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Properties

Label 16-162e8-1.1-c4e8-0-0
Degree 1616
Conductor 4.744×10174.744\times 10^{17}
Sign 11
Analytic cond. 6.18407×1096.18407\times 10^{9}
Root an. cond. 4.092174.09217
Motivic weight 44
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  + 16·4-s + 52·7-s − 572·13-s + 64·16-s − 248·19-s − 304·25-s + 832·28-s − 3.58e3·31-s − 1.88e4·37-s − 3.02e3·43-s + 1.49e3·49-s − 9.15e3·52-s + 4.14e3·61-s − 1.02e3·64-s + 1.20e4·67-s − 3.58e3·73-s − 3.96e3·76-s + 1.50e4·79-s − 2.97e4·91-s − 4.63e4·97-s − 4.86e3·100-s + 3.35e4·103-s + 8.90e4·109-s + 3.32e3·112-s − 4.42e4·121-s − 5.73e4·124-s + 127-s + ⋯
L(s)  = 1  + 4-s + 1.06·7-s − 3.38·13-s + 1/4·16-s − 0.686·19-s − 0.486·25-s + 1.06·28-s − 3.72·31-s − 13.7·37-s − 1.63·43-s + 0.621·49-s − 3.38·52-s + 1.11·61-s − 1/4·64-s + 2.69·67-s − 0.672·73-s − 0.686·76-s + 2.40·79-s − 3.59·91-s − 4.92·97-s − 0.486·100-s + 3.16·103-s + 7.49·109-s + 0.265·112-s − 3.01·121-s − 3.72·124-s + 6.20e−5·127-s + ⋯

Functional equation

Λ(s)=((28⋅332)s/2 ΓC(s)8 L(s)=( Λ(5−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}
Λ(s)=((28⋅332)s/2 ΓC(s+2)8 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s+2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 1616
Conductor: 28⋅3322^{8} \cdot 3^{32}
Sign: 11
Analytic conductor: 6.18407×1096.18407\times 10^{9}
Root analytic conductor: 4.092174.09217
Motivic weight: 44
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (16, 28⋅332, ( :[2]8), 1)(16,\ 2^{8} \cdot 3^{32} ,\ ( \ : [2]^{8} ),\ 1 )

