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Properties

Label 10-4140e5-1.1-c1e5-0-1
Degree $10$
Conductor $1.216\times 10^{18}$
Sign $1$
Analytic cond. $3.94809\times 10^{7}$
Root an. cond. $5.74961$
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  + 5·5-s + 4·7-s + 4·11-s + 2·13-s + 2·17-s + 6·19-s + 5·23-s + 15·25-s + 2·29-s + 8·31-s + 20·35-s + 8·37-s + 2·41-s + 18·43-s − 4·47-s − 3·49-s − 2·53-s + 20·55-s + 6·59-s + 10·61-s + 10·65-s + 16·67-s + 10·71-s + 18·73-s + 16·77-s + 14·79-s + 8·83-s + ⋯
L(s)  = 1  + 2.23·5-s + 1.51·7-s + 1.20·11-s + 0.554·13-s + 0.485·17-s + 1.37·19-s + 1.04·23-s + 3·25-s + 0.371·29-s + 1.43·31-s + 3.38·35-s + 1.31·37-s + 0.312·41-s + 2.74·43-s − 0.583·47-s − 3/7·49-s − 0.274·53-s + 2.69·55-s + 0.781·59-s + 1.28·61-s + 1.24·65-s + 1.95·67-s + 1.18·71-s + 2.10·73-s + 1.82·77-s + 1.57·79-s + 0.878·83-s + ⋯

Functional equation

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

Invariants

Degree: \(10\)
Conductor: \(2^{10} \cdot 3^{10} \cdot 5^{5} \cdot 23^{5}\)
Sign: $1$
Analytic conductor: \(3.94809\times 10^{7}\)
Root analytic conductor: \(5.74961\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((10,\ 2^{10} \cdot 3^{10} \cdot 5^{5} \cdot 23^{5} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(55.92035220\)
\(L(\frac12)\) \(\approx\) \(55.92035220\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_1$ \( ( 1 - T )^{5} \)
23$C_1$ \( ( 1 - T )^{5} \)
good7$C_2 \wr S_5$ \( 1 - 4 T + 19 T^{2} - 66 T^{3} + 177 T^{4} - 530 T^{5} + 177 p T^{6} - 66 p^{2} T^{7} + 19 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
11$C_2 \wr S_5$ \( 1 - 4 T + 29 T^{2} - 68 T^{3} + 436 T^{4} - 960 T^{5} + 436 p T^{6} - 68 p^{2} T^{7} + 29 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
13$C_2 \wr S_5$ \( 1 - 2 T + 15 T^{2} - 16 T^{3} + 112 T^{4} + 188 T^{5} + 112 p T^{6} - 16 p^{2} T^{7} + 15 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \)
17$C_2 \wr S_5$ \( 1 - 2 T + 41 T^{2} + 20 T^{3} + 45 p T^{4} + 1170 T^{5} + 45 p^{2} T^{6} + 20 p^{2} T^{7} + 41 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \)
19$C_2 \wr S_5$ \( 1 - 6 T + 77 T^{2} - 392 T^{3} + 2740 T^{4} - 10492 T^{5} + 2740 p T^{6} - 392 p^{2} T^{7} + 77 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} \)
29$C_2 \wr S_5$ \( 1 - 2 T + 107 T^{2} - 118 T^{3} + 5157 T^{4} - 3732 T^{5} + 5157 p T^{6} - 118 p^{2} T^{7} + 107 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \)
31$C_2 \wr S_5$ \( 1 - 8 T + 73 T^{2} - 344 T^{3} + 1969 T^{4} - 6064 T^{5} + 1969 p T^{6} - 344 p^{2} T^{7} + 73 p^{3} T^{8} - 8 p^{4} T^{9} + p^{5} T^{10} \)
37$C_2 \wr S_5$ \( 1 - 8 T + 145 T^{2} - 686 T^{3} + 8113 T^{4} - 28738 T^{5} + 8113 p T^{6} - 686 p^{2} T^{7} + 145 p^{3} T^{8} - 8 p^{4} T^{9} + p^{5} T^{10} \)
41$C_2 \wr S_5$ \( 1 - 2 T + 15 T^{2} - 282 T^{3} + 2285 T^{4} + 1648 T^{5} + 2285 p T^{6} - 282 p^{2} T^{7} + 15 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \)
43$C_2 \wr S_5$ \( 1 - 18 T + 203 T^{2} - 2144 T^{3} + 17950 T^{4} - 121564 T^{5} + 17950 p T^{6} - 2144 p^{2} T^{7} + 203 p^{3} T^{8} - 18 p^{4} T^{9} + p^{5} T^{10} \)
47$C_2 \wr S_5$ \( 1 + 4 T + 65 T^{2} + 620 T^{3} + 3124 T^{4} + 33264 T^{5} + 3124 p T^{6} + 620 p^{2} T^{7} + 65 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \)
53$C_2 \wr S_5$ \( 1 + 2 T + 93 T^{2} - 188 T^{3} + 4765 T^{4} - 24738 T^{5} + 4765 p T^{6} - 188 p^{2} T^{7} + 93 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
59$C_2 \wr S_5$ \( 1 - 6 T + 161 T^{2} - 738 T^{3} + 15065 T^{4} - 64116 T^{5} + 15065 p T^{6} - 738 p^{2} T^{7} + 161 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} \)
61$C_2 \wr S_5$ \( 1 - 10 T + 167 T^{2} - 1432 T^{3} + 14400 T^{4} - 99908 T^{5} + 14400 p T^{6} - 1432 p^{2} T^{7} + 167 p^{3} T^{8} - 10 p^{4} T^{9} + p^{5} T^{10} \)
67$C_2 \wr S_5$ \( 1 - 16 T + 287 T^{2} - 2078 T^{3} + 21265 T^{4} - 107186 T^{5} + 21265 p T^{6} - 2078 p^{2} T^{7} + 287 p^{3} T^{8} - 16 p^{4} T^{9} + p^{5} T^{10} \)
71$C_2 \wr S_5$ \( 1 - 10 T + 237 T^{2} - 2202 T^{3} + 29657 T^{4} - 212680 T^{5} + 29657 p T^{6} - 2202 p^{2} T^{7} + 237 p^{3} T^{8} - 10 p^{4} T^{9} + p^{5} T^{10} \)
73$C_2 \wr S_5$ \( 1 - 18 T + 443 T^{2} - 5192 T^{3} + 69816 T^{4} - 566324 T^{5} + 69816 p T^{6} - 5192 p^{2} T^{7} + 443 p^{3} T^{8} - 18 p^{4} T^{9} + p^{5} T^{10} \)
79$C_2 \wr S_5$ \( 1 - 14 T + 219 T^{2} - 2728 T^{3} + 29146 T^{4} - 260884 T^{5} + 29146 p T^{6} - 2728 p^{2} T^{7} + 219 p^{3} T^{8} - 14 p^{4} T^{9} + p^{5} T^{10} \)
83$C_2 \wr S_5$ \( 1 - 8 T + p T^{2} - 1108 T^{3} + 14061 T^{4} - 107580 T^{5} + 14061 p T^{6} - 1108 p^{2} T^{7} + p^{4} T^{8} - 8 p^{4} T^{9} + p^{5} T^{10} \)
89$C_2 \wr S_5$ \( 1 + 18 T + 485 T^{2} + 5928 T^{3} + 88354 T^{4} + 768876 T^{5} + 88354 p T^{6} + 5928 p^{2} T^{7} + 485 p^{3} T^{8} + 18 p^{4} T^{9} + p^{5} T^{10} \)
97$C_2 \wr S_5$ \( 1 - 6 T + 297 T^{2} - 3056 T^{3} + 39046 T^{4} - 479988 T^{5} + 39046 p T^{6} - 3056 p^{2} T^{7} + 297 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} \)
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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

