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Properties

Label 10-4205e5-1.1-c1e5-0-3
Degree 1010
Conductor 1.315×10181.315\times 10^{18}
Sign −1-1
Analytic cond. 4.26791×1074.26791\times 10^{7}
Root an. cond. 5.794575.79457
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 55

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 2·3-s + 5·5-s + 2·6-s − 3·7-s − 3·8-s − 3·9-s − 5·10-s − 2·11-s + 3·14-s − 10·15-s + 4·16-s − 19·17-s + 3·18-s + 19-s + 6·21-s + 2·22-s + 4·23-s + 6·24-s + 15·25-s + 7·27-s + 10·30-s − 18·31-s + 32-s + 4·33-s + 19·34-s − 15·35-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.15·3-s + 2.23·5-s + 0.816·6-s − 1.13·7-s − 1.06·8-s − 9-s − 1.58·10-s − 0.603·11-s + 0.801·14-s − 2.58·15-s + 16-s − 4.60·17-s + 0.707·18-s + 0.229·19-s + 1.30·21-s + 0.426·22-s + 0.834·23-s + 1.22·24-s + 3·25-s + 1.34·27-s + 1.82·30-s − 3.23·31-s + 0.176·32-s + 0.696·33-s + 3.25·34-s − 2.53·35-s + ⋯

Functional equation

Λ(s)=((55⋅2910)s/2 ΓC(s)5 L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{5} \cdot 29^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}
Λ(s)=((55⋅2910)s/2 ΓC(s+1/2)5 L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{5} \cdot 29^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}

Invariants

Degree: 1010
Conductor: 55⋅29105^{5} \cdot 29^{10}
Sign: −1-1
Analytic conductor: 4.26791×1074.26791\times 10^{7}
Root analytic conductor: 5.794575.79457
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 55
Selberg data: (10, 55⋅2910, ( :1/2,1/2,1/2,1/2,1/2), −1)(10,\ 5^{5} \cdot 29^{10} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ -1 )

