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Properties

Label 16-260e8-1.1-c1e8-0-1
Degree 1616
Conductor 2.088×10192.088\times 10^{19}
Sign 11
Analytic cond. 345.146345.146
Root an. cond. 1.440871.44087
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·9-s + 4·25-s − 24·29-s − 16·49-s − 8·61-s + 16·79-s + 16·81-s + 44·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯
L(s)  = 1  + 4/3·9-s + 4/5·25-s − 4.45·29-s − 2.28·49-s − 1.02·61-s + 1.80·79-s + 16/9·81-s + 4·121-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 + 4/13·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + ⋯

Functional equation

Λ(s)=((216⋅58⋅138)s/2 ΓC(s)8 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((216⋅58⋅138)s/2 ΓC(s+1/2)8 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{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⋅58⋅1382^{16} \cdot 5^{8} \cdot 13^{8}
Sign: 11
Analytic conductor: 345.146345.146
Root analytic conductor: 1.440871.44087
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (16, 216⋅58⋅138, ( :[1/2]8), 1)(16,\ 2^{16} \cdot 5^{8} \cdot 13^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 1.6939599941.693959994
L(12)L(\frac12) ≈\approx 1.6939599941.693959994
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
5 (1−2T2+p2T4)2 ( 1 - 2 T^{2} + p^{2} T^{4} )^{2}
13 (1−2T2+p2T4)2 ( 1 - 2 T^{2} + p^{2} T^{4} )^{2}
good3 (1−2T2−2T4−2p2T6+p4T8)2 ( 1 - 2 T^{2} - 2 T^{4} - 2 p^{2} T^{6} + p^{4} T^{8} )^{2}
7 (1+8T2+30T4+8p2T6+p4T8)2 ( 1 + 8 T^{2} + 30 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} )^{2}
11 (1−2pT2+342T4−2p3T6+p4T8)2 ( 1 - 2 p T^{2} + 342 T^{4} - 2 p^{3} T^{6} + p^{4} T^{8} )^{2}
17 (1−28T2+438T4−28p2T6+p4T8)2 ( 1 - 28 T^{2} + 438 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} )^{2}
19 (1−46T2+1062T4−46p2T6+p4T8)2 ( 1 - 46 T^{2} + 1062 T^{4} - 46 p^{2} T^{6} + p^{4} T^{8} )^{2}
23 (1−58T2+1710T4−58p2T6+p4T8)2 ( 1 - 58 T^{2} + 1710 T^{4} - 58 p^{2} T^{6} + p^{4} T^{8} )^{2}
29 (1+6T+46T2+6pT3+p2T4)4 ( 1 + 6 T + 46 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{4}
31 (1−22T2+342T4−22p2T6+p4T8)2 ( 1 - 22 T^{2} + 342 T^{4} - 22 p^{2} T^{6} + p^{4} T^{8} )^{2}
37 (1+80T2+3582T4+80p2T6+p4T8)2 ( 1 + 80 T^{2} + 3582 T^{4} + 80 p^{2} T^{6} + p^{4} T^{8} )^{2}
41 (1−76T2+4470T4−76p2T6+p4T8)2 ( 1 - 76 T^{2} + 4470 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} )^{2}
43 (1−106T2+6318T4−106p2T6+p4T8)2 ( 1 - 106 T^{2} + 6318 T^{4} - 106 p^{2} T^{6} + p^{4} T^{8} )^{2}
47 (1+56T2+4446T4+56p2T6+p4T8)2 ( 1 + 56 T^{2} + 4446 T^{4} + 56 p^{2} T^{6} + p^{4} T^{8} )^{2}
53 (1−172T2+12678T4−172p2T6+p4T8)2 ( 1 - 172 T^{2} + 12678 T^{4} - 172 p^{2} T^{6} + p^{4} T^{8} )^{2}
59 (1−46T2+7302T4−46p2T6+p4T8)2 ( 1 - 46 T^{2} + 7302 T^{4} - 46 p^{2} T^{6} + p^{4} T^{8} )^{2}
61 (1+2T+102T2+2pT3+p2T4)4 ( 1 + 2 T + 102 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{4}
67 (1+200T2+18222T4+200p2T6+p4T8)2 ( 1 + 200 T^{2} + 18222 T^{4} + 200 p^{2} T^{6} + p^{4} T^{8} )^{2}
71 (1−238T2+23718T4−238p2T6+p4T8)2 ( 1 - 238 T^{2} + 23718 T^{4} - 238 p^{2} T^{6} + p^{4} T^{8} )^{2}
73 (1+224T2+22446T4+224p2T6+p4T8)2 ( 1 + 224 T^{2} + 22446 T^{4} + 224 p^{2} T^{6} + p^{4} T^{8} )^{2}
79 (1−4T+78T2−4pT3+p2T4)4 ( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{4}
83 (1+56T2−4338T4+56p2T6+p4T8)2 ( 1 + 56 T^{2} - 4338 T^{4} + 56 p^{2} T^{6} + p^{4} T^{8} )^{2}
89 (1−50T2+p2T4)4 ( 1 - 50 T^{2} + p^{2} T^{4} )^{4}
97 (1+236T2+27366T4+236p2T6+p4T8)2 ( 1 + 236 T^{2} + 27366 T^{4} + 236 p^{2} T^{6} + 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

−5.49405522674696637756099548560, −5.14888512411359633695121673671, −5.14106146244585628738676197479, −4.82721571351949580683652488741, −4.73355821031781078324458696033, −4.69462248036237674970182723789, −4.57132868850683820827067915798, −4.52780398632788205743878440926, −4.03780666551540051716002148793, −3.90613938289059777092992060759, −3.85557078244062818850473660291, −3.60092490133216161012012192919, −3.51061612675524548810961982939, −3.40129154430712144561358409556, −3.20968770176302602339701251735, −3.07030047949319203839179817174, −2.57360796458670893059355337453, −2.51728384451142389207985567391, −2.17168941959692474062811147196, −2.11739811192374868269543172472, −1.74778547814625556394771400359, −1.54954256022427537684495640079, −1.52491595230567678578685632686, −0.956121926463777211496794486991, −0.39571677344501970876545734483, 0.39571677344501970876545734483, 0.956121926463777211496794486991, 1.52491595230567678578685632686, 1.54954256022427537684495640079, 1.74778547814625556394771400359, 2.11739811192374868269543172472, 2.17168941959692474062811147196, 2.51728384451142389207985567391, 2.57360796458670893059355337453, 3.07030047949319203839179817174, 3.20968770176302602339701251735, 3.40129154430712144561358409556, 3.51061612675524548810961982939, 3.60092490133216161012012192919, 3.85557078244062818850473660291, 3.90613938289059777092992060759, 4.03780666551540051716002148793, 4.52780398632788205743878440926, 4.57132868850683820827067915798, 4.69462248036237674970182723789, 4.73355821031781078324458696033, 4.82721571351949580683652488741, 5.14106146244585628738676197479, 5.14888512411359633695121673671, 5.49405522674696637756099548560

Graph of the ZZ-function along the critical line

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