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Properties

Label 8-1449e4-1.1-c1e4-0-3
Degree 88
Conductor 4.408×10124.408\times 10^{12}
Sign 11
Analytic cond. 17921.817921.8
Root an. cond. 3.401513.40151
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  − 2·4-s − 5·5-s + 4·7-s + 8-s + 5·11-s + 7·13-s + 2·16-s − 12·17-s + 3·19-s + 10·20-s + 4·23-s + 6·25-s − 8·28-s − 6·29-s + 4·31-s − 2·32-s − 20·35-s + 20·37-s − 5·40-s − 3·41-s + 9·43-s − 10·44-s − 7·47-s + 10·49-s − 14·52-s + 6·53-s − 25·55-s + ⋯
L(s)  = 1  − 4-s − 2.23·5-s + 1.51·7-s + 0.353·8-s + 1.50·11-s + 1.94·13-s + 1/2·16-s − 2.91·17-s + 0.688·19-s + 2.23·20-s + 0.834·23-s + 6/5·25-s − 1.51·28-s − 1.11·29-s + 0.718·31-s − 0.353·32-s − 3.38·35-s + 3.28·37-s − 0.790·40-s − 0.468·41-s + 1.37·43-s − 1.50·44-s − 1.02·47-s + 10/7·49-s − 1.94·52-s + 0.824·53-s − 3.37·55-s + ⋯

Functional equation

Λ(s)=((38⋅74⋅234)s/2 ΓC(s)4 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((38⋅74⋅234)s/2 ΓC(s+1/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 38⋅74⋅2343^{8} \cdot 7^{4} \cdot 23^{4}
Sign: 11
Analytic conductor: 17921.817921.8
Root analytic conductor: 3.401513.40151
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 38⋅74⋅234, ( :1/2,1/2,1/2,1/2), 1)(8,\ 3^{8} \cdot 7^{4} \cdot 23^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 3.2133252443.213325244
L(12)L(\frac12) ≈\approx 3.2133252443.213325244
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)
bad3 1 1
7C1C_1 (1−T)4 ( 1 - T )^{4}
23C1C_1 (1−T)4 ( 1 - T )^{4}
good2C2≀S4C_2 \wr S_4 1+pT2−T3+pT4−pT5+p3T6+p4T8 1 + p T^{2} - T^{3} + p T^{4} - p T^{5} + p^{3} T^{6} + p^{4} T^{8}
5C2≀S4C_2 \wr S_4 1+pT+19T2+61T3+148T4+61pT5+19p2T6+p4T7+p4T8 1 + p T + 19 T^{2} + 61 T^{3} + 148 T^{4} + 61 p T^{5} + 19 p^{2} T^{6} + p^{4} T^{7} + p^{4} T^{8}
11C2≀S4C_2 \wr S_4 1−5T+35T2−100T3+460T4−100pT5+35p2T6−5p3T7+p4T8 1 - 5 T + 35 T^{2} - 100 T^{3} + 460 T^{4} - 100 p T^{5} + 35 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8}
13C2≀S4C_2 \wr S_4 1−7T+41T2−121T3+492T4−121pT5+41p2T6−7p3T7+p4T8 1 - 7 T + 41 T^{2} - 121 T^{3} + 492 T^{4} - 121 p T^{5} + 41 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8}
17C2≀S4C_2 \wr S_4 1+12T+105T2+602T3+2888T4+602pT5+105p2T6+12p3T7+p4T8 1 + 12 T + 105 T^{2} + 602 T^{3} + 2888 T^{4} + 602 p T^{5} + 105 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8}
19C2≀S4C_2 \wr S_4 1−3T+39T2−120T3+812T4−120pT5+39p2T6−3p3T7+p4T8 1 - 3 T + 39 T^{2} - 120 T^{3} + 812 T^{4} - 120 p T^{5} + 39 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8}
29C2≀S4C_2 \wr S_4 1+6T+85T2+258T3+2812T4+258pT5+85p2T6+6p3T7+p4T8 1 + 6 T + 85 T^{2} + 258 T^{3} + 2812 T^{4} + 258 p T^{5} + 85 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8}
31C2≀S4C_2 \wr S_4 1−4T+113T2−326T3+5068T4−326pT5+113p2T6−4p3T7+p4T8 1 - 4 T + 113 T^{2} - 326 T^{3} + 5068 T^{4} - 326 p T^{5} + 113 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8}
37C2≀S4C_2 \wr S_4 1−20T+239T2−1940T3+13080T4−1940pT5+239p2T6−20p3T7+p4T8 1 - 20 T + 239 T^{2} - 1940 T^{3} + 13080 T^{4} - 1940 p T^{5} + 239 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8}
41C2≀S4C_2 \wr S_4 1+3T+137T2+298T3+7838T4+298pT5+137p2T6+3p3T7+p4T8 1 + 3 T + 137 T^{2} + 298 T^{3} + 7838 T^{4} + 298 p T^{5} + 137 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8}
43C2≀S4C_2 \wr S_4 1−9T+133T2−977T3+8012T4−977pT5+133p2T6−9p3T7+p4T8 1 - 9 T + 133 T^{2} - 977 T^{3} + 8012 T^{4} - 977 p T^{5} + 133 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8}
47C2≀S4C_2 \wr S_4 1+7T+194T2+959T3+13786T4+959pT5+194p2T6+7p3T7+p4T8 1 + 7 T + 194 T^{2} + 959 T^{3} + 13786 T^{4} + 959 p T^{5} + 194 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8}
53C2≀S4C_2 \wr S_4 1−6T+144T2−705T3+9444T4−705pT5+144p2T6−6p3T7+p4T8 1 - 6 T + 144 T^{2} - 705 T^{3} + 9444 T^{4} - 705 p T^{5} + 144 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8}
59C2≀S4C_2 \wr S_4 1−2T−42T2−91T3+5646T4−91pT5−42p2T6−2p3T7+p4T8 1 - 2 T - 42 T^{2} - 91 T^{3} + 5646 T^{4} - 91 p T^{5} - 42 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8}
61C2≀S4C_2 \wr S_4 1−24T+416T2−4583T3+42024T4−4583pT5+416p2T6−24p3T7+p4T8 1 - 24 T + 416 T^{2} - 4583 T^{3} + 42024 T^{4} - 4583 p T^{5} + 416 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8}
67C2≀S4C_2 \wr S_4 1−T+127T2−225T3+12248T4−225pT5+127p2T6−p3T7+p4T8 1 - T + 127 T^{2} - 225 T^{3} + 12248 T^{4} - 225 p T^{5} + 127 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8}
71C2≀S4C_2 \wr S_4 1−17T+363T2−3749T3+41528T4−3749pT5+363p2T6−17p3T7+p4T8 1 - 17 T + 363 T^{2} - 3749 T^{3} + 41528 T^{4} - 3749 p T^{5} + 363 p^{2} T^{6} - 17 p^{3} T^{7} + p^{4} T^{8}
73C2≀S4C_2 \wr S_4 1−16T+197T2−1234T3+11448T4−1234pT5+197p2T6−16p3T7+p4T8 1 - 16 T + 197 T^{2} - 1234 T^{3} + 11448 T^{4} - 1234 p T^{5} + 197 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8}
79C2≀S4C_2 \wr S_4 1+10T+299T2+2156T3+34888T4+2156pT5+299p2T6+10p3T7+p4T8 1 + 10 T + 299 T^{2} + 2156 T^{3} + 34888 T^{4} + 2156 p T^{5} + 299 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8}
83C2≀S4C_2 \wr S_4 1+8T+137T2−458T3+1708T4−458pT5+137p2T6+8p3T7+p4T8 1 + 8 T + 137 T^{2} - 458 T^{3} + 1708 T^{4} - 458 p T^{5} + 137 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8}
89C2≀S4C_2 \wr S_4 1−3T+243T2−1181T3+27320T4−1181pT5+243p2T6−3p3T7+p4T8 1 - 3 T + 243 T^{2} - 1181 T^{3} + 27320 T^{4} - 1181 p T^{5} + 243 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8}
97C2≀S4C_2 \wr S_4 1+2T+301T2+226T3+39580T4+226pT5+301p2T6+2p3T7+p4T8 1 + 2 T + 301 T^{2} + 226 T^{3} + 39580 T^{4} + 226 p T^{5} + 301 p^{2} T^{6} + 2 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

