↑

Properties

Label 6-2475e3-1.1-c1e3-0-2
Degree 66
Conductor 1516092187515160921875
Sign 11
Analytic cond. 7718.927718.92
Root an. cond. 4.445554.44555
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3·2-s + 4·4-s + 8·7-s − 4·8-s + 3·11-s + 6·13-s − 24·14-s + 3·16-s − 4·17-s − 2·19-s − 9·22-s − 12·23-s − 18·26-s + 32·28-s + 8·29-s + 8·31-s + 32-s + 12·34-s + 4·37-s + 6·38-s + 8·41-s + 8·43-s + 12·44-s + 36·46-s − 8·47-s + 27·49-s + 24·52-s + ⋯
L(s)  = 1  − 2.12·2-s + 2·4-s + 3.02·7-s − 1.41·8-s + 0.904·11-s + 1.66·13-s − 6.41·14-s + 3/4·16-s − 0.970·17-s − 0.458·19-s − 1.91·22-s − 2.50·23-s − 3.53·26-s + 6.04·28-s + 1.48·29-s + 1.43·31-s + 0.176·32-s + 2.05·34-s + 0.657·37-s + 0.973·38-s + 1.24·41-s + 1.21·43-s + 1.80·44-s + 5.30·46-s − 1.16·47-s + 27/7·49-s + 3.32·52-s + ⋯