−4.90361575746663998989685793178, −4.78535092781026978522172288015, −4.70899864484118438677909233458, −4.64484905541205043103253117664, −4.58658192230373695524631545996, −4.07000330528055237724083539442, −3.87061606167528162462517371709, −3.76673764769669862155933461542, −3.75401153155864613992329865068, −3.75041019627346759919978199615, −3.08200078225406413264561959210, −2.94340654299619777424573183212, −2.93468719703186432104853487762, −2.77086373348407089923421084432, −2.70724901188590614156795148511, −2.18577620187739833051955038335, −2.06620689135795090059865489172, −1.87841174631998414917874934252, −1.81439403720821503003579893097, −1.76655044520155341346497847415, −1.10546118893969112856178848140, −1.00936089847603733380098820065, −0.968539965046280998779571369819, −0.73145949031201432934140032293, −0.71010563717108415910656754494, 0.71010563717108415910656754494, 0.73145949031201432934140032293, 0.968539965046280998779571369819, 1.00936089847603733380098820065, 1.10546118893969112856178848140, 1.76655044520155341346497847415, 1.81439403720821503003579893097, 1.87841174631998414917874934252, 2.06620689135795090059865489172, 2.18577620187739833051955038335, 2.70724901188590614156795148511, 2.77086373348407089923421084432, 2.93468719703186432104853487762, 2.94340654299619777424573183212, 3.08200078225406413264561959210, 3.75041019627346759919978199615, 3.75401153155864613992329865068, 3.76673764769669862155933461542, 3.87061606167528162462517371709, 4.07000330528055237724083539442, 4.58658192230373695524631545996, 4.64484905541205043103253117664, 4.70899864484118438677909233458, 4.78535092781026978522172288015, 4.90361575746663998989685793178

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