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Properties

Label 16-1152e8-1.1-c3e8-0-0
Degree 1616
Conductor 3.102×10243.102\times 10^{24}
Sign 11
Analytic cond. 4.55562×10144.55562\times 10^{14}
Root an. cond. 8.244408.24440
Motivic weight 33
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  − 240·17-s + 328·25-s + 816·41-s − 536·49-s − 1.93e3·73-s − 7.34e3·89-s − 2.96e3·97-s + 7.92e3·113-s + 8.02e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 296·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
L(s)  = 1  − 3.42·17-s + 2.62·25-s + 3.10·41-s − 1.56·49-s − 3.10·73-s − 8.74·89-s − 3.09·97-s + 6.59·113-s + 6.02·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 0.134·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + 0.000361·197-s + 0.000356·199-s + 0.000326·211-s + ⋯

Functional equation

Λ(s)=((256⋅316)s/2 ΓC(s)8 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}
Λ(s)=((256⋅316)s/2 ΓC(s+3/2)8 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 1616
Conductor: 256⋅3162^{56} \cdot 3^{16}
Sign: 11
Analytic conductor: 4.55562×10144.55562\times 10^{14}
Root analytic conductor: 8.244408.24440
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (16, 256⋅316, ( :[3/2]8), 1)(16,\ 2^{56} \cdot 3^{16} ,\ ( \ : [3/2]^{8} ),\ 1 )

Particular Values

L(2)L(2) ≈\approx 0.13301925250.1330192525
L(12)L(\frac12) ≈\approx 0.13301925250.1330192525
L(52)L(\frac{5}{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
good5 (1−164T2+19542T4−164p6T6+p12T8)2 ( 1 - 164 T^{2} + 19542 T^{4} - 164 p^{6} T^{6} + p^{12} T^{8} )^{2}
7 (1+268T2−41658T4+268p6T6+p12T8)2 ( 1 + 268 T^{2} - 41658 T^{4} + 268 p^{6} T^{6} + p^{12} T^{8} )^{2}
11 (1−4012T2+7272246T4−4012p6T6+p12T8)2 ( 1 - 4012 T^{2} + 7272246 T^{4} - 4012 p^{6} T^{6} + p^{12} T^{8} )^{2}
13 (1−148T2+3687126T4−148p6T6+p12T8)2 ( 1 - 148 T^{2} + 3687126 T^{4} - 148 p^{6} T^{6} + p^{12} T^{8} )^{2}
17 (1+60T+6118T2+60p3T3+p6T4)4 ( 1 + 60 T + 6118 T^{2} + 60 p^{3} T^{3} + p^{6} T^{4} )^{4}
19 (1−17932T2+171826710T4−17932p6T6+p12T8)2 ( 1 - 17932 T^{2} + 171826710 T^{4} - 17932 p^{6} T^{6} + p^{12} T^{8} )^{2}
23 (1−4900T2+206522790T4−4900p6T6+p12T8)2 ( 1 - 4900 T^{2} + 206522790 T^{4} - 4900 p^{6} T^{6} + p^{12} T^{8} )^{2}
29 (1−56516T2+1986668214T4−56516p6T6+p12T8)2 ( 1 - 56516 T^{2} + 1986668214 T^{4} - 56516 p^{6} T^{6} + p^{12} T^{8} )^{2}
31 (1+63916T2+2595558p2T4+63916p6T6+p12T8)2 ( 1 + 63916 T^{2} + 2595558 p^{2} T^{4} + 63916 p^{6} T^{6} + p^{12} T^{8} )^{2}
37 (1−95476T2+7028163510T4−95476p6T6+p12T8)2 ( 1 - 95476 T^{2} + 7028163510 T^{4} - 95476 p^{6} T^{6} + p^{12} T^{8} )^{2}
41 (1−204T+806pT2−204p3T3+p6T4)4 ( 1 - 204 T + 806 p T^{2} - 204 p^{3} T^{3} + p^{6} T^{4} )^{4}
43 (1−273964T2+31085635254T4−273964p6T6+p12T8)2 ( 1 - 273964 T^{2} + 31085635254 T^{4} - 273964 p^{6} T^{6} + p^{12} T^{8} )^{2}
47 (1+112892T2+23215757766T4+112892p6T6+p12T8)2 ( 1 + 112892 T^{2} + 23215757766 T^{4} + 112892 p^{6} T^{6} + p^{12} T^{8} )^{2}
53 (1−507620T2+107623720470T4−507620p6T6+p12T8)2 ( 1 - 507620 T^{2} + 107623720470 T^{4} - 507620 p^{6} T^{6} + p^{12} T^{8} )^{2}
59 (1−121324T2−26790709386T4−121324p6T6+p12T8)2 ( 1 - 121324 T^{2} - 26790709386 T^{4} - 121324 p^{6} T^{6} + p^{12} T^{8} )^{2}
61 (1+132332T2+107394800406T4+132332p6T6+p12T8)2 ( 1 + 132332 T^{2} + 107394800406 T^{4} + 132332 p^{6} T^{6} + p^{12} T^{8} )^{2}
67 (1−364108T2+112096368726T4−364108p6T6+p12T8)2 ( 1 - 364108 T^{2} + 112096368726 T^{4} - 364108 p^{6} T^{6} + p^{12} T^{8} )^{2}
71 (1+4612pT2+275267100390T4+4612p7T6+p12T8)2 ( 1 + 4612 p T^{2} + 275267100390 T^{4} + 4612 p^{7} T^{6} + p^{12} T^{8} )^{2}
73 (1+484T+541686T2+484p3T3+p6T4)4 ( 1 + 484 T + 541686 T^{2} + 484 p^{3} T^{3} + p^{6} T^{4} )^{4}
79 (1+932332T2+435081520806T4+932332p6T6+p12T8)2 ( 1 + 932332 T^{2} + 435081520806 T^{4} + 932332 p^{6} T^{6} + p^{12} T^{8} )^{2}
83 (1−539404T2+296977663254T4−539404p6T6+p12T8)2 ( 1 - 539404 T^{2} + 296977663254 T^{4} - 539404 p^{6} T^{6} + p^{12} T^{8} )^{2}
89 (1+1836T+2086774T2+1836p3T3+p6T4)4 ( 1 + 1836 T + 2086774 T^{2} + 1836 p^{3} T^{3} + p^{6} T^{4} )^{4}
97 (1+740T+1943814T2+740p3T3+p6T4)4 ( 1 + 740 T + 1943814 T^{2} + 740 p^{3} T^{3} + p^{6} T^{4} )^{4}
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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

