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Properties

Label 16-3276e8-1.1-c1e8-0-3
Degree 1616
Conductor 1.327×10281.327\times 10^{28}
Sign 11
Analytic cond. 2.19264×10112.19264\times 10^{11}
Root an. cond. 5.114585.11458
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  + 2·13-s − 4·17-s + 6·23-s + 21·25-s + 2·29-s − 6·43-s − 4·49-s − 22·53-s + 8·61-s − 26·79-s − 28·101-s − 32·107-s − 50·113-s + 52·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 16·169-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  + 0.554·13-s − 0.970·17-s + 1.25·23-s + 21/5·25-s + 0.371·29-s − 0.914·43-s − 4/7·49-s − 3.02·53-s + 1.02·61-s − 2.92·79-s − 2.78·101-s − 3.09·107-s − 4.70·113-s + 4.72·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.23·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯

Functional equation

Λ(s)=((216⋅316⋅78⋅138)s/2 ΓC(s)8 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 7^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((216⋅316⋅78⋅138)s/2 ΓC(s+1/2)8 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 7^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 1616
Conductor: 216⋅316⋅78⋅1382^{16} \cdot 3^{16} \cdot 7^{8} \cdot 13^{8}
Sign: 11
Analytic conductor: 2.19264×10112.19264\times 10^{11}
Root analytic conductor: 5.114585.11458
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (16, 216⋅316⋅78⋅138, ( :[1/2]8), 1)(16,\ 2^{16} \cdot 3^{16} \cdot 7^{8} \cdot 13^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 2.3012367312.301236731
L(12)L(\frac12) ≈\approx 2.3012367312.301236731
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)
bad2 1 1
3 1 1
7 (1+T2)4 ( 1 + T^{2} )^{4}
13 1−2T+20T2−94T3+278T4−94pT5+20p2T6−2p3T7+p4T8 1 - 2 T + 20 T^{2} - 94 T^{3} + 278 T^{4} - 94 p T^{5} + 20 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8}
good5 1−21T2+249T4−2006T6+11626T8−2006p2T10+249p4T12−21p6T14+p8T16 1 - 21 T^{2} + 249 T^{4} - 2006 T^{6} + 11626 T^{8} - 2006 p^{2} T^{10} + 249 p^{4} T^{12} - 21 p^{6} T^{14} + p^{8} T^{16}
11 1−52T2+1368T4−24280T6+312502T8−24280p2T10+1368p4T12−52p6T14+p8T16 1 - 52 T^{2} + 1368 T^{4} - 24280 T^{6} + 312502 T^{8} - 24280 p^{2} T^{10} + 1368 p^{4} T^{12} - 52 p^{6} T^{14} + p^{8} T^{16}
17 (1+2T+16T2+24T3+10T4+24pT5+16p2T6+2p3T7+p4T8)2 ( 1 + 2 T + 16 T^{2} + 24 T^{3} + 10 T^{4} + 24 p T^{5} + 16 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2}
19 1−81T2+3529T4−102890T6+2243374T8−102890p2T10+3529p4T12−81p6T14+p8T16 1 - 81 T^{2} + 3529 T^{4} - 102890 T^{6} + 2243374 T^{8} - 102890 p^{2} T^{10} + 3529 p^{4} T^{12} - 81 p^{6} T^{14} + p^{8} T^{16}
23 (1−3T+79T2−160T3+2540T4−160pT5+79p2T6−3p3T7+p4T8)2 ( 1 - 3 T + 79 T^{2} - 160 T^{3} + 2540 T^{4} - 160 p T^{5} + 79 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} )^{2}
29 (1−T+33T2−2T3+1414T4−2pT5+33p2T6−p3T7+p4T8)2 ( 1 - T + 33 T^{2} - 2 T^{3} + 1414 T^{4} - 2 p T^{5} + 33 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} )^{2}
31 1−125T2+8493T4−384510T6+13439406T8−384510p2T10+8493p4T12−125p6T14+p8T16 1 - 125 T^{2} + 8493 T^{4} - 384510 T^{6} + 13439406 T^{8} - 384510 p^{2} T^{10} + 8493 p^{4} T^{12} - 125 p^{6} T^{14} + p^{8} T^{16}
37 1−84T2+2216T4+61384T6−5827706T8+61384p2T10+2216p4T12−84p6T14+p8T16 1 - 84 T^{2} + 2216 T^{4} + 61384 T^{6} - 5827706 T^{8} + 61384 p^{2} T^{10} + 2216 p^{4} T^{12} - 84 p^{6} T^{14} + p^{8} T^{16}
41 1−172T2+16612T4−1090964T6+51790710T8−1090964p2T10+16612p4T12−172p6T14+p8T16 1 - 172 T^{2} + 16612 T^{4} - 1090964 T^{6} + 51790710 T^{8} - 1090964 p^{2} T^{10} + 16612 p^{4} T^{12} - 172 p^{6} T^{14} + p^{8} T^{16}
43 (1+3T+87T2+20T3+4016T4+20pT5+87p2T6+3p3T7+p4T8)2 ( 1 + 3 T + 87 T^{2} + 20 T^{3} + 4016 T^{4} + 20 p T^{5} + 87 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} )^{2}
47 1−137T2+13681T4−938098T6+50458758T8−938098p2T10+13681p4T12−137p6T14+p8T16 1 - 137 T^{2} + 13681 T^{4} - 938098 T^{6} + 50458758 T^{8} - 938098 p^{2} T^{10} + 13681 p^{4} T^{12} - 137 p^{6} T^{14} + p^{8} T^{16}
53 (1+11T+209T2+1686T3+16482T4+1686pT5+209p2T6+11p3T7+p4T8)2 ( 1 + 11 T + 209 T^{2} + 1686 T^{3} + 16482 T^{4} + 1686 p T^{5} + 209 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} )^{2}
59 1−292T2+39492T4−3405100T6+222712566T8−3405100p2T10+39492p4T12−292p6T14+p8T16 1 - 292 T^{2} + 39492 T^{4} - 3405100 T^{6} + 222712566 T^{8} - 3405100 p^{2} T^{10} + 39492 p^{4} T^{12} - 292 p^{6} T^{14} + p^{8} T^{16}
61 (1−4T+132T2−1068T3+8646T4−1068pT5+132p2T6−4p3T7+p4T8)2 ( 1 - 4 T + 132 T^{2} - 1068 T^{3} + 8646 T^{4} - 1068 p T^{5} + 132 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2}
67 1−168T2+18332T4−1592024T6+121444582T8−1592024p2T10+18332p4T12−168p6T14+p8T16 1 - 168 T^{2} + 18332 T^{4} - 1592024 T^{6} + 121444582 T^{8} - 1592024 p^{2} T^{10} + 18332 p^{4} T^{12} - 168 p^{6} T^{14} + p^{8} T^{16}
71 1−160T2+10920T4−867532T6+79016406T8−867532p2T10+10920p4T12−160p6T14+p8T16 1 - 160 T^{2} + 10920 T^{4} - 867532 T^{6} + 79016406 T^{8} - 867532 p^{2} T^{10} + 10920 p^{4} T^{12} - 160 p^{6} T^{14} + p^{8} T^{16}
73 1−305T2+45549T4−4497786T6+354140538T8−4497786p2T10+45549p4T12−305p6T14+p8T16 1 - 305 T^{2} + 45549 T^{4} - 4497786 T^{6} + 354140538 T^{8} - 4497786 p^{2} T^{10} + 45549 p^{4} T^{12} - 305 p^{6} T^{14} + p^{8} T^{16}
79 (1+13T+219T2+2704T3+22084T4+2704pT5+219p2T6+13p3T7+p4T8)2 ( 1 + 13 T + 219 T^{2} + 2704 T^{3} + 22084 T^{4} + 2704 p T^{5} + 219 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} )^{2}
83 1−117T2+15309T4−1167406T6+97224854T8−1167406p2T10+15309p4T12−117p6T14+p8T16 1 - 117 T^{2} + 15309 T^{4} - 1167406 T^{6} + 97224854 T^{8} - 1167406 p^{2} T^{10} + 15309 p^{4} T^{12} - 117 p^{6} T^{14} + p^{8} T^{16}
89 1−273T2+54405T4−7250626T6+745491194T8−7250626p2T10+54405p4T12−273p6T14+p8T16 1 - 273 T^{2} + 54405 T^{4} - 7250626 T^{6} + 745491194 T^{8} - 7250626 p^{2} T^{10} + 54405 p^{4} T^{12} - 273 p^{6} T^{14} + p^{8} T^{16}
97 1−329T2+69397T4−9804594T6+1094792986T8−9804594p2T10+69397p4T12−329p6T14+p8T16 1 - 329 T^{2} + 69397 T^{4} - 9804594 T^{6} + 1094792986 T^{8} - 9804594 p^{2} T^{10} + 69397 p^{4} T^{12} - 329 p^{6} T^{14} + p^{8} T^{16}
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   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

