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Properties

Label 16-31e16-1.1-c1e8-0-8
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 5·4-s + 2·7-s − 16·8-s + 8·9-s + 4·14-s − 16-s + 16·18-s − 2·19-s − 20·25-s − 10·28-s + 48·32-s − 40·36-s − 4·38-s − 26·41-s − 24·47-s + 15·49-s − 40·50-s − 32·56-s + 34·59-s + 16·63-s + 71·64-s + 48·67-s + 14·71-s − 128·72-s + 10·76-s + 49·81-s + ⋯
L(s)  = 1  + 1.41·2-s − 5/2·4-s + 0.755·7-s − 5.65·8-s + 8/3·9-s + 1.06·14-s − 1/4·16-s + 3.77·18-s − 0.458·19-s − 4·25-s − 1.88·28-s + 8.48·32-s − 6.66·36-s − 0.648·38-s − 4.06·41-s − 3.50·47-s + 15/7·49-s − 5.65·50-s − 4.27·56-s + 4.42·59-s + 2.01·63-s + 71/8·64-s + 5.86·67-s + 1.66·71-s − 15.0·72-s + 1.14·76-s + 49/9·81-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 4.3324497704.332449770
L(12)L(\frac12) ≈\approx 4.3324497704.332449770
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+p2T2−pT3+9T4−p2T5+p4T6−p3T7+p4T8)2 ( 1 - T + p^{2} T^{2} - p T^{3} + 9 T^{4} - p^{2} T^{5} + p^{4} T^{6} - p^{3} T^{7} + p^{4} T^{8} )^{2}
3 1−8T2+5pT4+62T6−371T8+62p2T10+5p5T12−8p6T14+p8T16 1 - 8 T^{2} + 5 p T^{4} + 62 T^{6} - 371 T^{8} + 62 p^{2} T^{10} + 5 p^{5} T^{12} - 8 p^{6} T^{14} + p^{8} T^{16}
5 (1+pT2+p2T4)4 ( 1 + p T^{2} + p^{2} T^{4} )^{4}
7 (1−T−6T2+13T3+29T4+13pT5−6p2T6−p3T7+p4T8)2 ( 1 - T - 6 T^{2} + 13 T^{3} + 29 T^{4} + 13 p T^{5} - 6 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} )^{2}
11 1−4T2−105T4+904T6+9089T8+904p2T10−105p4T12−4p6T14+p8T16 1 - 4 T^{2} - 105 T^{4} + 904 T^{6} + 9089 T^{8} + 904 p^{2} T^{10} - 105 p^{4} T^{12} - 4 p^{6} T^{14} + p^{8} T^{16}
13 1−44T2+1047T4−18922T6+274925T8−18922p2T10+1047p4T12−44p6T14+p8T16 1 - 44 T^{2} + 1047 T^{4} - 18922 T^{6} + 274925 T^{8} - 18922 p^{2} T^{10} + 1047 p^{4} T^{12} - 44 p^{6} T^{14} + p^{8} T^{16}
17 1−42T2+575T4+7398T6−301751T8+7398p2T10+575p4T12−42p6T14+p8T16 1 - 42 T^{2} + 575 T^{4} + 7398 T^{6} - 301751 T^{8} + 7398 p^{2} T^{10} + 575 p^{4} T^{12} - 42 p^{6} T^{14} + p^{8} T^{16}
19 (1+T−18T2−37T3+305T4−37pT5−18p2T6+p3T7+p4T8)2 ( 1 + T - 18 T^{2} - 37 T^{3} + 305 T^{4} - 37 p T^{5} - 18 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} )^{2}
23 1+26T2+1047T4+22918T6+827705T8+22918p2T10+1047p4T12+26p6T14+p8T16 1 + 26 T^{2} + 1047 T^{4} + 22918 T^{6} + 827705 T^{8} + 22918 p^{2} T^{10} + 1047 p^{4} T^{12} + 26 p^{6} T^{14} + p^{8} T^{16}
29 1−66T2+1895T4−6234T6−570791T8−6234p2T10+1895p4T12−66p6T14+p8T16 1 - 66 T^{2} + 1895 T^{4} - 6234 T^{6} - 570791 T^{8} - 6234 p^{2} T^{10} + 1895 p^{4} T^{12} - 66 p^{6} T^{14} + p^{8} T^{16}
37 (1+56T2+p2T4)4 ( 1 + 56 T^{2} + p^{2} T^{4} )^{4}
41 (1+13T+28T2−169T3−945T4−169pT5+28p2T6+13p3T7+p4T8)2 ( 1 + 13 T + 28 T^{2} - 169 T^{3} - 945 T^{4} - 169 p T^{5} + 28 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} )^{2}
43 1+66T2+6287T4+242718T6+14670025T8+242718p2T10+6287p4T12+66p6T14+p8T16 1 + 66 T^{2} + 6287 T^{4} + 242718 T^{6} + 14670025 T^{8} + 242718 p^{2} T^{10} + 6287 p^{4} T^{12} + 66 p^{6} T^{14} + p^{8} T^{16}
47 (1+12T+97T2+930T3+8581T4+930pT5+97p2T6+12p3T7+p4T8)2 ( 1 + 12 T + 97 T^{2} + 930 T^{3} + 8581 T^{4} + 930 p T^{5} + 97 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2}
53 1−178T2+16635T4−1189868T6+70020149T8−1189868p2T10+16635p4T12−178p6T14+p8T16 1 - 178 T^{2} + 16635 T^{4} - 1189868 T^{6} + 70020149 T^{8} - 1189868 p^{2} T^{10} + 16635 p^{4} T^{12} - 178 p^{6} T^{14} + p^{8} T^{16}
59 (1−17T+190T2−1987T3+18729T4−1987pT5+190p2T6−17p3T7+p4T8)2 ( 1 - 17 T + 190 T^{2} - 1987 T^{3} + 18729 T^{4} - 1987 p T^{5} + 190 p^{2} T^{6} - 17 p^{3} T^{7} + p^{4} T^{8} )^{2}
61 (1+30T2+62T4+30p2T6+p4T8)2 ( 1 + 30 T^{2} + 62 T^{4} + 30 p^{2} T^{6} + p^{4} T^{8} )^{2}
67 (1−6T+pT2)8 ( 1 - 6 T + p T^{2} )^{8}
71 (1−7T−2T2−569T3+8925T4−569pT5−2p2T6−7p3T7+p4T8)2 ( 1 - 7 T - 2 T^{2} - 569 T^{3} + 8925 T^{4} - 569 p T^{5} - 2 p^{2} T^{6} - 7 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+40T2+13719T4−51610T6+81749501T8−51610p2T10+13719p4T12+40p6T14+p8T16 1 + 40 T^{2} + 13719 T^{4} - 51610 T^{6} + 81749501 T^{8} - 51610 p^{2} T^{10} + 13719 p^{4} T^{12} + 40 p^{6} T^{14} + p^{8} T^{16}
83 1−156T2+17447T4−1647048T6+136747105T8−1647048p2T10+17447p4T12−156p6T14+p8T16 1 - 156 T^{2} + 17447 T^{4} - 1647048 T^{6} + 136747105 T^{8} - 1647048 p^{2} T^{10} + 17447 p^{4} T^{12} - 156 p^{6} T^{14} + p^{8} T^{16}
89 1−268T2+24303T4−825026T6+14225405T8−825026p2T10+24303p4T12−268p6T14+p8T16 1 - 268 T^{2} + 24303 T^{4} - 825026 T^{6} + 14225405 T^{8} - 825026 p^{2} T^{10} + 24303 p^{4} T^{12} - 268 p^{6} T^{14} + p^{8} T^{16}
97 (1−7T−48T2+1015T3−2449T4+1015pT5−48p2T6−7p3T7+p4T8)2 ( 1 - 7 T - 48 T^{2} + 1015 T^{3} - 2449 T^{4} + 1015 p T^{5} - 48 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} 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