Particular Values

L(52)L(\frac{5}{2}) ≈\approx 0.14235399910.1423539991
L(12)L(\frac12) ≈\approx 0.14235399910.1423539991
L(3)L(3) 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−p3T2+p6T4)2 ( 1 - p^{3} T^{2} + p^{6} T^{4} )^{2}
3 1 1
good5 1+304T2+314737T4−305085584T6−139261263584T8−305085584p8T10+314737p16T12+304p24T14+p32T16 1 + 304 T^{2} + 314737 T^{4} - 305085584 T^{6} - 139261263584 T^{8} - 305085584 p^{8} T^{10} + 314737 p^{16} T^{12} + 304 p^{24} T^{14} + p^{32} T^{16}
7 (1−26T+268T2+114244T3−7221413T4+114244p4T5+268p8T6−26p12T7+p16T8)2 ( 1 - 26 T + 268 T^{2} + 114244 T^{3} - 7221413 T^{4} + 114244 p^{4} T^{5} + 268 p^{8} T^{6} - 26 p^{12} T^{7} + p^{16} T^{8} )^{2}
11 1+44200T2+1068005038T4+20195740240000T6+319619471776329283T8+20195740240000p8T10+1068005038p16T12+44200p24T14+p32T16 1 + 44200 T^{2} + 1068005038 T^{4} + 20195740240000 T^{6} + 319619471776329283 T^{8} + 20195740240000 p^{8} T^{10} + 1068005038 p^{16} T^{12} + 44200 p^{24} T^{14} + p^{32} T^{16}
13 (1+22pT+9517T2+333454pT3+2072347564T4+333454p5T5+9517p8T6+22p13T7+p16T8)2 ( 1 + 22 p T + 9517 T^{2} + 333454 p T^{3} + 2072347564 T^{4} + 333454 p^{5} T^{5} + 9517 p^{8} T^{6} + 22 p^{13} T^{7} + p^{16} T^{8} )^{2}
17 (1−267736T2+30823871559T4−267736p8T6+p16T8)2 ( 1 - 267736 T^{2} + 30823871559 T^{4} - 267736 p^{8} T^{6} + p^{16} T^{8} )^{2}
19 (1+62T+211680T2+62p4T3+p8T4)4 ( 1 + 62 T + 211680 T^{2} + 62 p^{4} T^{3} + p^{8} T^{4} )^{4}
23 1−33536T2−77046473090T4+2630927174926336T6− 1 - 33536 T^{2} - 77046473090 T^{4} + 2630927174926336 T^{6} - 10 ⁣⋯ ⁣4110\!\cdots\!41T8+2630927174926336p8T10−77046473090p16T12−33536p24T14+p32T16 T^{8} + 2630927174926336 p^{8} T^{10} - 77046473090 p^{16} T^{12} - 33536 p^{24} T^{14} + p^{32} T^{16}
29 1+343288T2−592173740039T4−99715701245339432T6+ 1 + 343288 T^{2} - 592173740039 T^{4} - 99715701245339432 T^{6} + 19 ⁣⋯ ⁣8419\!\cdots\!84T8−99715701245339432p8T10−592173740039p16T12+343288p24T14+p32T16 T^{8} - 99715701245339432 p^{8} T^{10} - 592173740039 p^{16} T^{12} + 343288 p^{24} T^{14} + p^{32} T^{16}
31 (1+1792T+640138T2+1297558528T3+2792128129891T4+1297558528p4T5+640138p8T6+1792p12T7+p16T8)2 ( 1 + 1792 T + 640138 T^{2} + 1297558528 T^{3} + 2792128129891 T^{4} + 1297558528 p^{4} T^{5} + 640138 p^{8} T^{6} + 1792 p^{12} T^{7} + p^{16} T^{8} )^{2}
37 (1+4700T+9270795T2+4700p4T3+p8T4)4 ( 1 + 4700 T + 9270795 T^{2} + 4700 p^{4} T^{3} + p^{8} T^{4} )^{4}
41 1+8941048T2+44849615538094T4+ 1 + 8941048 T^{2} + 44849615538094 T^{4} + 17 ⁣⋯ ⁣6417\!\cdots\!64T6+ T^{6} + 52 ⁣⋯ ⁣7552\!\cdots\!75T8+ T^{8} + 17 ⁣⋯ ⁣6417\!\cdots\!64p8T10+44849615538094p16T12+8941048p24T14+p32T16 p^{8} T^{10} + 44849615538094 p^{16} T^{12} + 8941048 p^{24} T^{14} + p^{32} T^{16}
43 (1+1510T−4281644T2−416545580T3+23573075883835T4−416545580p4T5−4281644p8T6+1510p12T7+p16T8)2 ( 1 + 1510 T - 4281644 T^{2} - 416545580 T^{3} + 23573075883835 T^{4} - 416545580 p^{4} T^{5} - 4281644 p^{8} T^{6} + 1510 p^{12} T^{7} + p^{16} T^{8} )^{2}
47 1+7387156T2+18121663956442T4−82545289029903849968T6− 1 + 7387156 T^{2} + 18121663956442 T^{4} - 82545289029903849968 T^{6} - 53 ⁣⋯ ⁣8153\!\cdots\!81T8−82545289029903849968p8T10+18121663956442p16T12+7387156p24T14+p32T16 T^{8} - 82545289029903849968 p^{8} T^{10} + 18121663956442 p^{16} T^{12} + 7387156 p^{24} T^{14} + p^{32} T^{16}
53 (1−28489720T2+325176478293714T4−28489720p8T6+p16T8)2 ( 1 - 28489720 T^{2} + 325176478293714 T^{4} - 28489720 p^{8} T^{6} + p^{16} T^{8} )^{2}
59 1+44516212T2+1194314285673178T4+ 1 + 44516212 T^{2} + 1194314285673178 T^{4} + 21 ⁣⋯ ⁣8821\!\cdots\!88T6+ T^{6} + 30 ⁣⋯ ⁣9130\!\cdots\!91T8+ T^{8} + 21 ⁣⋯ ⁣8821\!\cdots\!88p8T10+1194314285673178p16T12+44516212p24T14+p32T16 p^{8} T^{10} + 1194314285673178 p^{16} T^{12} + 44516212 p^{24} T^{14} + p^{32} T^{16}
61 (1−2072T−21145367T2+4668487432T3+395969820934576T4+4668487432p4T5−21145367p8T6−2072p12T7+p16T8)2 ( 1 - 2072 T - 21145367 T^{2} + 4668487432 T^{3} + 395969820934576 T^{4} + 4668487432 p^{4} T^{5} - 21145367 p^{8} T^{6} - 2072 p^{12} T^{7} + p^{16} T^{8} )^{2}
67 (1−6038T+21578188T2+153503989468T3−919333783837829T4+153503989468p4T5+21578188p8T6−6038p12T7+p16T8)2 ( 1 - 6038 T + 21578188 T^{2} + 153503989468 T^{3} - 919333783837829 T^{4} + 153503989468 p^{4} T^{5} + 21578188 p^{8} T^{6} - 6038 p^{12} T^{7} + p^{16} T^{8} )^{2}
71 (1−26476960T2+845007854749314T4−26476960p8T6+p16T8)2 ( 1 - 26476960 T^{2} + 845007854749314 T^{4} - 26476960 p^{8} T^{6} + p^{16} T^{8} )^{2}
73 (1+896T−162489T2+896p4T3+p8T4)4 ( 1 + 896 T - 162489 T^{2} + 896 p^{4} T^{3} + p^{8} T^{4} )^{4}
79 (1−7502T−35614316T2−104984173316T3+4735346523875515T4−104984173316p4T5−35614316p8T6−7502p12T7+p16T8)2 ( 1 - 7502 T - 35614316 T^{2} - 104984173316 T^{3} + 4735346523875515 T^{4} - 104984173316 p^{4} T^{5} - 35614316 p^{8} T^{6} - 7502 p^{12} T^{7} + p^{16} T^{8} )^{2}
83 1+185096692T2+21192092541154858T4+ 1 + 185096692 T^{2} + 21192092541154858 T^{4} + 15 ⁣⋯ ⁣0815\!\cdots\!08T6+ T^{6} + 88 ⁣⋯ ⁣3188\!\cdots\!31T8+ T^{8} + 15 ⁣⋯ ⁣0815\!\cdots\!08p8T10+21192092541154858p16T12+185096692p24T14+p32T16 p^{8} T^{10} + 21192092541154858 p^{16} T^{12} + 185096692 p^{24} T^{14} + p^{32} T^{16}
89 (1−116789296T2+8841733583204319T4−116789296p8T6+p16T8)2 ( 1 - 116789296 T^{2} + 8841733583204319 T^{4} - 116789296 p^{8} T^{6} + p^{16} T^{8} )^{2}
97 (1+23152T+227547466T2+3042382927552T3+39861818336661859T4+3042382927552p4T5+227547466p8T6+23152p12T7+p16T8)2 ( 1 + 23152 T + 227547466 T^{2} + 3042382927552 T^{3} + 39861818336661859 T^{4} + 3042382927552 p^{4} T^{5} + 227547466 p^{8} T^{6} + 23152 p^{12} T^{7} + p^{16} T^{8} )^{2}
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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