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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)
bad5C1C_1 (1−T)5 ( 1 - T )^{5}
29 1 1
good2C2≀S5C_2 \wr S_5 1+T+T2+p2T3+3T4+T5+3pT6+p4T7+p3T8+p4T9+p5T10 1 + T + T^{2} + p^{2} T^{3} + 3 T^{4} + T^{5} + 3 p T^{6} + p^{4} T^{7} + p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10}
3C2≀S5C_2 \wr S_5 1+2T+7T2+13T3+10pT4+55T5+10p2T6+13p2T7+7p3T8+2p4T9+p5T10 1 + 2 T + 7 T^{2} + 13 T^{3} + 10 p T^{4} + 55 T^{5} + 10 p^{2} T^{6} + 13 p^{2} T^{7} + 7 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10}
7C2≀S5C_2 \wr S_5 1+3T+15T2+29T3+23pT4+339T5+23p2T6+29p2T7+15p3T8+3p4T9+p5T10 1 + 3 T + 15 T^{2} + 29 T^{3} + 23 p T^{4} + 339 T^{5} + 23 p^{2} T^{6} + 29 p^{2} T^{7} + 15 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10}
11C2≀S5C_2 \wr S_5 1+2T+38T2+42T3+659T4+507T5+659pT6+42p2T7+38p3T8+2p4T9+p5T10 1 + 2 T + 38 T^{2} + 42 T^{3} + 659 T^{4} + 507 T^{5} + 659 p T^{6} + 42 p^{2} T^{7} + 38 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10}
13C2≀S5C_2 \wr S_5 1+20T2+42T3+209T4+97pT5+209pT6+42p2T7+20p3T8+p5T10 1 + 20 T^{2} + 42 T^{3} + 209 T^{4} + 97 p T^{5} + 209 p T^{6} + 42 p^{2} T^{7} + 20 p^{3} T^{8} + p^{5} T^{10}
17C2≀S5C_2 \wr S_5 1+19T+220T2+1732T3+10416T4+48237T5+10416pT6+1732p2T7+220p3T8+19p4T9+p5T10 1 + 19 T + 220 T^{2} + 1732 T^{3} + 10416 T^{4} + 48237 T^{5} + 10416 p T^{6} + 1732 p^{2} T^{7} + 220 p^{3} T^{8} + 19 p^{4} T^{9} + p^{5} T^{10}
19C2≀S5C_2 \wr S_5 1−T+78T2−35T3+2627T4−625T5+2627pT6−35p2T7+78p3T8−p4T9+p5T10 1 - T + 78 T^{2} - 35 T^{3} + 2627 T^{4} - 625 T^{5} + 2627 p T^{6} - 35 p^{2} T^{7} + 78 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10}
23C2≀S5C_2 \wr S_5 1−4T+96T2−335T3+4040T4−11267T5+4040pT6−335p2T7+96p3T8−4p4T9+p5T10 1 - 4 T + 96 T^{2} - 335 T^{3} + 4040 T^{4} - 11267 T^{5} + 4040 p T^{6} - 335 p^{2} T^{7} + 96 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10}
31C2≀S5C_2 \wr S_5 1+18T+275T2+2595T3+21254T4+126505T5+21254pT6+2595p2T7+275p3T8+18p4T9+p5T10 1 + 18 T + 275 T^{2} + 2595 T^{3} + 21254 T^{4} + 126505 T^{5} + 21254 p T^{6} + 2595 p^{2} T^{7} + 275 p^{3} T^{8} + 18 p^{4} T^{9} + p^{5} T^{10}
37C2≀S5C_2 \wr S_5 1+2T+80T2+375T3+2290T4+22265T5+2290pT6+375p2T7+80p3T8+2p4T9+p5T10 1 + 2 T + 80 T^{2} + 375 T^{3} + 2290 T^{4} + 22265 T^{5} + 2290 p T^{6} + 375 p^{2} T^{7} + 80 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10}
41C2≀S5C_2 \wr S_5 1+2T+109T2+269T3+6604T4+14767T5+6604pT6+269p2T7+109p3T8+2p4T9+p5T10 1 + 2 T + 109 T^{2} + 269 T^{3} + 6604 T^{4} + 14767 T^{5} + 6604 p T^{6} + 269 p^{2} T^{7} + 109 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10}
43C2≀S5C_2 \wr S_5 1+22T+350T2+3762T3+34123T4+241123T5+34123pT6+3762p2T7+350p3T8+22p4T9+p5T10 1 + 22 T + 350 T^{2} + 3762 T^{3} + 34123 T^{4} + 241123 T^{5} + 34123 p T^{6} + 3762 p^{2} T^{7} + 350 p^{3} T^{8} + 22 p^{4} T^{9} + p^{5} T^{10}
47C2≀S5C_2 \wr S_5 1−2T+182T2−322T3+15091T4−22153T5+15091pT6−322p2T7+182p3T8−2p4T9+p5T10 1 - 2 T + 182 T^{2} - 322 T^{3} + 15091 T^{4} - 22153 T^{5} + 15091 p T^{6} - 322 p^{2} T^{7} + 182 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10}
53C2≀S5C_2 \wr S_5 1+14T+128T2+1603T3+13482T4+85069T5+13482pT6+1603p2T7+128p3T8+14p4T9+p5T10 1 + 14 T + 128 T^{2} + 1603 T^{3} + 13482 T^{4} + 85069 T^{5} + 13482 p T^{6} + 1603 p^{2} T^{7} + 128 p^{3} T^{8} + 14 p^{4} T^{9} + p^{5} T^{10}
59C2≀S5C_2 \wr S_5 1−20T+384T2−4494T3+49003T4−390251T5+49003pT6−4494p2T7+384p3T8−20p4T9+p5T10 1 - 20 T + 384 T^{2} - 4494 T^{3} + 49003 T^{4} - 390251 T^{5} + 49003 p T^{6} - 4494 p^{2} T^{7} + 384 p^{3} T^{8} - 20 p^{4} T^{9} + p^{5} T^{10}
61C2≀S5C_2 \wr S_5 1+5T+58T2+320T3+1846T4+40683T5+1846pT6+320p2T7+58p3T8+5p4T9+p5T10 1 + 5 T + 58 T^{2} + 320 T^{3} + 1846 T^{4} + 40683 T^{5} + 1846 p T^{6} + 320 p^{2} T^{7} + 58 p^{3} T^{8} + 5 p^{4} T^{9} + p^{5} T^{10}
67C2≀S5C_2 \wr S_5 1+21T+392T2+5058T3+54770T4+490501T5+54770pT6+5058p2T7+392p3T8+21p4T9+p5T10 1 + 21 T + 392 T^{2} + 5058 T^{3} + 54770 T^{4} + 490501 T^{5} + 54770 p T^{6} + 5058 p^{2} T^{7} + 392 p^{3} T^{8} + 21 p^{4} T^{9} + p^{5} T^{10}
71C2≀S5C_2 \wr S_5 1+4T+243T2+459T3+26400T4+27539T5+26400pT6+459p2T7+243p3T8+4p4T9+p5T10 1 + 4 T + 243 T^{2} + 459 T^{3} + 26400 T^{4} + 27539 T^{5} + 26400 p T^{6} + 459 p^{2} T^{7} + 243 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10}
73C2≀S5C_2 \wr S_5 1+28T+451T2+4697T3+40258T4+321827T5+40258pT6+4697p2T7+451p3T8+28p4T9+p5T10 1 + 28 T + 451 T^{2} + 4697 T^{3} + 40258 T^{4} + 321827 T^{5} + 40258 p T^{6} + 4697 p^{2} T^{7} + 451 p^{3} T^{8} + 28 p^{4} T^{9} + p^{5} T^{10}
79C2≀S5C_2 \wr S_5 1+3T+148T2+1016T3+14230T4+124135T5+14230pT6+1016p2T7+148p3T8+3p4T9+p5T10 1 + 3 T + 148 T^{2} + 1016 T^{3} + 14230 T^{4} + 124135 T^{5} + 14230 p T^{6} + 1016 p^{2} T^{7} + 148 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10}
83C2≀S5C_2 \wr S_5 1−8T+51T2−1107T3+10766T4−49059T5+10766pT6−1107p2T7+51p3T8−8p4T9+p5T10 1 - 8 T + 51 T^{2} - 1107 T^{3} + 10766 T^{4} - 49059 T^{5} + 10766 p T^{6} - 1107 p^{2} T^{7} + 51 p^{3} T^{8} - 8 p^{4} T^{9} + p^{5} T^{10}
89C2≀S5C_2 \wr S_5 1−34T+481T2−1879T3−38568T4+652859T5−38568pT6−1879p2T7+481p3T8−34p4T9+p5T10 1 - 34 T + 481 T^{2} - 1879 T^{3} - 38568 T^{4} + 652859 T^{5} - 38568 p T^{6} - 1879 p^{2} T^{7} + 481 p^{3} T^{8} - 34 p^{4} T^{9} + p^{5} T^{10}
97C2≀S5C_2 \wr S_5 1+9T+339T2+3194T3+54021T4+452547T5+54021pT6+3194p2T7+339p3T8+9p4T9+p5T10 1 + 9 T + 339 T^{2} + 3194 T^{3} + 54021 T^{4} + 452547 T^{5} + 54021 p T^{6} + 3194 p^{2} T^{7} + 339 p^{3} T^{8} + 9 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

