Properties

Label 16-1620e8-1.1-c2e8-0-0
Degree $16$
Conductor $4.744\times 10^{25}$
Sign $1$
Analytic cond. $1.44145\times 10^{13}$
Root an. cond. $6.64392$
Motivic weight $2$
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  − 48·19-s + 30·25-s − 136·31-s − 68·49-s − 296·61-s + 312·79-s + 592·109-s − 324·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 388·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯
L(s)  = 1  − 2.52·19-s + 6/5·25-s − 4.38·31-s − 1.38·49-s − 4.85·61-s + 3.94·79-s + 5.43·109-s − 2.67·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 2.29·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + 0.00448·223-s + ⋯

Functional equation

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

Invariants

Degree: \(16\)
Conductor: \(2^{16} \cdot 3^{32} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(1.44145\times 10^{13}\)
Root analytic conductor: \(6.64392\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1620} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{16} \cdot 3^{32} \cdot 5^{8} ,\ ( \ : [1]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.2425206024\)
\(L(\frac12)\) \(\approx\) \(0.2425206024\)
\(L(2)\) 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 \( 1 \)
5 \( 1 - 6 p T^{2} + 11 p^{2} T^{4} - 6 p^{5} T^{6} + p^{8} T^{8} \)
good7 \( ( 1 + 34 T^{2} - 1245 T^{4} + 34 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
11 \( ( 1 + 162 T^{2} + 11603 T^{4} + 162 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
13 \( ( 1 + 194 T^{2} + 9075 T^{4} + 194 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
17 \( ( 1 - 402 T^{2} + p^{4} T^{4} )^{4} \)
19 \( ( 1 + 6 T + p^{2} T^{2} )^{8} \)
23 \( ( 1 - 1038 T^{2} + 797603 T^{4} - 1038 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
29 \( ( 1 + 962 T^{2} + 218163 T^{4} + 962 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
31 \( ( 1 + 34 T + 195 T^{2} + 34 p^{2} T^{3} + p^{4} T^{4} )^{4} \)
37 \( ( 1 - 802 T^{2} + p^{4} T^{4} )^{4} \)
41 \( ( 1 + 3042 T^{2} + 6428003 T^{4} + 3042 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
43 \( ( 1 + 2914 T^{2} + 5072595 T^{4} + 2914 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
47 \( ( 1 - 4398 T^{2} + 14462723 T^{4} - 4398 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
53 \( ( 1 + 3998 T^{2} + p^{4} T^{4} )^{4} \)
59 \( ( 1 - 2718 T^{2} - 4729837 T^{4} - 2718 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
61 \( ( 1 - 47 T + p^{2} T^{2} )^{4}( 1 + 121 T + p^{2} T^{2} )^{4} \)
67 \( ( 1 + 514 T^{2} - 19886925 T^{4} + 514 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
71 \( ( 1 - 7202 T^{2} + p^{4} T^{4} )^{4} \)
73 \( ( 1 - 7522 T^{2} + p^{4} T^{4} )^{4} \)
79 \( ( 1 - 78 T - 157 T^{2} - 78 p^{2} T^{3} + p^{4} T^{4} )^{4} \)
83 \( ( 1 - 3198 T^{2} - 37231117 T^{4} - 3198 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
89 \( ( 1 - 15522 T^{2} + p^{4} T^{4} )^{4} \)
97 \( ( 1 + 17794 T^{2} + 228097155 T^{4} + 17794 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
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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

−3.58752440991198709434534511758, −3.48309795429317477501662916643, −3.39712503917245192754917286265, −3.37962813660079851686207312879, −3.30993111547073370092424224781, −3.30138062992587912599954109700, −3.24499585474696822617405587688, −2.74083283639085502242290794595, −2.71331063725859911847606084985, −2.48075981528634319808845770473, −2.31496900390626574167587563930, −2.29096900695283609188987385238, −2.25823820793768436679213305407, −2.14530454207462955793769418553, −1.76677040384922465576207309128, −1.63918984811085404407385668628, −1.57262201412259184762121007835, −1.50335977623397322346618022703, −1.45491846827417809694650204337, −1.04182500290628080580830634959, −0.914517795649753385210500789644, −0.49481356376974540261332416849, −0.46954036144229912448394498586, −0.33885002129136205542496753310, −0.04421477625999639498248510872, 0.04421477625999639498248510872, 0.33885002129136205542496753310, 0.46954036144229912448394498586, 0.49481356376974540261332416849, 0.914517795649753385210500789644, 1.04182500290628080580830634959, 1.45491846827417809694650204337, 1.50335977623397322346618022703, 1.57262201412259184762121007835, 1.63918984811085404407385668628, 1.76677040384922465576207309128, 2.14530454207462955793769418553, 2.25823820793768436679213305407, 2.29096900695283609188987385238, 2.31496900390626574167587563930, 2.48075981528634319808845770473, 2.71331063725859911847606084985, 2.74083283639085502242290794595, 3.24499585474696822617405587688, 3.30138062992587912599954109700, 3.30993111547073370092424224781, 3.37962813660079851686207312879, 3.39712503917245192754917286265, 3.48309795429317477501662916643, 3.58752440991198709434534511758

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

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