↑

Properties

Label 16-31e16-1.1-c1e8-0-3
Degree 1616
Conductor 7.274×10237.274\times 10^{23}
Sign 11
Analytic cond. 1.20227×1071.20227\times 10^{7}
Root an. cond. 2.770132.77013
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  + 2·2-s + 5·4-s − 8·7-s + 4·8-s − 2·9-s − 16·14-s + 4·16-s − 4·18-s + 8·19-s − 40·25-s − 40·28-s − 22·32-s − 10·36-s + 16·38-s + 4·41-s − 24·47-s + 30·49-s − 80·50-s − 32·56-s − 16·59-s + 16·63-s − 44·64-s − 32·67-s − 16·71-s − 8·72-s + 40·76-s + 9·81-s + ⋯
L(s)  = 1  + 1.41·2-s + 5/2·4-s − 3.02·7-s + 1.41·8-s − 2/3·9-s − 4.27·14-s + 16-s − 0.942·18-s + 1.83·19-s − 8·25-s − 7.55·28-s − 3.88·32-s − 5/3·36-s + 2.59·38-s + 0.624·41-s − 3.50·47-s + 30/7·49-s − 11.3·50-s − 4.27·56-s − 2.08·59-s + 2.01·63-s − 5.5·64-s − 3.90·67-s − 1.89·71-s − 0.942·72-s + 4.58·76-s + 81-s + ⋯

