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Properties

Label 8-95e8-1.1-c1e4-0-2
Degree 88
Conductor 6.634×10156.634\times 10^{15}
Sign 11
Analytic cond. 2.69710×1072.69710\times 10^{7}
Root an. cond. 8.489108.48910
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  + 2-s + 3·3-s − 4-s + 3·6-s + 4·7-s + 2·8-s − 9-s − 2·11-s − 3·12-s + 7·13-s + 4·14-s + 3·16-s + 17-s − 18-s + 12·21-s − 2·22-s − 2·23-s + 6·24-s + 7·26-s − 11·27-s − 4·28-s + 29-s − 2·32-s − 6·33-s + 34-s + 36-s − 2·37-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.73·3-s − 1/2·4-s + 1.22·6-s + 1.51·7-s + 0.707·8-s − 1/3·9-s − 0.603·11-s − 0.866·12-s + 1.94·13-s + 1.06·14-s + 3/4·16-s + 0.242·17-s − 0.235·18-s + 2.61·21-s − 0.426·22-s − 0.417·23-s + 1.22·24-s + 1.37·26-s − 2.11·27-s − 0.755·28-s + 0.185·29-s − 0.353·32-s − 1.04·33-s + 0.171·34-s + 1/6·36-s − 0.328·37-s + ⋯

