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Properties

Label 10-6384e5-1.1-c1e5-0-0
Degree $10$
Conductor $1.060\times 10^{19}$
Sign $1$
Analytic cond. $3.44230\times 10^{8}$
Root an. cond. $7.13978$
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  − 5·3-s − 2·5-s − 5·7-s + 15·9-s + 2·11-s + 8·13-s + 10·15-s − 2·17-s − 5·19-s + 25·21-s − 2·23-s − 5·25-s − 35·27-s − 8·31-s − 10·33-s + 10·35-s + 2·37-s − 40·39-s + 2·41-s − 20·43-s − 30·45-s + 26·47-s + 15·49-s + 10·51-s + 4·53-s − 4·55-s + 25·57-s + ⋯
L(s)  = 1  − 2.88·3-s − 0.894·5-s − 1.88·7-s + 5·9-s + 0.603·11-s + 2.21·13-s + 2.58·15-s − 0.485·17-s − 1.14·19-s + 5.45·21-s − 0.417·23-s − 25-s − 6.73·27-s − 1.43·31-s − 1.74·33-s + 1.69·35-s + 0.328·37-s − 6.40·39-s + 0.312·41-s − 3.04·43-s − 4.47·45-s + 3.79·47-s + 15/7·49-s + 1.40·51-s + 0.549·53-s − 0.539·55-s + 3.31·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{5} \cdot 7^{5} \cdot 19^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{5} \cdot 7^{5} \cdot 19^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(2^{20} \cdot 3^{5} \cdot 7^{5} \cdot 19^{5}\)
Sign: $1$
Analytic conductor: \(3.44230\times 10^{8}\)
Root analytic conductor: \(7.13978\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((10,\ 2^{20} \cdot 3^{5} \cdot 7^{5} \cdot 19^{5} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.3280175105\)
\(L(\frac12)\) \(\approx\) \(0.3280175105\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{5} \)
7$C_1$ \( ( 1 + T )^{5} \)
19$C_1$ \( ( 1 + T )^{5} \)
good5$C_2 \wr S_5$ \( 1 + 2 T + 9 T^{2} + 32 T^{3} + 78 T^{4} + 172 T^{5} + 78 p T^{6} + 32 p^{2} T^{7} + 9 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
11$C_2 \wr S_5$ \( 1 - 2 T + 23 T^{2} - 72 T^{3} + 410 T^{4} - 908 T^{5} + 410 p T^{6} - 72 p^{2} T^{7} + 23 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \)
13$C_2 \wr S_5$ \( 1 - 8 T + 57 T^{2} - 304 T^{3} + 1410 T^{4} - 5456 T^{5} + 1410 p T^{6} - 304 p^{2} T^{7} + 57 p^{3} T^{8} - 8 p^{4} T^{9} + p^{5} T^{10} \)
17$C_2 \wr S_5$ \( 1 + 2 T + 13 T^{2} - 40 T^{3} + 326 T^{4} + 652 T^{5} + 326 p T^{6} - 40 p^{2} T^{7} + 13 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
23$C_2 \wr S_5$ \( 1 + 2 T + 83 T^{2} + 168 T^{3} + 3338 T^{4} + 5420 T^{5} + 3338 p T^{6} + 168 p^{2} T^{7} + 83 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
29$C_2 \wr S_5$ \( 1 + 89 T^{2} - 80 T^{3} + 3566 T^{4} - 4616 T^{5} + 3566 p T^{6} - 80 p^{2} T^{7} + 89 p^{3} T^{8} + p^{5} T^{10} \)
31$C_2 \wr S_5$ \( 1 + 8 T + 147 T^{2} + 848 T^{3} + 8818 T^{4} + 37712 T^{5} + 8818 p T^{6} + 848 p^{2} T^{7} + 147 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} \)
37$C_2 \wr S_5$ \( 1 - 2 T + 65 T^{2} - 120 T^{3} + 2498 T^{4} - 2796 T^{5} + 2498 p T^{6} - 120 p^{2} T^{7} + 65 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \)
41$C_2 \wr S_5$ \( 1 - 2 T + 133 T^{2} - 424 T^{3} + 8130 T^{4} - 27948 T^{5} + 8130 p T^{6} - 424 p^{2} T^{7} + 133 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \)
43$C_2 \wr S_5$ \( 1 + 20 T + 247 T^{2} + 48 p T^{3} + 14346 T^{4} + 90360 T^{5} + 14346 p T^{6} + 48 p^{3} T^{7} + 247 p^{3} T^{8} + 20 p^{4} T^{9} + p^{5} T^{10} \)
47$C_2 \wr S_5$ \( 1 - 26 T + 459 T^{2} - 5480 T^{3} + 52838 T^{4} - 396604 T^{5} + 52838 p T^{6} - 5480 p^{2} T^{7} + 459 p^{3} T^{8} - 26 p^{4} T^{9} + p^{5} T^{10} \)
53$C_2 \wr S_5$ \( 1 - 4 T + 73 T^{2} - 600 T^{3} + 6374 T^{4} - 20752 T^{5} + 6374 p T^{6} - 600 p^{2} T^{7} + 73 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
59$C_2 \wr S_5$ \( 1 - 4 T + 167 T^{2} - 816 T^{3} + 16250 T^{4} - 62296 T^{5} + 16250 p T^{6} - 816 p^{2} T^{7} + 167 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
61$C_2 \wr S_5$ \( 1 - 10 T + 217 T^{2} - 1496 T^{3} + 20802 T^{4} - 111964 T^{5} + 20802 p T^{6} - 1496 p^{2} T^{7} + 217 p^{3} T^{8} - 10 p^{4} T^{9} + p^{5} T^{10} \)
67$C_2 \wr S_5$ \( 1 + 10 T + 199 T^{2} + 1272 T^{3} + 21074 T^{4} + 120732 T^{5} + 21074 p T^{6} + 1272 p^{2} T^{7} + 199 p^{3} T^{8} + 10 p^{4} T^{9} + p^{5} T^{10} \)
71$C_2 \wr S_5$ \( 1 + 10 T + 271 T^{2} + 1944 T^{3} + 32282 T^{4} + 179116 T^{5} + 32282 p T^{6} + 1944 p^{2} T^{7} + 271 p^{3} T^{8} + 10 p^{4} T^{9} + p^{5} T^{10} \)
73$C_2 \wr S_5$ \( 1 - 10 T + 245 T^{2} - 1176 T^{3} + 21778 T^{4} - 65148 T^{5} + 21778 p T^{6} - 1176 p^{2} T^{7} + 245 p^{3} T^{8} - 10 p^{4} T^{9} + p^{5} T^{10} \)
79$C_2 \wr S_5$ \( 1 + 14 T + 323 T^{2} + 3912 T^{3} + 46370 T^{4} + 448212 T^{5} + 46370 p T^{6} + 3912 p^{2} T^{7} + 323 p^{3} T^{8} + 14 p^{4} T^{9} + p^{5} T^{10} \)
83$C_2 \wr S_5$ \( 1 - 34 T + 663 T^{2} - 8416 T^{3} + 85166 T^{4} - 768716 T^{5} + 85166 p T^{6} - 8416 p^{2} T^{7} + 663 p^{3} T^{8} - 34 p^{4} T^{9} + p^{5} T^{10} \)
89$C_2 \wr S_5$ \( 1 - 10 T + 133 T^{2} - 1256 T^{3} + 20082 T^{4} - 179484 T^{5} + 20082 p T^{6} - 1256 p^{2} T^{7} + 133 p^{3} T^{8} - 10 p^{4} T^{9} + p^{5} T^{10} \)
97$C_2 \wr S_5$ \( 1 + 16 T + 173 T^{2} + 1664 T^{3} + 22834 T^{4} + 216544 T^{5} + 22834 p T^{6} + 1664 p^{2} T^{7} + 173 p^{3} T^{8} + 16 p^{4} T^{9} + p^{5} T^{10} \)
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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