−6.75482314270277489328946716004, −6.54188593670510091225262653045, −6.26527625400123745309812827712, −6.21593634501715843295262002119, −6.18857255069453120421485758447, −5.61542048682272155027672847041, −5.33389264968437426012314506799, −5.21730686776054707791465178814, −4.87612297281730863126342357551, −4.67677228673703421201738031280, −4.44001302761881934070506937183, −4.24006939781376570562638045585, −4.14542886113682980717603561464, −3.88364053689209564595947871843, −3.82074467726599430102288110264, −3.47262107097778259514272468392, −3.44467052638250643823348991228, −2.66325502998751394349493692560, −2.60555125049593158330926400402, −2.17753329320047194668244540053, −1.76591910803781584328621050263, −1.61667524175534746634977226468, −0.857669742248525348815589827704, −0.828487297195982993622951530321, −0.55750122907402016844331512209, 0.55750122907402016844331512209, 0.828487297195982993622951530321, 0.857669742248525348815589827704, 1.61667524175534746634977226468, 1.76591910803781584328621050263, 2.17753329320047194668244540053, 2.60555125049593158330926400402, 2.66325502998751394349493692560, 3.44467052638250643823348991228, 3.47262107097778259514272468392, 3.82074467726599430102288110264, 3.88364053689209564595947871843, 4.14542886113682980717603561464, 4.24006939781376570562638045585, 4.44001302761881934070506937183, 4.67677228673703421201738031280, 4.87612297281730863126342357551, 5.21730686776054707791465178814, 5.33389264968437426012314506799, 5.61542048682272155027672847041, 6.18857255069453120421485758447, 6.21593634501715843295262002119, 6.26527625400123745309812827712, 6.54188593670510091225262653045, 6.75482314270277489328946716004

Graph of the ZZ-function along the critical line