−3.73383295787935081753443259970, −3.71310039081462585177292609242, −3.46750259161953231436724651718, −3.35637100326392822492134091137, −3.09723682831516419102028626658, −2.92298887870634147017861010240, −2.85243219494961954132908540998, −2.85077827660327784830161955048, −2.74695332104695814375032805201, −2.69994288669671897950507309541, −2.56640593021480509127586301485, −2.13809632665568597118602319348, −2.01149314010461355207566147560, −1.95623543902953692579732440371, −1.93727046918475540583331458278, −1.69051032914352187361069855606, −1.49508914497466477179276019444, −1.31750218326804521209734162842, −1.16329946430623431297983349548, −1.03831991580905578803413510145, −0.74132435380500799694113562094, −0.59200752991177752465189296636, −0.55318733233634350709028126579, −0.18704147047753559682033196515, −0.03387559755214427791727877428, 0.03387559755214427791727877428, 0.18704147047753559682033196515, 0.55318733233634350709028126579, 0.59200752991177752465189296636, 0.74132435380500799694113562094, 1.03831991580905578803413510145, 1.16329946430623431297983349548, 1.31750218326804521209734162842, 1.49508914497466477179276019444, 1.69051032914352187361069855606, 1.93727046918475540583331458278, 1.95623543902953692579732440371, 2.01149314010461355207566147560, 2.13809632665568597118602319348, 2.56640593021480509127586301485, 2.69994288669671897950507309541, 2.74695332104695814375032805201, 2.85077827660327784830161955048, 2.85243219494961954132908540998, 2.92298887870634147017861010240, 3.09723682831516419102028626658, 3.35637100326392822492134091137, 3.46750259161953231436724651718, 3.71310039081462585177292609242, 3.73383295787935081753443259970

Graph of the ZZ-function along the critical line

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