↑

Properties

Label 6-7400e3-1.1-c1e3-0-0
Degree 66
Conductor 405224000000405224000000
Sign 11
Analytic cond. 206312.206312.
Root an. cond. 7.686957.68695
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  − 2·3-s − 7·7-s − 9-s − 3·13-s + 4·17-s + 8·19-s + 14·21-s − 9·23-s + 7·27-s − 9·29-s + 17·31-s + 3·37-s + 6·39-s − 16·41-s + 4·43-s − 11·47-s + 18·49-s − 8·51-s + 3·53-s − 16·57-s − 2·59-s + 15·61-s + 7·63-s + 5·67-s + 18·69-s − 5·71-s + 6·73-s + ⋯
L(s)  = 1  − 1.15·3-s − 2.64·7-s − 1/3·9-s − 0.832·13-s + 0.970·17-s + 1.83·19-s + 3.05·21-s − 1.87·23-s + 1.34·27-s − 1.67·29-s + 3.05·31-s + 0.493·37-s + 0.960·39-s − 2.49·41-s + 0.609·43-s − 1.60·47-s + 18/7·49-s − 1.12·51-s + 0.412·53-s − 2.11·57-s − 0.260·59-s + 1.92·61-s + 0.881·63-s + 0.610·67-s + 2.16·69-s − 0.593·71-s + 0.702·73-s + ⋯

Functional equation

Λ(s)=((29⋅56⋅373)s/2 ΓC(s)3 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 5^{6} \cdot 37^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((29⋅56⋅373)s/2 ΓC(s+1/2)3 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 5^{6} \cdot 37^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 66
Conductor: 29⋅56⋅3732^{9} \cdot 5^{6} \cdot 37^{3}
Sign: 11
Analytic conductor: 206312.206312.
Root analytic conductor: 7.686957.68695
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (6, 29⋅56⋅373, ( :1/2,1/2,1/2), 1)(6,\ 2^{9} \cdot 5^{6} \cdot 37^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 1.1656858561.165685856
L(12)L(\frac12) ≈\approx 1.1656858561.165685856
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
5 1 1
37C1C_1 (1−T)3 ( 1 - T )^{3}
good3S4×C2S_4\times C_2 1+2T+5T2+5T3+5pT4+2p2T5+p3T6 1 + 2 T + 5 T^{2} + 5 T^{3} + 5 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6}
7S4×C2S_4\times C_2 1+pT+31T2+94T3+31pT4+p3T5+p3T6 1 + p T + 31 T^{2} + 94 T^{3} + 31 p T^{4} + p^{3} T^{5} + p^{3} T^{6}
11S4×C2S_4\times C_2 1−3T2+27T3−3pT4+p3T6 1 - 3 T^{2} + 27 T^{3} - 3 p T^{4} + p^{3} T^{6}
13S4×C2S_4\times C_2 1+3T+6T2+16T3+6pT4+3p2T5+p3T6 1 + 3 T + 6 T^{2} + 16 T^{3} + 6 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6}
17S4×C2S_4\times C_2 1−4T+31T2−120T3+31pT4−4p2T5+p3T6 1 - 4 T + 31 T^{2} - 120 T^{3} + 31 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6}
19S4×C2S_4\times C_2 1−8T+53T2−240T3+53pT4−8p2T5+p3T6 1 - 8 T + 53 T^{2} - 240 T^{3} + 53 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6}
23S4×C2S_4\times C_2 1+9T+4pT2+428T3+4p2T4+9p2T5+p3T6 1 + 9 T + 4 p T^{2} + 428 T^{3} + 4 p^{2} T^{4} + 9 p^{2} T^{5} + p^{3} T^{6}
29S4×C2S_4\times C_2 1+9T+110T2+536T3+110pT4+9p2T5+p3T6 1 + 9 T + 110 T^{2} + 536 T^{3} + 110 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6}
31S4×C2S_4\times C_2 1−17T+184T2−1202T3+184pT4−17p2T5+p3T6 1 - 17 T + 184 T^{2} - 1202 T^{3} + 184 p T^{4} - 17 p^{2} T^{5} + p^{3} T^{6}
41S4×C2S_4\times C_2 1+16T+193T2+1359T3+193pT4+16p2T5+p3T6 1 + 16 T + 193 T^{2} + 1359 T^{3} + 193 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6}
43S4×C2S_4\times C_2 1−4T+9T2−112T3+9pT4−4p2T5+p3T6 1 - 4 T + 9 T^{2} - 112 T^{3} + 9 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6}
47S4×C2S_4\times C_2 1+11T+131T2+1030T3+131pT4+11p2T5+p3T6 1 + 11 T + 131 T^{2} + 1030 T^{3} + 131 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6}
53S4×C2S_4\times C_2 1−3T+59T2−610T3+59pT4−3p2T5+p3T6 1 - 3 T + 59 T^{2} - 610 T^{3} + 59 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6}
59S4×C2S_4\times C_2 1+2T+53T2+220T3+53pT4+2p2T5+p3T6 1 + 2 T + 53 T^{2} + 220 T^{3} + 53 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6}
61S4×C2S_4\times C_2 1−15T+212T2−1778T3+212pT4−15p2T5+p3T6 1 - 15 T + 212 T^{2} - 1778 T^{3} + 212 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6}
67S4×C2S_4\times C_2 1−5T+22T2+274T3+22pT4−5p2T5+p3T6 1 - 5 T + 22 T^{2} + 274 T^{3} + 22 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6}
71S4×C2S_4\times C_2 1+5T+189T2+714T3+189pT4+5p2T5+p3T6 1 + 5 T + 189 T^{2} + 714 T^{3} + 189 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6}
73S4×C2S_4\times C_2 1−6T+195T2−839T3+195pT4−6p2T5+p3T6 1 - 6 T + 195 T^{2} - 839 T^{3} + 195 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6}
79S4×C2S_4\times C_2 1+T+218T2+126T3+218pT4+p2T5+p3T6 1 + T + 218 T^{2} + 126 T^{3} + 218 p T^{4} + p^{2} T^{5} + p^{3} T^{6}
83S4×C2S_4\times C_2 1−9T+173T2−1606T3+173pT4−9p2T5+p3T6 1 - 9 T + 173 T^{2} - 1606 T^{3} + 173 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6}
89S4×C2S_4\times C_2 1−16T+3pT2−2784T3+3p2T4−16p2T5+p3T6 1 - 16 T + 3 p T^{2} - 2784 T^{3} + 3 p^{2} T^{4} - 16 p^{2} T^{5} + p^{3} T^{6}
97S4×C2S_4\times C_2 1+47T2−256T3+47pT4+p3T6 1 + 47 T^{2} - 256 T^{3} + 47 p T^{4} + 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