Functional equation

Λ(s)=((36⋅56⋅113)s/2 ΓC(s)3 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{6} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((36⋅56⋅113)s/2 ΓC(s+1/2)3 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{6} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 66
Conductor: 36⋅56⋅1133^{6} \cdot 5^{6} \cdot 11^{3}
Sign: 11
Analytic conductor: 7718.927718.92
Root analytic conductor: 4.445554.44555
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (6, 36⋅56⋅113, ( :1/2,1/2,1/2), 1)(6,\ 3^{6} \cdot 5^{6} \cdot 11^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 2.2085300792.208530079
L(12)L(\frac12) ≈\approx 2.2085300792.208530079
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
5 1 1
11C1C_1 (1−T)3 ( 1 - T )^{3}
good2S4×C2S_4\times C_2 1+3T+5T2+7T3+5pT4+3p2T5+p3T6 1 + 3 T + 5 T^{2} + 7 T^{3} + 5 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6}
7S4×C2S_4\times C_2 1−8T+37T2−116T3+37pT4−8p2T5+p3T6 1 - 8 T + 37 T^{2} - 116 T^{3} + 37 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6}
13S4×C2S_4\times C_2 1−6T+11T2−8T3+11pT4−6p2T5+p3T6 1 - 6 T + 11 T^{2} - 8 T^{3} + 11 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6}
17S4×C2S_4\times C_2 1+4T+23T2+20T3+23pT4+4p2T5+p3T6 1 + 4 T + 23 T^{2} + 20 T^{3} + 23 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6}
19S4×C2S_4\times C_2 1+2T+5T2−108T3+5pT4+2p2T5+p3T6 1 + 2 T + 5 T^{2} - 108 T^{3} + 5 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6}
23C2C_2 (1+4T+pT2)3 ( 1 + 4 T + p T^{2} )^{3}
29S4×C2S_4\times C_2 1−8T+3pT2−432T3+3p2T4−8p2T5+p3T6 1 - 8 T + 3 p T^{2} - 432 T^{3} + 3 p^{2} T^{4} - 8 p^{2} T^{5} + p^{3} T^{6}
31S4×C2S_4\times C_2 1−8T+101T2−480T3+101pT4−8p2T5+p3T6 1 - 8 T + 101 T^{2} - 480 T^{3} + 101 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6}
37S4×C2S_4\times C_2 1−4T+95T2−264T3+95pT4−4p2T5+p3T6 1 - 4 T + 95 T^{2} - 264 T^{3} + 95 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6}
41S4×C2S_4\times C_2 1−8T+3pT2−624T3+3p2T4−8p2T5+p3T6 1 - 8 T + 3 p T^{2} - 624 T^{3} + 3 p^{2} T^{4} - 8 p^{2} T^{5} + p^{3} T^{6}
43S4×C2S_4\times C_2 1−8T+145T2−692T3+145pT4−8p2T5+p3T6 1 - 8 T + 145 T^{2} - 692 T^{3} + 145 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6}
47S4×C2S_4\times C_2 1+8T+125T2+592T3+125pT4+8p2T5+p3T6 1 + 8 T + 125 T^{2} + 592 T^{3} + 125 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6}
53S4×C2S_4\times C_2 1−8T+127T2−576T3+127pT4−8p2T5+p3T6 1 - 8 T + 127 T^{2} - 576 T^{3} + 127 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6}
59S4×C2S_4\times C_2 1−8T+113T2−1024T3+113pT4−8p2T5+p3T6 1 - 8 T + 113 T^{2} - 1024 T^{3} + 113 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6}
61S4×C2S_4\times C_2 1−2T+131T2−204T3+131pT4−2p2T5+p3T6 1 - 2 T + 131 T^{2} - 204 T^{3} + 131 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6}
67S4×C2S_4\times C_2 1−12T+185T2−1288T3+185pT4−12p2T5+p3T6 1 - 12 T + 185 T^{2} - 1288 T^{3} + 185 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6}
71S4×C2S_4\times C_2 1−12T+245T2−1688T3+245pT4−12p2T5+p3T6 1 - 12 T + 245 T^{2} - 1688 T^{3} + 245 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6}
73S4×C2S_4\times C_2 1−18T+279T2−2536T3+279pT4−18p2T5+p3T6 1 - 18 T + 279 T^{2} - 2536 T^{3} + 279 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6}
79S4×C2S_4\times C_2 1+6T+233T2+940T3+233pT4+6p2T5+p3T6 1 + 6 T + 233 T^{2} + 940 T^{3} + 233 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6}
83S4×C2S_4\times C_2 1−2T+245T2−328T3+245pT4−2p2T5+p3T6 1 - 2 T + 245 T^{2} - 328 T^{3} + 245 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6}
89S4×C2S_4\times C_2 1+2T+255T2+348T3+255pT4+2p2T5+p3T6 1 + 2 T + 255 T^{2} + 348 T^{3} + 255 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6}
97S4×C2S_4\times C_2 1+8T+259T2+1424T3+259pT4+8p2T5+p3T6 1 + 8 T + 259 T^{2} + 1424 T^{3} + 259 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6}
show more
show less
   L(s)=∏p ∏j=16(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−8.313550274550052274072416963525, −7.71961054666166689607552563894, −7.60933599002272806508545558447, −7.52213522491542998239252820619, −6.96801570104968158819303396632, −6.68238811525108223837190753596, −6.24867010664876310423898326668, −6.23457926262712208796389110711, −6.10320096753296583924939336118, −5.72652633179915019160833579688, −5.11090574914218431648885106409, −4.90640372836009974697442855441, −4.85498820906525652532797636383, −4.22545949719542738448259963100, −4.12566059157906538560690873193, −4.06498697054004920083203930398, −3.52725488187448770204150681972, −3.10559347532886370234082178054, −2.35274810372861076627246468867, −2.18552927336778953146604969676, −2.08670824598092730451087395858, −1.68191823746829392101927104267, −1.08868185482885373761069613152, −0.848118109209223973030578593831, −0.65765036634972510776457339643, 0.65765036634972510776457339643, 0.848118109209223973030578593831, 1.08868185482885373761069613152, 1.68191823746829392101927104267, 2.08670824598092730451087395858, 2.18552927336778953146604969676, 2.35274810372861076627246468867, 3.10559347532886370234082178054, 3.52725488187448770204150681972, 4.06498697054004920083203930398, 4.12566059157906538560690873193, 4.22545949719542738448259963100, 4.85498820906525652532797636383, 4.90640372836009974697442855441, 5.11090574914218431648885106409, 5.72652633179915019160833579688, 6.10320096753296583924939336118, 6.23457926262712208796389110711, 6.24867010664876310423898326668, 6.68238811525108223837190753596, 6.96801570104968158819303396632, 7.52213522491542998239252820619, 7.60933599002272806508545558447, 7.71961054666166689607552563894, 8.313550274550052274072416963525

Graph of the ZZ-function along the critical line