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Properties

Label 16-7500e8-1.1-c1e8-0-0
Degree $16$
Conductor $1.001\times 10^{31}$
Sign $1$
Analytic cond. $1.65465\times 10^{14}$
Root an. cond. $7.73872$
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  − 4·9-s − 2·11-s + 10·19-s + 8·29-s − 18·31-s + 32·49-s − 2·59-s − 4·61-s − 40·71-s + 6·79-s + 10·81-s − 30·89-s + 8·99-s + 44·101-s − 14·109-s − 77·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 59·169-s − 40·171-s + ⋯
L(s)  = 1  − 4/3·9-s − 0.603·11-s + 2.29·19-s + 1.48·29-s − 3.23·31-s + 32/7·49-s − 0.260·59-s − 0.512·61-s − 4.74·71-s + 0.675·79-s + 10/9·81-s − 3.17·89-s + 0.804·99-s + 4.37·101-s − 1.34·109-s − 7·121-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 + 4.53·169-s − 3.05·171-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{32}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{16} \cdot 3^{8} \cdot 5^{32}\)
Sign: $1$
Analytic conductor: \(1.65465\times 10^{14}\)
Root analytic conductor: \(7.73872\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{16} \cdot 3^{8} \cdot 5^{32} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.1023280108\)
\(L(\frac12)\) \(\approx\) \(0.1023280108\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( ( 1 + T^{2} )^{4} \)
5 \( 1 \)
good7 \( 1 - 32 T^{2} + 390 T^{4} - 2287 T^{6} + 10949 T^{8} - 2287 p^{2} T^{10} + 390 p^{4} T^{12} - 32 p^{6} T^{14} + p^{8} T^{16} \)
11 \( ( 1 + T + 40 T^{2} + 29 T^{3} + 639 T^{4} + 29 p T^{5} + 40 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} )^{2} \)
13 \( 1 - 59 T^{2} + 1512 T^{4} - 23737 T^{6} + 308255 T^{8} - 23737 p^{2} T^{10} + 1512 p^{4} T^{12} - 59 p^{6} T^{14} + p^{8} T^{16} \)
17 \( 1 - 52 T^{2} + 1710 T^{4} - 37487 T^{6} + 703529 T^{8} - 37487 p^{2} T^{10} + 1710 p^{4} T^{12} - 52 p^{6} T^{14} + p^{8} T^{16} \)
19 \( ( 1 - 5 T + 51 T^{2} - 10 p T^{3} + 1361 T^{4} - 10 p^{2} T^{5} + 51 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
23 \( 1 - 95 T^{2} + 212 p T^{4} - 174845 T^{6} + 4637071 T^{8} - 174845 p^{2} T^{10} + 212 p^{5} T^{12} - 95 p^{6} T^{14} + p^{8} T^{16} \)
29 \( ( 1 - 4 T + 22 T^{2} + 223 T^{3} - 1005 T^{4} + 223 p T^{5} + 22 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
31 \( ( 1 + 9 T + 130 T^{2} + 711 T^{3} + 189 p T^{4} + 711 p T^{5} + 130 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
37 \( 1 - 200 T^{2} + 19566 T^{4} - 1225075 T^{6} + 53629481 T^{8} - 1225075 p^{2} T^{10} + 19566 p^{4} T^{12} - 200 p^{6} T^{14} + p^{8} T^{16} \)
41 \( ( 1 + 94 T^{2} + 240 T^{3} + 4191 T^{4} + 240 p T^{5} + 94 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
43 \( 1 + 10 T^{2} + 591 T^{4} - 41380 T^{6} - 1432939 T^{8} - 41380 p^{2} T^{10} + 591 p^{4} T^{12} + 10 p^{6} T^{14} + p^{8} T^{16} \)
47 \( 1 - 152 T^{2} + 14830 T^{4} - 1007627 T^{6} + 53628469 T^{8} - 1007627 p^{2} T^{10} + 14830 p^{4} T^{12} - 152 p^{6} T^{14} + p^{8} T^{16} \)
53 \( 1 - 328 T^{2} + 50670 T^{4} - 4809563 T^{6} + 307064009 T^{8} - 4809563 p^{2} T^{10} + 50670 p^{4} T^{12} - 328 p^{6} T^{14} + p^{8} T^{16} \)
