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Properties

Label 32-80e16-1.1-c1e16-0-0
Degree $32$
Conductor $2.815\times 10^{30}$
Sign $1$
Analytic cond. $0.000768906$
Root an. cond. $0.799251$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s − 8·5-s + 8·11-s + 6·16-s − 8·19-s + 16·20-s + 32·25-s − 16·29-s + 16·31-s − 16·44-s − 48·49-s − 64·55-s − 24·59-s − 12·64-s + 16·76-s + 16·79-s − 48·80-s + 8·81-s + 64·95-s − 64·100-s − 16·101-s − 16·109-s + 32·116-s + 32·121-s − 32·124-s − 72·125-s + 127-s + ⋯
L(s)  = 1  − 4-s − 3.57·5-s + 2.41·11-s + 3/2·16-s − 1.83·19-s + 3.57·20-s + 32/5·25-s − 2.97·29-s + 2.87·31-s − 2.41·44-s − 6.85·49-s − 8.62·55-s − 3.12·59-s − 3/2·64-s + 1.83·76-s + 1.80·79-s − 5.36·80-s + 8/9·81-s + 6.56·95-s − 6.39·100-s − 1.59·101-s − 1.53·109-s + 2.97·116-s + 2.90·121-s − 2.87·124-s − 6.43·125-s + 0.0887·127-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(2^{64} \cdot 5^{16}\)
Sign: $1$
Analytic conductor: \(0.000768906\)
Root analytic conductor: \(0.799251\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 2^{64} \cdot 5^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.1382615416\)
\(L(\frac12)\) \(\approx\) \(0.1382615416\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + p T^{2} - p T^{4} - p^{2} T^{6} + p^{2} T^{8} - p^{4} T^{10} - p^{5} T^{12} + p^{7} T^{14} + p^{8} T^{16} \)
5 \( ( 1 + 4 T + 8 T^{2} + 4 T^{3} - 14 T^{4} + 4 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
good3 \( 1 - 8 T^{4} - 20 p T^{8} + 424 T^{12} + 3334 T^{16} + 424 p^{4} T^{20} - 20 p^{9} T^{24} - 8 p^{12} T^{28} + p^{16} T^{32} \)
7 \( ( 1 + 24 T^{2} + 340 T^{4} + 72 p^{2} T^{6} + 28410 T^{8} + 72 p^{4} T^{10} + 340 p^{4} T^{12} + 24 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
11 \( ( 1 - 4 T + 8 T^{2} - 36 T^{3} + 92 T^{4} - 68 T^{5} + 184 T^{6} - 324 T^{7} - 1370 T^{8} - 324 p T^{9} + 184 p^{2} T^{10} - 68 p^{3} T^{11} + 92 p^{4} T^{12} - 36 p^{5} T^{13} + 8 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
13 \( ( 1 - 140 T^{4} + 49734 T^{8} - 140 p^{4} T^{12} + p^{8} T^{16} )^{2} \)
17 \( ( 1 - 40 T^{2} + 870 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
19 \( ( 1 + 4 T + 8 T^{2} - 28 T^{3} - 484 T^{4} - 220 T^{5} + 3384 T^{6} + 22692 T^{7} + 191206 T^{8} + 22692 p T^{9} + 3384 p^{2} T^{10} - 220 p^{3} T^{11} - 484 p^{4} T^{12} - 28 p^{5} T^{13} + 8 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
23 \( ( 1 + 152 T^{2} + 10564 T^{4} + 442664 T^{6} + 12324922 T^{8} + 442664 p^{2} T^{10} + 10564 p^{4} T^{12} + 152 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
29 \( ( 1 + 8 T + 32 T^{2} - 72 T^{3} + 380 T^{4} + 12232 T^{5} + 88288 T^{6} + 148152 T^{7} - 290138 T^{8} + 148152 p T^{9} + 88288 p^{2} T^{10} + 12232 p^{3} T^{11} + 380 p^{4} T^{12} - 72 p^{5} T^{13} + 32 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
31 \( ( 1 - 4 T + 52 T^{2} - 292 T^{3} + 1510 T^{4} - 292 p T^{5} + 52 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{4} \)
37 \( ( 1 - 92 T^{4} + 1830438 T^{8} - 92 p^{4} T^{12} + p^{8} T^{16} )^{2} \)
41 \( ( 1 - 136 T^{2} + 9988 T^{4} - 572632 T^{6} + 26690182 T^{8} - 572632 p^{2} T^{10} + 9988 p^{4} T^{12} - 136 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
43 \( 1 + 4568 T^{4} + 15432388 T^{8} + 32835546440 T^{12} + 67805368306822 T^{16} + 32835546440 p^{4} T^{20} + 15432388 p^{8} T^{24} + 4568 p^{12} T^{28} + p^{16} T^{32} \)
47 \( ( 1 - 296 T^{2} + 41044 T^{4} - 3480152 T^{6} + 197533306 T^{8} - 3480152 p^{2} T^{10} + 41044 p^{4} T^{12} - 296 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