−6.82597383886115103152097747077, −6.62476670334853610059785180050, −6.50519004640076053306026871112, −6.29225074822902450509485758747, −5.94693968480935463685694953958, −5.91467031455581180554891968681, −5.75664377606120889453259489507, −5.25819526777832721885376784105, −5.11488878697100925241818021955, −5.09971396338078411086741366277, −4.61782844692307296945102744552, −4.46630307893789394838827068730, −3.91398495037311008660499534178, −3.64943228517592097414866611642, −3.51335728165785725100706819935, −3.42144629669058363788685205009, −2.92275536711509294072318885703, −2.75076857123395160049850922252, −2.68195605252461722984835362331, −2.15355087273357745853983803431, −1.65860456085650906516836146068, −1.51516477875460610649956303060, −0.63139176988533939824238816023, −0.58579249176455241752766890652, −0.38871566194525147371078479803, 0.38871566194525147371078479803, 0.58579249176455241752766890652, 0.63139176988533939824238816023, 1.51516477875460610649956303060, 1.65860456085650906516836146068, 2.15355087273357745853983803431, 2.68195605252461722984835362331, 2.75076857123395160049850922252, 2.92275536711509294072318885703, 3.42144629669058363788685205009, 3.51335728165785725100706819935, 3.64943228517592097414866611642, 3.91398495037311008660499534178, 4.46630307893789394838827068730, 4.61782844692307296945102744552, 5.09971396338078411086741366277, 5.11488878697100925241818021955, 5.25819526777832721885376784105, 5.75664377606120889453259489507, 5.91467031455581180554891968681, 5.94693968480935463685694953958, 6.29225074822902450509485758747, 6.50519004640076053306026871112, 6.62476670334853610059785180050, 6.82597383886115103152097747077

Graph of the ZZ-function along the critical line