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Properties

Label 10-4140e5-1.1-c1e5-0-1
Degree 1010
Conductor 1.216×10181.216\times 10^{18}
Sign 11
Analytic cond. 3.94809×1073.94809\times 10^{7}
Root an. cond. 5.749615.74961
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

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

Λ(s)=((210⋅310⋅55⋅235)s/2 ΓC(s)5 L(s)=( Λ(2−s)\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}
Λ(s)=((210⋅310⋅55⋅235)s/2 ΓC(s+1/2)5 L(s)=( Λ(1−s)\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: 1010
Conductor: 210⋅310⋅55⋅2352^{10} \cdot 3^{10} \cdot 5^{5} \cdot 23^{5}
Sign: 11
Analytic conductor: 3.94809×1073.94809\times 10^{7}
Root analytic conductor: 5.749615.74961
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (10, 210⋅310⋅55⋅235, ( :1/2,1/2,1/2,1/2,1/2), 1)(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)L(1) ≈\approx 55.9203522055.92035220
L(12)L(\frac12) ≈\approx 55.9203522055.92035220
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppGal⁡(Fp)\Gal(F_p)Fp(T)F_p(T)
bad2 1 1
3 1 1
5C1C_1 (1−T)5 ( 1 - T )^{5}
23C1C_1 (1−T)5 ( 1 - T )^{5}
good7C2≀S5C_2 \wr S_5 1−4T+19T2−66T3+177T4−530T5+177pT6−66p2T7+19p3T8−4p4T9+p5T10 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}
11C2≀S5C_2 \wr S_5 1−4T+29T2−68T3+436T4−960T5+436pT6−68p2T7+29p3T8−4p4T9+p5T10 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}
13C2≀S5C_2 \wr S_5 1−2T+15T2−16T3+112T4+188T5+112pT6−16p2T7+15p3T8−2p4T9+p5T10 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}
17C2≀S5C_2 \wr S_5 1−2T+41T2+20T3+45pT4+1170T5+45p2T6+20p2T7+41p3T8−2p4T9+p5T10 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}
19C2≀S5C_2 \wr S_5 1−6T+77T2−392T3+2740T4−10492T5+2740pT6−392p2T7+77p3T8−6p4T9+p5T10 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}
29C2≀S5C_2 \wr S_5 1−2T+107T2−118T3+5157T4−3732T5+5157pT6−118p2T7+107p3T8−2p4T9+p5T10 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}
31C2≀S5C_2 \wr S_5 1−8T+73T2−344T3+1969T4−6064T5+1969pT6−344p2T7+73p3T8−8p4T9+p5T10 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}
37C2≀S5C_2 \wr S_5 1−8T+145T2−686T3+8113T4−28738T5+8113pT6−686p2T7+145p3T8−8p4T9+p5T10 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}
41C2≀S5C_2 \wr S_5 1−2T+15T2−282T3+2285T4+1648T5+2285pT6−282p2T7+15p3T8−2p4T9+p5T10 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}
43C2≀S5C_2 \wr S_5 1−18T+203T2−2144T3+17950T4−121564T5+17950pT6−2144p2T7+203p3T8−18p4T9+p5T10 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}
47C2≀S5C_2 \wr S_5 1+4T+65T2+620T3+3124T4+33264T5+3124pT6+620p2T7+65p3T8+4p4T9+p5T10 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}
53C2≀S5C_2 \wr S_5 1+2T+93T2−188T3+4765T4−24738T5+4765pT6−188p2T7+93p3T8+2p4T9+p5T10 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}
59C2≀S5C_2 \wr S_5 1−6T+161T2−738T3+15065T4−64116T5+15065pT6−738p2T7+161p3T8−6p4T9+p5T10 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}
61C2≀S5C_2 \wr S_5 1−10T+167T2−1432T3+14400T4−99908T5+14400pT6−1432p2T7+167p3T8−10p4T9+p5T10 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}
67C2≀S5C_2 \wr S_5 1−16T+287T2−2078T3+21265T4−107186T5+21265pT6−2078p2T7+287p3T8−16p4T9+p5T10 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}
71C2≀S5C_2 \wr S_5 1−10T+237T2−2202T3+29657T4−212680T5+29657pT6−2202p2T7+237p3T8−10p4T9+p5T10 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}
73C2≀S5C_2 \wr S_5 1−18T+443T2−5192T3+69816T4−566324T5+69816pT6−5192p2T7+443p3T8−18p4T9+p5T10 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}
79C2≀S5C_2 \wr S_5 1−14T+219T2−2728T3+29146T4−260884T5+29146pT6−2728p2T7+219p3T8−14p4T9+p5T10 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}
83C2≀S5C_2 \wr S_5 1−8T+pT2−1108T3+14061T4−107580T5+14061pT6−1108p2T7+p4T8−8p4T9+p5T10 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}
89C2≀S5C_2 \wr S_5 1+18T+485T2+5928T3+88354T4+768876T5+88354pT6+5928p2T7+485p3T8+18p4T9+p5T10 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}
97C2≀S5C_2 \wr S_5 1−6T+297T2−3056T3+39046T4−479988T5+39046pT6−3056p2T7+297p3T8−6p4T9+p5T10 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)=∏p ∏j=110(1−αj,p p−s)−1L(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 ZZ-function along the critical line