53 \( 1 + 8488 T^{4} + 28367644 T^{8} + 65683864984 T^{12} + 173166077304070 T^{16} + 65683864984 p^{4} T^{20} + 28367644 p^{8} T^{24} + 8488 p^{12} T^{28} + p^{16} T^{32} \)
59 \( ( 1 + 12 T + 72 T^{2} + 396 T^{3} + 6748 T^{4} + 83052 T^{5} + 589176 T^{6} + 3781452 T^{7} + 22178598 T^{8} + 3781452 p T^{9} + 589176 p^{2} T^{10} + 83052 p^{3} T^{11} + 6748 p^{4} T^{12} + 396 p^{5} T^{13} + 72 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
61 \( ( 1 + 2162 T^{4} + p^{4} T^{8} )^{4} \)
67 \( 1 + 6296 T^{4} + 28867012 T^{8} + 143453549192 T^{12} + 987214874886406 T^{16} + 143453549192 p^{4} T^{20} + 28867012 p^{8} T^{24} + 6296 p^{12} T^{28} + p^{16} T^{32} \)
71 \( ( 1 - 312 T^{2} + 52060 T^{4} - 5897736 T^{6} + 485188230 T^{8} - 5897736 p^{2} T^{10} + 52060 p^{4} T^{12} - 312 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
73 \( ( 1 + 128 T^{2} + 11692 T^{4} + 338816 T^{6} + 16421926 T^{8} + 338816 p^{2} T^{10} + 11692 p^{4} T^{12} + 128 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
79 \( ( 1 - 4 T + 196 T^{2} - 484 T^{3} + 18118 T^{4} - 484 p T^{5} + 196 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{4} \)
83 \( 1 - 24808 T^{4} + 301625668 T^{8} - 2745693101560 T^{12} + 20938093531246342 T^{16} - 2745693101560 p^{4} T^{20} + 301625668 p^{8} T^{24} - 24808 p^{12} T^{28} + p^{16} T^{32} \)
89 \( ( 1 - 504 T^{2} + 120796 T^{4} - 18233160 T^{6} + 1917111942 T^{8} - 18233160 p^{2} T^{10} + 120796 p^{4} T^{12} - 504 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
97 \( ( 1 - 576 T^{2} + 159724 T^{4} - 27489216 T^{6} + 3203946342 T^{8} - 27489216 p^{2} T^{10} + 159724 p^{4} T^{12} - 576 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−4.60585392816244410577943273729, −4.28343193271277613432706933154, −4.24506280589737075028477561108, −4.20043566813844357610118406853, −4.19170700488718924150141187347, −4.10604551183961593520974466299, −4.03081793542534553811673865677, −3.84241418047159126184184669106, −3.84069826645771333305387753811, −3.62287289909574280848490993344, −3.57954630959318027825433039589, −3.32155890344846506280339010064, −3.27636019614546309662413379690, −3.21651712146688656948704997612, −2.98427119637115740645264346864, −2.92195415866566169542857055225, −2.90833175246742863955935257584, −2.85960730156526136200675067883, −2.34958294078332596318156431949, −2.13047183919042619871583100321, −1.80668676801464873908425654601, −1.71650780813667433124270790332, −1.59597343247988219945523923530, −1.52746321359538754853897119772, −0.71469653344265967894585319826, 0.71469653344265967894585319826, 1.52746321359538754853897119772, 1.59597343247988219945523923530, 1.71650780813667433124270790332, 1.80668676801464873908425654601, 2.13047183919042619871583100321, 2.34958294078332596318156431949, 2.85960730156526136200675067883, 2.90833175246742863955935257584, 2.92195415866566169542857055225, 2.98427119637115740645264346864, 3.21651712146688656948704997612, 3.27636019614546309662413379690, 3.32155890344846506280339010064, 3.57954630959318027825433039589, 3.62287289909574280848490993344, 3.84069826645771333305387753811, 3.84241418047159126184184669106, 4.03081793542534553811673865677, 4.10604551183961593520974466299, 4.19170700488718924150141187347, 4.20043566813844357610118406853, 4.24506280589737075028477561108, 4.28343193271277613432706933154, 4.60585392816244410577943273729

Graph of the $Z$-function along the critical line

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