59 \( ( 1 + T + 97 T^{2} + 218 T^{3} + 6825 T^{4} + 218 p T^{5} + 97 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} )^{2} \)
61 \( ( 1 + 2 T + 148 T^{2} + 764 T^{3} + 10165 T^{4} + 764 p T^{5} + 148 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
67 \( 1 - 202 T^{2} + 22315 T^{4} - 1681732 T^{6} + 112260709 T^{8} - 1681732 p^{2} T^{10} + 22315 p^{4} T^{12} - 202 p^{6} T^{14} + p^{8} T^{16} \)
71 \( ( 1 + 20 T + 334 T^{2} + 3385 T^{3} + 33471 T^{4} + 3385 p T^{5} + 334 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
73 \( 1 - 158 T^{2} + 4735 T^{4} + 639412 T^{6} - 75033251 T^{8} + 639412 p^{2} T^{10} + 4735 p^{4} T^{12} - 158 p^{6} T^{14} + p^{8} T^{16} \)
79 \( ( 1 - 3 T + 130 T^{2} + 567 T^{3} + 6699 T^{4} + 567 p T^{5} + 130 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
83 \( 1 - 580 T^{2} + 153066 T^{4} - 23994800 T^{6} + 2442557651 T^{8} - 23994800 p^{2} T^{10} + 153066 p^{4} T^{12} - 580 p^{6} T^{14} + p^{8} T^{16} \)
89 \( ( 1 + 15 T + 386 T^{2} + 3915 T^{3} + 52911 T^{4} + 3915 p T^{5} + 386 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
97 \( 1 - 380 T^{2} + 77106 T^{4} - 10616800 T^{6} + 1143390491 T^{8} - 10616800 p^{2} T^{10} + 77106 p^{4} T^{12} - 380 p^{6} T^{14} + p^{8} T^{16} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−3.07216068103367636387873485654, −3.02915785969883678441164815471, −3.01565311789095173838006280430, −2.89603754961245393033766123751, −2.84221950045537320423766417852, −2.69903437106796502491675153233, −2.57304852765651932870674522858, −2.56978697251351836306978245355, −2.25195397208124724303182785650, −2.23385252125040264504759251815, −2.06102239810051673830391937656, −1.95514384798287765140330418212, −1.95511334041563168843364298943, −1.76740550420837030815953667472, −1.52245246961098847665838456627, −1.50254873826807905607202347679, −1.40321080822204991115960764086, −1.09282416721942664408284934566, −1.05664966787453395142415412510, −0.943014542728720522841855419825, −0.914327710670402886986720505260, −0.54268260749868070564455459361, −0.46719554701026393159259236004, −0.26029167982790399538179866666, −0.02545836718785390804262432021, 0.02545836718785390804262432021, 0.26029167982790399538179866666, 0.46719554701026393159259236004, 0.54268260749868070564455459361, 0.914327710670402886986720505260, 0.943014542728720522841855419825, 1.05664966787453395142415412510, 1.09282416721942664408284934566, 1.40321080822204991115960764086, 1.50254873826807905607202347679, 1.52245246961098847665838456627, 1.76740550420837030815953667472, 1.95511334041563168843364298943, 1.95514384798287765140330418212, 2.06102239810051673830391937656, 2.23385252125040264504759251815, 2.25195397208124724303182785650, 2.56978697251351836306978245355, 2.57304852765651932870674522858, 2.69903437106796502491675153233, 2.84221950045537320423766417852, 2.89603754961245393033766123751, 3.01565311789095173838006280430, 3.02915785969883678441164815471, 3.07216068103367636387873485654

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

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