C_2 \wr S_5$ \( 1 - 8 T + 313 T^{2} - 3760 T^{3} + 43334 T^{4} - 579024 T^{5} + 43334 p T^{6} - 3760 p^{2} T^{7} + 313 p^{3} T^{8} - 8 p^{4} T^{9} + p^{5} T^{10} \)
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

−4.31200151388813041918371951764, −4.21071162778156798931014871677, −4.20281252449408683236140833305, −4.14236618183931777645538345078, −3.90254779641570082860597331351, −3.68183160137058929057553483277, −3.60599521298646817845975881874, −3.57501215913349413776621631070, −3.46059325924665562834732104434, −3.16082321397427373006292418074, −2.91902880629184358501316294680, −2.64806692057089703019525312092, −2.60536847620827166720106523175, −2.56399640259234639467548118934, −2.23198625118616367925005824542, −2.07004711388864479692031643526, −1.86314391387657025371408245253, −1.60610180456851709967054703843, −1.59944121109028231427853471114, −1.41525172964452349384319530777, −1.04098793486253116509606413594, −0.789532545732488832147249260937, −0.70732303832711008265485724378, −0.67654009814019598512140254832, −0.31422576490895026381145820510, 0.31422576490895026381145820510, 0.67654009814019598512140254832, 0.70732303832711008265485724378, 0.789532545732488832147249260937, 1.04098793486253116509606413594, 1.41525172964452349384319530777, 1.59944121109028231427853471114, 1.60610180456851709967054703843, 1.86314391387657025371408245253, 2.07004711388864479692031643526, 2.23198625118616367925005824542, 2.56399640259234639467548118934, 2.60536847620827166720106523175, 2.64806692057089703019525312092, 2.91902880629184358501316294680, 3.16082321397427373006292418074, 3.46059325924665562834732104434, 3.57501215913349413776621631070, 3.60599521298646817845975881874, 3.68183160137058929057553483277, 3.90254779641570082860597331351, 4.14236618183931777645538345078, 4.20281252449408683236140833305, 4.21071162778156798931014871677, 4.31200151388813041918371951764

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