−3.67063405331076510340095833844, −3.39508543152180396243458827032, −3.32089168728891205471997290263, −3.15595059815724933436593019100, −3.04290282253881187147083052461, −2.98453566555373910082216441658, −2.77398016703014255493118419931, −2.72928974954659336799779295943, −2.69180527314960563627560834455, −2.60973049736296790119946064418, −2.47842425404481916264562900416, −2.40342699490560689873850161496, −2.20056182311185154659689413356, −1.91519043399298284939048364015, −1.67770148007758647227028006340, −1.54774990506521917859998408974, −1.53583311093972841324108008246, −1.37637974374407496676731427861, −1.33447250392610322844320176183, −1.28610722456293395051990019899, −0.885989048193185327027851672189, −0.860040924826318634128989553093, −0.52578354477728068428559483920, −0.34562550069372353586130543541, −0.12690236858152335507266055452, 0.12690236858152335507266055452, 0.34562550069372353586130543541, 0.52578354477728068428559483920, 0.860040924826318634128989553093, 0.885989048193185327027851672189, 1.28610722456293395051990019899, 1.33447250392610322844320176183, 1.37637974374407496676731427861, 1.53583311093972841324108008246, 1.54774990506521917859998408974, 1.67770148007758647227028006340, 1.91519043399298284939048364015, 2.20056182311185154659689413356, 2.40342699490560689873850161496, 2.47842425404481916264562900416, 2.60973049736296790119946064418, 2.69180527314960563627560834455, 2.72928974954659336799779295943, 2.77398016703014255493118419931, 2.98453566555373910082216441658, 3.04290282253881187147083052461, 3.15595059815724933436593019100, 3.32089168728891205471997290263, 3.39508543152180396243458827032, 3.67063405331076510340095833844

Graph of the ZZ-function along the critical line

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