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Properties

Label 16-250e8-1.1-c1e8-0-4
Degree $16$
Conductor $1.526\times 10^{19}$
Sign $1$
Analytic cond. $252.195$
Root an. cond. $1.41289$
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-s + 5·9-s + 6·11-s + 20·19-s − 30·29-s + 6·31-s + 5·36-s + 26·41-s + 6·44-s + 20·49-s + 20·59-s − 4·61-s + 6·71-s + 20·76-s − 20·79-s − 6·81-s − 20·89-s + 30·99-s − 4·101-s − 30·109-s − 30·116-s + 11·121-s + 6·124-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  + 1/2·4-s + 5/3·9-s + 1.80·11-s + 4.58·19-s − 5.57·29-s + 1.07·31-s + 5/6·36-s + 4.06·41-s + 0.904·44-s + 20/7·49-s + 2.60·59-s − 0.512·61-s + 0.712·71-s + 2.29·76-s − 2.25·79-s − 2/3·81-s − 2.11·89-s + 3.01·99-s − 0.398·101-s − 2.87·109-s − 2.78·116-s + 121-s + 0.538·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{8} \cdot 5^{24}\)
Sign: $1$
Analytic conductor: \(252.195\)
Root analytic conductor: \(1.41289\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{8} \cdot 5^{24} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(5.516065560\)
\(L(\frac12)\) \(\approx\) \(5.516065560\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
5 \( 1 \)
good3 \( 1 - 5 T^{2} + 31 T^{4} - 95 T^{6} + 376 T^{8} - 95 p^{2} T^{10} + 31 p^{4} T^{12} - 5 p^{6} T^{14} + p^{8} T^{16} \)
7 \( ( 1 - 5 T^{2} + p^{2} T^{4} )^{4} \)
11 \( ( 1 - 3 T + 8 T^{2} - 51 T^{3} + 265 T^{4} - 51 p T^{5} + 8 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
13 \( 1 + 25 T^{2} + 456 T^{4} + 7175 T^{6} + 102311 T^{8} + 7175 p^{2} T^{10} + 456 p^{4} T^{12} + 25 p^{6} T^{14} + p^{8} T^{16} \)
17 \( 1 - 65 T^{2} + 2451 T^{4} - 64795 T^{6} + 1258016 T^{8} - 64795 p^{2} T^{10} + 2451 p^{4} T^{12} - 65 p^{6} T^{14} + p^{8} T^{16} \)
19 \( ( 1 - 10 T + 21 T^{2} + 70 T^{3} - 469 T^{4} + 70 p T^{5} + 21 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
23 \( 1 + 65 T^{2} + 1056 T^{4} - 31025 T^{6} - 1507729 T^{8} - 31025 p^{2} T^{10} + 1056 p^{4} T^{12} + 65 p^{6} T^{14} + p^{8} T^{16} \)
29 \( ( 1 + 15 T + 56 T^{2} - 315 T^{3} - 3629 T^{4} - 315 p T^{5} + 56 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
31 \( ( 1 - 3 T - 12 T^{2} - 131 T^{3} + 1365 T^{4} - 131 p T^{5} - 12 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
37 \( 1 - 5 T^{2} - 1029 T^{4} - 31375 T^{6} + 1924256 T^{8} - 31375 p^{2} T^{10} - 1029 p^{4} T^{12} - 5 p^{6} T^{14} + p^{8} T^{16} \)
41 \( ( 1 - 13 T + 28 T^{2} + 169 T^{3} - 945 T^{4} + 169 p T^{5} + 28 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
43 \( ( 1 - 130 T^{2} + 7603 T^{4} - 130 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
47 \( 1 + 125 T^{2} + 6456 T^{4} + 306235 T^{6} + 16576151 T^{8} + 306235 p^{2} T^{10} + 6456 p^{4} T^{12} + 125 p^{6} T^{14} + p^{8} T^{16} \)
53 \( 1 + 170 T^{2} + 8331 T^{4} - 254300 T^{6} - 38650699 T^{8} - 254300 p^{2} T^{10} + 8331 p^{4} T^{12} + 170 p^{6} T^{14} + p^{8} T^{16} \)
59 \( ( 1 - 10 T + T^{2} - 200 T^{3} + 5061 T^{4} - 200 p T^{5} + p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
61 \( ( 1 + 2 T + 3 T^{2} + 424 T^{3} + 4265 T^{4} + 424 p T^{5} + 3 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
67 \( 1 + 145 T^{2} + 3576 T^{4} - 627745 T^{6} - 69013609 T^{8} - 627745 p^{2} T^{10} + 3576 p^{4} T^{12} + 145 p^{6} T^{14} + p^{8} T^{16} \)
71 \( ( 1 - 3 T - 62 T^{2} + 399 T^{3} + 3205 T^{4} + 399 p T^{5} - 62 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
73 \( 1 + 70 T^{2} + 5211 T^{4} + 210800 T^{6} - 2106139 T^{8} + 210800 p^{2} T^{10} + 5211 p^{4} T^{12} + 70 p^{6} T^{14} + p^{8} T^{16} \)
79 \( ( 1 + 10 T - 39 T^{2} - 10 p T^{3} - 2839 T^{4} - 10 p^{2} T^{5} - 39 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
83 \( 1 - 115 T^{2} + 16776 T^{4} - 1448045 T^{6} + 171068591 T^{8} - 1448045 p^{2} T^{10} + 16776 p^{4} T^{12} - 115 p^{6} T^{14} + p^{8} T^{16} \)
89 \( ( 1 + 10 T - 29 T^{2} + 200 T^{3} + 10101 T^{4} + 200 p T^{5} - 29 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
97 \( 1 + 70 T^{2} - 549 T^{4} - 768340 T^{6} - 41383339 T^{8} - 768340 p^{2} T^{10} - 549 p^{4} T^{12} + 70 p^{6} T^{14} + p^{8} T^{16} \)
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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

−5.68216944181054893102906862252, −5.21424529529663670703220538639, −5.16536624018022619918366350927, −5.11985097790070323125380914353, −5.04374163086090432545981250444, −4.43938634878455591738178291575, −4.38198616228745248864822649722, −4.35541275715992879365664468035, −4.21614085590050743232153156815, −3.95034299686036942159072292257, −3.93255648615023935854378425144, −3.71992476184533651231796234164, −3.53409756889326006433309838110, −3.47923890316461281412482621684, −3.12111601673179735293861415038, −2.96921416374152910795517659433, −2.81591016590616022669371027351, −2.46915054419044815812603744465, −2.31635545798386494666393888730, −2.09378425469208818846876058417, −1.81267306037006791034545598334, −1.45507448123385519112442549006, −1.25572305089395129833709086484, −1.09375105037554201008796845030, −0.837638982495163801991920243724, 0.837638982495163801991920243724, 1.09375105037554201008796845030, 1.25572305089395129833709086484, 1.45507448123385519112442549006, 1.81267306037006791034545598334, 2.09378425469208818846876058417, 2.31635545798386494666393888730, 2.46915054419044815812603744465, 2.81591016590616022669371027351, 2.96921416374152910795517659433, 3.12111601673179735293861415038, 3.47923890316461281412482621684, 3.53409756889326006433309838110, 3.71992476184533651231796234164, 3.93255648615023935854378425144, 3.95034299686036942159072292257, 4.21614085590050743232153156815, 4.35541275715992879365664468035, 4.38198616228745248864822649722, 4.43938634878455591738178291575, 5.04374163086090432545981250444, 5.11985097790070323125380914353, 5.16536624018022619918366350927, 5.21424529529663670703220538639, 5.68216944181054893102906862252

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

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