−5.38793340403305927336471181093, −5.36193691472971951280787891314, −5.20008067352100838750521085099, −4.95953019754265346570286813753, −4.90851761830961027198381936671, −4.51299758414817032817752545057, −4.47252696455220729216181350272, −4.40089748220770563737607140320, −4.28500756698698299578505139928, −3.80340182641658453953101075652, −3.51747982214993566318402831087, −3.46462412890037505664121717579, −3.40941524390359957177212550349, −3.12805402784365255079366956767, −3.03673990796860739830719538903, −2.72497896670605168756404664843, −2.53852058217544949438741529867, −2.49792810825375743361365602135, −2.27421235976264044436882644364, −1.91749211285794864864701837129, −1.90017899413792410491872737844, −1.80014133748797939287248327236, −1.40744837257987688700519077899, −1.20282522586832668508859049040, −0.887083036256129725478936855607, 0, 0, 0, 0, 0, 0.887083036256129725478936855607, 1.20282522586832668508859049040, 1.40744837257987688700519077899, 1.80014133748797939287248327236, 1.90017899413792410491872737844, 1.91749211285794864864701837129, 2.27421235976264044436882644364, 2.49792810825375743361365602135, 2.53852058217544949438741529867, 2.72497896670605168756404664843, 3.03673990796860739830719538903, 3.12805402784365255079366956767, 3.40941524390359957177212550349, 3.46462412890037505664121717579, 3.51747982214993566318402831087, 3.80340182641658453953101075652, 4.28500756698698299578505139928, 4.40089748220770563737607140320, 4.47252696455220729216181350272, 4.51299758414817032817752545057, 4.90851761830961027198381936671, 4.95953019754265346570286813753, 5.20008067352100838750521085099, 5.36193691472971951280787891314, 5.38793340403305927336471181093

Graph of the ZZ-function along the critical line