Properties

Label 16-1050e8-1.1-c1e8-0-7
Degree $16$
Conductor $1.477\times 10^{24}$
Sign $1$
Analytic cond. $2.44191\times 10^{7}$
Root an. cond. $2.89556$
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  + 8·9-s − 2·16-s − 24·29-s + 24·31-s − 56·59-s − 32·61-s + 30·81-s − 56·89-s + 64·121-s + 127-s + 131-s + 137-s + 139-s − 16·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
L(s)  = 1  + 8/3·9-s − 1/2·16-s − 4.45·29-s + 4.31·31-s − 7.29·59-s − 4.09·61-s + 10/3·81-s − 5.93·89-s + 5.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 4/3·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{8} \cdot 3^{8} \cdot 5^{16} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(2.44191\times 10^{7}\)
Root analytic conductor: \(2.89556\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{8} \cdot 3^{8} \cdot 5^{16} \cdot 7^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.5824142080\)
\(L(\frac12)\) \(\approx\) \(0.5824142080\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + T^{4} )^{2} \)
3 \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{2} \)
5 \( 1 \)
7 \( ( 1 + T^{4} )^{2} \)
good11 \( ( 1 - 32 T^{2} + 478 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
13 \( 1 - 28 T^{4} + 52198 T^{8} - 28 p^{4} T^{12} + p^{8} T^{16} \)
17 \( ( 1 - 16 T^{2} + p^{2} T^{4} )^{2}( 1 + 16 T^{2} + p^{2} T^{4} )^{2} \)
19 \( ( 1 - 48 T^{2} + 1118 T^{4} - 48 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
23 \( 1 - 508 T^{4} + 4678 T^{8} - 508 p^{4} T^{12} + p^{8} T^{16} \)
29 \( ( 1 + 6 T + 62 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{4} \)
31 \( ( 1 - 6 T + 66 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{4} \)
37 \( 1 - 604 T^{4} - 32474 T^{8} - 604 p^{4} T^{12} + p^{8} T^{16} \)
41 \( ( 1 - 56 T^{2} + 3166 T^{4} - 56 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
43 \( 1 + 5444 T^{4} + 13734886 T^{8} + 5444 p^{4} T^{12} + p^{8} T^{16} \)
47 \( 1 - 5788 T^{4} + 16086598 T^{8} - 5788 p^{4} T^{12} + p^{8} T^{16} \)
53 \( 1 - 7708 T^{4} + 28156198 T^{8} - 7708 p^{4} T^{12} + p^{8} T^{16} \)
59 \( ( 1 + 14 T + 122 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{4} \)
61 \( ( 1 + 8 T + 58 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{4} \)
67 \( 1 - 12988 T^{4} + 81854758 T^{8} - 12988 p^{4} T^{12} + p^{8} T^{16} \)
71 \( ( 1 + 84 T^{2} + 8966 T^{4} + 84 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
73 \( 1 - 1148 T^{4} + 2279238 T^{8} - 1148 p^{4} T^{12} + p^{8} T^{16} \)
79 \( ( 1 - 124 T^{2} + 11206 T^{4} - 124 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
83 \( 1 + 10436 T^{4} + 85152166 T^{8} + 10436 p^{4} T^{12} + p^{8} T^{16} \)
89 \( ( 1 + 14 T + 222 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{4} \)
97 \( 1 - 21436 T^{4} + 227134086 T^{8} - 21436 p^{4} T^{12} + p^{8} T^{16} \)
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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

−4.42248701126860740966365060264, −4.25253685131402855723918585626, −4.16332725913613375356569476426, −3.90967220035381126949981681493, −3.63104443747150019741927788365, −3.45509577812791646638580324361, −3.43761864260800206155872613224, −3.31115845460641395005949076788, −3.27262001149074591175751181679, −3.24709796797310013658675335816, −2.78743837406760573471705585904, −2.70737364707928382071434079493, −2.61735953997163248131059011254, −2.50105507914648378023827823958, −2.36110228153840807525461952117, −1.84870562463776408508834455426, −1.84169251230660354208071275303, −1.84123788760882676365365835728, −1.67419083260778784383569779392, −1.34454325390965214090742185165, −1.26448955480865647643216324782, −1.14015342842950694613175368867, −1.07425915295009402376606286008, −0.30610487291024506905471696072, −0.12195826207085402574685112126, 0.12195826207085402574685112126, 0.30610487291024506905471696072, 1.07425915295009402376606286008, 1.14015342842950694613175368867, 1.26448955480865647643216324782, 1.34454325390965214090742185165, 1.67419083260778784383569779392, 1.84123788760882676365365835728, 1.84169251230660354208071275303, 1.84870562463776408508834455426, 2.36110228153840807525461952117, 2.50105507914648378023827823958, 2.61735953997163248131059011254, 2.70737364707928382071434079493, 2.78743837406760573471705585904, 3.24709796797310013658675335816, 3.27262001149074591175751181679, 3.31115845460641395005949076788, 3.43761864260800206155872613224, 3.45509577812791646638580324361, 3.63104443747150019741927788365, 3.90967220035381126949981681493, 4.16332725913613375356569476426, 4.25253685131402855723918585626, 4.42248701126860740966365060264

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

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