Functional equation

Λ(s)=((3116)s/2 ΓC(s)8 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(31^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((3116)s/2 ΓC(s+1/2)8 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(31^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 1616
Conductor: 311631^{16}
Sign: 11
Analytic conductor: 1.20227×1071.20227\times 10^{7}
Root analytic conductor: 2.770132.77013
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (16, 3116, ( :[1/2]8), 1)(16,\ 31^{16} ,\ ( \ : [1/2]^{8} ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 0.54358510900.5435851090
L(12)L(\frac12) ≈\approx 0.54358510900.5435851090
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}
ppFp(T)F_p(T)
bad31 1 1
good2 (1−T−T2+3T3−T4+3pT5−p2T6−p3T7+p4T8)2 ( 1 - T - T^{2} + 3 T^{3} - T^{4} + 3 p T^{5} - p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} )^{2}
3 1+2T2−5T4−28T6−11T8−28p2T10−5p4T12+2p6T14+p8T16 1 + 2 T^{2} - 5 T^{4} - 28 T^{6} - 11 T^{8} - 28 p^{2} T^{10} - 5 p^{4} T^{12} + 2 p^{6} T^{14} + p^{8} T^{16}
5 (1+pT2)8 ( 1 + p T^{2} )^{8}
7 (1+4T+9T2+8T3−31T4+8pT5+9p2T6+4p3T7+p4T8)2 ( 1 + 4 T + 9 T^{2} + 8 T^{3} - 31 T^{4} + 8 p T^{5} + 9 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2}
11 1−14T2+75T4+644T6−18091T8+644p2T10+75p4T12−14p6T14+p8T16 1 - 14 T^{2} + 75 T^{4} + 644 T^{6} - 18091 T^{8} + 644 p^{2} T^{10} + 75 p^{4} T^{12} - 14 p^{6} T^{14} + p^{8} T^{16}
13 1−24T2+407T4−5712T6+68305T8−5712p2T10+407p4T12−24p6T14+p8T16 1 - 24 T^{2} + 407 T^{4} - 5712 T^{6} + 68305 T^{8} - 5712 p^{2} T^{10} + 407 p^{4} T^{12} - 24 p^{6} T^{14} + p^{8} T^{16}
17 1−32T2+735T4−14272T6+244289T8−14272p2T10+735p4T12−32p6T14+p8T16 1 - 32 T^{2} + 735 T^{4} - 14272 T^{6} + 244289 T^{8} - 14272 p^{2} T^{10} + 735 p^{4} T^{12} - 32 p^{6} T^{14} + p^{8} T^{16}
19 (1−4T−3T2+88T3−295T4+88pT5−3p2T6−4p3T7+p4T8)2 ( 1 - 4 T - 3 T^{2} + 88 T^{3} - 295 T^{4} + 88 p T^{5} - 3 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2}
23 1−14T2−333T4+12068T6+7205T8+12068p2T10−333p4T12−14p6T14+p8T16 1 - 14 T^{2} - 333 T^{4} + 12068 T^{6} + 7205 T^{8} + 12068 p^{2} T^{10} - 333 p^{4} T^{12} - 14 p^{6} T^{14} + p^{8} T^{16}
29 1−56T2+2295T4−81424T6+2629649T8−81424p2T10+2295p4T12−56p6T14+p8T16 1 - 56 T^{2} + 2295 T^{4} - 81424 T^{6} + 2629649 T^{8} - 81424 p^{2} T^{10} + 2295 p^{4} T^{12} - 56 p^{6} T^{14} + p^{8} T^{16}
37 (1+56T2+p2T4)4 ( 1 + 56 T^{2} + p^{2} T^{4} )^{4}
41 (1−2T−37T2+156T3+1205T4+156pT5−37p2T6−2p3T7+p4T8)2 ( 1 - 2 T - 37 T^{2} + 156 T^{3} + 1205 T^{4} + 156 p T^{5} - 37 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2}
43 1−14T2−1653T4+49028T6+2370005T8+49028p2T10−1653p4T12−14p6T14+p8T16 1 - 14 T^{2} - 1653 T^{4} + 49028 T^{6} + 2370005 T^{8} + 49028 p^{2} T^{10} - 1653 p^{4} T^{12} - 14 p^{6} T^{14} + p^{8} T^{16}
47 (1+12T+97T2+600T3+2641T4+600pT5+97p2T6+12p3T7+p4T8)2 ( 1 + 12 T + 97 T^{2} + 600 T^{3} + 2641 T^{4} + 600 p T^{5} + 97 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2}
53 1−88T2+4935T4−187088T6+2601329T8−187088p2T10+4935p4T12−88p6T14+p8T16 1 - 88 T^{2} + 4935 T^{4} - 187088 T^{6} + 2601329 T^{8} - 187088 p^{2} T^{10} + 4935 p^{4} T^{12} - 88 p^{6} T^{14} + p^{8} T^{16}
59 (1+8T+5T2−432T3−3751T4−432pT5+5p2T6+8p3T7+p4T8)2 ( 1 + 8 T + 5 T^{2} - 432 T^{3} - 3751 T^{4} - 432 p T^{5} + 5 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2}
61 (1+120T2+p2T4)4 ( 1 + 120 T^{2} + p^{2} T^{4} )^{4}
67 (1+4T+pT2)8 ( 1 + 4 T + p T^{2} )^{8}
71 (1+8T−7T2−624T3−4495T4−624pT5−7p2T6+8p3T7+p4T8)2 ( 1 + 8 T - 7 T^{2} - 624 T^{3} - 4495 T^{4} - 624 p T^{5} - 7 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2}
73 1−128T2+11055T4−732928T6+34902689T8−732928p2T10+11055p4T12−128p6T14+p8T16 1 - 128 T^{2} + 11055 T^{4} - 732928 T^{6} + 34902689 T^{8} - 732928 p^{2} T^{10} + 11055 p^{4} T^{12} - 128 p^{6} T^{14} + p^{8} T^{16}
79 1−30T2−5341T4+347460T6+22909381T8+347460p2T10−5341p4T12−30p6T14+p8T16 1 - 30 T^{2} - 5341 T^{4} + 347460 T^{6} + 22909381 T^{8} + 347460 p^{2} T^{10} - 5341 p^{4} T^{12} - 30 p^{6} T^{14} + p^{8} T^{16}
83 1+34T2−5733T4−429148T6+24903605T8−429148p2T10−5733p4T12+34p6T14+p8T16 1 + 34 T^{2} - 5733 T^{4} - 429148 T^{6} + 24903605 T^{8} - 429148 p^{2} T^{10} - 5733 p^{4} T^{12} + 34 p^{6} T^{14} + p^{8} T^{16}
89 1−128T2+8463T4−69376T6−58155295T8−69376p2T10+8463p4T12−128p6T14+p8T16 1 - 128 T^{2} + 8463 T^{4} - 69376 T^{6} - 58155295 T^{8} - 69376 p^{2} T^{10} + 8463 p^{4} T^{12} - 128 p^{6} T^{14} + p^{8} T^{16}
97 (1+8T−33T2−1040T3−5119T4−1040pT5−33p2T6+8p3T7+p4T8)2 ( 1 + 8 T - 33 T^{2} - 1040 T^{3} - 5119 T^{4} - 1040 p T^{5} - 33 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2}
show more
show less
   L(s)=∏p ∏j=116(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−4.20920402739178094077432048899, −4.10264067400641663459788834061, −4.00613070358330865625264534731, −3.85189044362347456404444368818, −3.64339140380163334650393725983, −3.49372927019269652497706819915, −3.38895091248599708654057576514, −3.38012084922844866881492975228, −3.30749823697700474063424828720, −3.30541539917666214436583216260, −3.08903106456124521169085803083, −2.80101347493079498441657035596, −2.67538618661698730870904822945, −2.61233695252139389758908225888, −2.45206236445472566432441286515, −2.31867655238823765203183921213, −1.89654529688565128923031207327, −1.87932386944512873731836542569, −1.71691447868741558980655395382, −1.71376050206020056652030416325, −1.46741319083444709556948620869, −1.31113221548804069886282760625, −0.42069047049783314087943704927, −0.34733634611187430549729242984, −0.15113960954661166720935292978, 0.15113960954661166720935292978, 0.34733634611187430549729242984, 0.42069047049783314087943704927, 1.31113221548804069886282760625, 1.46741319083444709556948620869, 1.71376050206020056652030416325, 1.71691447868741558980655395382, 1.87932386944512873731836542569, 1.89654529688565128923031207327, 2.31867655238823765203183921213, 2.45206236445472566432441286515, 2.61233695252139389758908225888, 2.67538618661698730870904822945, 2.80101347493079498441657035596, 3.08903106456124521169085803083, 3.30541539917666214436583216260, 3.30749823697700474063424828720, 3.38012084922844866881492975228, 3.38895091248599708654057576514, 3.49372927019269652497706819915, 3.64339140380163334650393725983, 3.85189044362347456404444368818, 4.00613070358330865625264534731, 4.10264067400641663459788834061, 4.20920402739178094077432048899

Graph of the ZZ-function along the critical line

Plot not available for L-functions of degree greater than 10.