−4.18643962869516729082908886012, −4.16231692111490310939547617750, −4.01164100645533765330520596091, −3.98927581358723747433060299358, −3.83552227282569033900427694556, −3.78523006995041160247245904983, −3.77797185115916013025697908259, −3.48764152835142104301866320006, −3.38343691580079268664831829000, −3.33281305165544701233278071018, −3.22316670130601093682400870099, −2.86431119317105014928732676100, −2.68938325104183145343888789853, −2.26889770184985744718150736994, −2.22461815705677917084077702266, −2.18818382507276559931892493590, −2.06840203876604331012134715632, −1.95146203857152594286689073823, −1.70069504281993286228078324862, −1.45948924702432612467962836253, −1.38027559817080793611718707090, −0.72472226765662158350568612095, −0.70190553900694786941739771434, −0.50238138823720498035102120434, −0.36296353502557454738909330259, 0.36296353502557454738909330259, 0.50238138823720498035102120434, 0.70190553900694786941739771434, 0.72472226765662158350568612095, 1.38027559817080793611718707090, 1.45948924702432612467962836253, 1.70069504281993286228078324862, 1.95146203857152594286689073823, 2.06840203876604331012134715632, 2.18818382507276559931892493590, 2.22461815705677917084077702266, 2.26889770184985744718150736994, 2.68938325104183145343888789853, 2.86431119317105014928732676100, 3.22316670130601093682400870099, 3.33281305165544701233278071018, 3.38343691580079268664831829000, 3.48764152835142104301866320006, 3.77797185115916013025697908259, 3.78523006995041160247245904983, 3.83552227282569033900427694556, 3.98927581358723747433060299358, 4.01164100645533765330520596091, 4.16231692111490310939547617750, 4.18643962869516729082908886012

Graph of the ZZ-function along the critical line

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