−5.17391955478848289890213021080, −5.15669432275966114804094319915, −5.11260245857849206341162471118, −4.79982622171108095295116690168, −4.63878414352447064977494468098, −4.04934378741390053639303379918, −4.02687955801441480136905574007, −3.85072386453772780672842706826, −3.65326343234008419766652731186, −3.58845529223399943831746453426, −3.41130791316374760248805599604, −3.28135416269961450335083021817, −3.04033118768955115601207867377, −2.68161791228087572915073556520, −2.46671011225977705969861453771, −2.17048088875929241793692025480, −2.01184541473869320880390228628, −1.84751062786432203001577928666, −1.70035321516695241219496604682, −1.70018912987514863128351857132, −1.69094880290039602065955543483, −1.06693972246287747964692662658, −0.35420835029935112297326025529, −0.23352924314046519199533659772, −0.088888185955104982266063468310, 0.088888185955104982266063468310, 0.23352924314046519199533659772, 0.35420835029935112297326025529, 1.06693972246287747964692662658, 1.69094880290039602065955543483, 1.70018912987514863128351857132, 1.70035321516695241219496604682, 1.84751062786432203001577928666, 2.01184541473869320880390228628, 2.17048088875929241793692025480, 2.46671011225977705969861453771, 2.68161791228087572915073556520, 3.04033118768955115601207867377, 3.28135416269961450335083021817, 3.41130791316374760248805599604, 3.58845529223399943831746453426, 3.65326343234008419766652731186, 3.85072386453772780672842706826, 4.02687955801441480136905574007, 4.04934378741390053639303379918, 4.63878414352447064977494468098, 4.79982622171108095295116690168, 5.11260245857849206341162471118, 5.15669432275966114804094319915, 5.17391955478848289890213021080

Graph of the ZZ-function along the critical line

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