Functional equation

Λ(s)=((58⋅198)s/2 ΓC(s)4 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((58⋅198)s/2 ΓC(s+1/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 58⋅1985^{8} \cdot 19^{8}
Sign: 11
Analytic conductor: 2.69710×1072.69710\times 10^{7}
Root analytic conductor: 8.489108.48910
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 58⋅198, ( :1/2,1/2,1/2,1/2), 1)(8,\ 5^{8} \cdot 19^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 19.3964790619.39647906
L(12)L(\frac12) ≈\approx 19.3964790619.39647906
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)
bad5 1 1
19 1 1
good2C2≀S4C_2 \wr S_4 1−T+pT2−5T3+3pT4−5pT5+p3T6−p3T7+p4T8 1 - T + p T^{2} - 5 T^{3} + 3 p T^{4} - 5 p T^{5} + p^{3} T^{6} - p^{3} T^{7} + p^{4} T^{8}
3C2≀S4C_2 \wr S_4 1−pT+10T2−22T3+44T4−22pT5+10p2T6−p4T7+p4T8 1 - p T + 10 T^{2} - 22 T^{3} + 44 T^{4} - 22 p T^{5} + 10 p^{2} T^{6} - p^{4} T^{7} + p^{4} T^{8}
7C2≀S4C_2 \wr S_4 1−4T+27T2−69T3+272T4−69pT5+27p2T6−4p3T7+p4T8 1 - 4 T + 27 T^{2} - 69 T^{3} + 272 T^{4} - 69 p T^{5} + 27 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8}
11C2≀S4C_2 \wr S_4 1+2T+19T2+47T3+179T4+47pT5+19p2T6+2p3T7+p4T8 1 + 2 T + 19 T^{2} + 47 T^{3} + 179 T^{4} + 47 p T^{5} + 19 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8}
13C2≀S4C_2 \wr S_4 1−7T+59T2−250T3+1180T4−250pT5+59p2T6−7p3T7+p4T8 1 - 7 T + 59 T^{2} - 250 T^{3} + 1180 T^{4} - 250 p T^{5} + 59 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8}
17C2≀S4C_2 \wr S_4 1−T+38T2−4pT3+822T4−4p2T5+38p2T6−p3T7+p4T8 1 - T + 38 T^{2} - 4 p T^{3} + 822 T^{4} - 4 p^{2} T^{5} + 38 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8}
23C2≀S4C_2 \wr S_4 1+2T+75T2+145T3+2398T4+145pT5+75p2T6+2p3T7+p4T8 1 + 2 T + 75 T^{2} + 145 T^{3} + 2398 T^{4} + 145 p T^{5} + 75 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8}
29C2≀S4C_2 \wr S_4 1−T+53T2−281T3+1251T4−281pT5+53p2T6−p3T7+p4T8 1 - T + 53 T^{2} - 281 T^{3} + 1251 T^{4} - 281 p T^{5} + 53 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8}
31C2≀S4C_2 \wr S_4 1+57T2+5T3+2675T4+5pT5+57p2T6+p4T8 1 + 57 T^{2} + 5 T^{3} + 2675 T^{4} + 5 p T^{5} + 57 p^{2} T^{6} + p^{4} T^{8}
37C2≀S4C_2 \wr S_4 1+2T+117T2+99T3+5802T4+99pT5+117p2T6+2p3T7+p4T8 1 + 2 T + 117 T^{2} + 99 T^{3} + 5802 T^{4} + 99 p T^{5} + 117 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8}
41C2≀S4C_2 \wr S_4 1−8T+77T2−13T3+714T4−13pT5+77p2T6−8p3T7+p4T8 1 - 8 T + 77 T^{2} - 13 T^{3} + 714 T^{4} - 13 p T^{5} + 77 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8}
43C2≀S4C_2 \wr S_4 1+T+74T2−68T3+3460T4−68pT5+74p2T6+p3T7+p4T8 1 + T + 74 T^{2} - 68 T^{3} + 3460 T^{4} - 68 p T^{5} + 74 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8}
47C2≀S4C_2 \wr S_4 1−12T+133T2−763T3+5768T4−763pT5+133p2T6−12p3T7+p4T8 1 - 12 T + 133 T^{2} - 763 T^{3} + 5768 T^{4} - 763 p T^{5} + 133 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8}
53C2≀S4C_2 \wr S_4 1+5T+126T2+568T3+7684T4+568pT5+126p2T6+5p3T7+p4T8 1 + 5 T + 126 T^{2} + 568 T^{3} + 7684 T^{4} + 568 p T^{5} + 126 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8}
59C2≀S4C_2 \wr S_4 1−5T+111T2−385T3+8011T4−385pT5+111p2T6−5p3T7+p4T8 1 - 5 T + 111 T^{2} - 385 T^{3} + 8011 T^{4} - 385 p T^{5} + 111 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8}
61C2≀S4C_2 \wr S_4 1+114T2+88T3+9515T4+88pT5+114p2T6+p4T8 1 + 114 T^{2} + 88 T^{3} + 9515 T^{4} + 88 p T^{5} + 114 p^{2} T^{6} + p^{4} T^{8}
67C2≀S4C_2 \wr S_4 1−4T+232T2−828T3+22174T4−828pT5+232p2T6−4p3T7+p4T8 1 - 4 T + 232 T^{2} - 828 T^{3} + 22174 T^{4} - 828 p T^{5} + 232 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8}
71C2≀S4C_2 \wr S_4 1+20T+375T2+4165T3+42925T4+4165pT5+375p2T6+20p3T7+p4T8 1 + 20 T + 375 T^{2} + 4165 T^{3} + 42925 T^{4} + 4165 p T^{5} + 375 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8}
73C2≀S4C_2 \wr S_4 1−20T+387T2−57pT3+44118T4−57p2T5+387p2T6−20p3T7+p4T8 1 - 20 T + 387 T^{2} - 57 p T^{3} + 44118 T^{4} - 57 p^{2} T^{5} + 387 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8}
79C2≀S4C_2 \wr S_4 1+17T+388T2+4009T3+48638T4+4009pT5+388p2T6+17p3T7+p4T8 1 + 17 T + 388 T^{2} + 4009 T^{3} + 48638 T^{4} + 4009 p T^{5} + 388 p^{2} T^{6} + 17 p^{3} T^{7} + p^{4} T^{8}
83C2≀S4C_2 \wr S_4 1+T+270T2+194T3+31408T4+194pT5+270p2T6+p3T7+p4T8 1 + T + 270 T^{2} + 194 T^{3} + 31408 T^{4} + 194 p T^{5} + 270 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8}
89C2≀S4C_2 \wr S_4 1+11T+266T2+1549T3+27690T4+1549pT5+266p2T6+11p3T7+p4T8 1 + 11 T + 266 T^{2} + 1549 T^{3} + 27690 T^{4} + 1549 p T^{5} + 266 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8}
97C2≀S4C_2 \wr S_4 1−T+122T2−1096T3+12292T4−1096pT5+122p2T6−p3T7+p4T8 1 - T + 122 T^{2} - 1096 T^{3} + 12292 T^{4} - 1096 p T^{5} + 122 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8}
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   L(s)=∏p ∏j=18(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−5.49619519517840036477797165948, −5.22647917266468075162048015474, −5.12226906379260178962265908387, −4.79646293763180028201366383522, −4.56226506936702915316399510193, −4.42633147690899610238170169119, −4.29229782423517606501374844596, −4.22016515258556046400663224144, −4.01865613232690361214707742783, −3.60962161464734653995688031979, −3.56275660414593274498909694785, −3.43748432181038534560541903929, −3.22755732048517901531965320170, −2.98549848272529657281961996394, −2.68683150592785329526508798278, −2.53586442675143376565979766784, −2.39842534605707709598620779330, −2.31109042727214224982455867084, −1.86559124902586477304134607381, −1.56755634933443402349990924665, −1.41033245981883767143394211320, −1.37072949960239552813855742296, −1.03175658801345090660419141278, −0.43993708233029008854818624015, −0.42699641486266994228717594978, 0.42699641486266994228717594978, 0.43993708233029008854818624015, 1.03175658801345090660419141278, 1.37072949960239552813855742296, 1.41033245981883767143394211320, 1.56755634933443402349990924665, 1.86559124902586477304134607381, 2.31109042727214224982455867084, 2.39842534605707709598620779330, 2.53586442675143376565979766784, 2.68683150592785329526508798278, 2.98549848272529657281961996394, 3.22755732048517901531965320170, 3.43748432181038534560541903929, 3.56275660414593274498909694785, 3.60962161464734653995688031979, 4.01865613232690361214707742783, 4.22016515258556046400663224144, 4.29229782423517606501374844596, 4.42633147690899610238170169119, 4.56226506936702915316399510193, 4.79646293763180028201366383522, 5.12226906379260178962265908387, 5.22647917266468075162048015474, 5.49619519517840036477797165948

Graph of the ZZ-function along the critical line