−4.49919041497010732753465987712, −4.48930476882124282803637710296, −4.44046591573599378308665993380, −4.28653054112453111691406603636, −4.19921261962602187664557649066, −3.83402133383364309198559209104, −3.78791615265428549941336694980, −3.66079010128630687790906645005, −3.65419169143230312425766815124, −3.60304467203770059505512798966, −3.25159647159962005654596355864, −2.88997114071676526237611020342, −2.76647288009502644132400556475, −2.62914911744548290906027354813, −2.54217506892764501543328033973, −1.82915002875615828721646305766, −1.81173588924814516266101391912, −1.80298273763158818813662424946, −1.76301617134691467785093461596, −1.45903464840294614065694305411, −0.77940342341145294245560495960, −0.69768603543590329219666474989, −0.66240686522731780146336473024, −0.60746792158654045205217971617, −0.12205572937481378192866232192, 0.12205572937481378192866232192, 0.60746792158654045205217971617, 0.66240686522731780146336473024, 0.69768603543590329219666474989, 0.77940342341145294245560495960, 1.45903464840294614065694305411, 1.76301617134691467785093461596, 1.80298273763158818813662424946, 1.81173588924814516266101391912, 1.82915002875615828721646305766, 2.54217506892764501543328033973, 2.62914911744548290906027354813, 2.76647288009502644132400556475, 2.88997114071676526237611020342, 3.25159647159962005654596355864, 3.60304467203770059505512798966, 3.65419169143230312425766815124, 3.66079010128630687790906645005, 3.78791615265428549941336694980, 3.83402133383364309198559209104, 4.19921261962602187664557649066, 4.28653054112453111691406603636, 4.44046591573599378308665993380, 4.48930476882124282803637710296, 4.49919041497010732753465987712

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