Properties

Label 16-300e8-1.1-c1e8-0-4
Degree $16$
Conductor $6.561\times 10^{19}$
Sign $1$
Analytic cond. $1084.39$
Root an. cond. $1.54774$
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 + 9-s + 4·13-s + 3·16-s − 36-s + 8·37-s + 18·49-s − 4·52-s − 20·61-s − 9·64-s − 36·73-s − 7·81-s − 12·97-s + 44·109-s + 4·117-s − 30·121-s + 127-s + 131-s + 137-s + 139-s + 3·144-s − 8·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  − 1/2·4-s + 1/3·9-s + 1.10·13-s + 3/4·16-s − 1/6·36-s + 1.31·37-s + 18/7·49-s − 0.554·52-s − 2.56·61-s − 9/8·64-s − 4.21·73-s − 7/9·81-s − 1.21·97-s + 4.21·109-s + 0.369·117-s − 2.72·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1/4·144-s − 0.657·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯

Functional equation

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

Invariants

Degree: \(16\)
Conductor: \(2^{16} \cdot 3^{8} \cdot 5^{16}\)
Sign: $1$
Analytic conductor: \(1084.39\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{300} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{16} \cdot 3^{8} \cdot 5^{16} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.537305868\)
\(L(\frac12)\) \(\approx\) \(1.537305868\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T^{2} - p T^{4} + p^{2} T^{6} + p^{4} T^{8} \)
3 \( 1 - T^{2} + 8 T^{4} - p^{2} T^{6} + p^{4} T^{8} \)
5 \( 1 \)
good7 \( ( 1 - 9 T^{2} + 108 T^{4} - 9 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
11 \( ( 1 + 15 T^{2} + 288 T^{4} + 15 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
13 \( ( 1 - T + 16 T^{2} - p T^{3} + p^{2} T^{4} )^{4} \)
17 \( ( 1 - 9 T^{2} - 232 T^{4} - 9 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
19 \( ( 1 - 50 T^{2} + 1183 T^{4} - 50 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
23 \( ( 1 + 64 T^{2} + 1918 T^{4} + 64 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
29 \( ( 1 - 80 T^{2} + 3118 T^{4} - 80 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
31 \( ( 1 - 69 T^{2} + 2856 T^{4} - 69 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
37 \( ( 1 - 2 T + 34 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{4} \)
41 \( ( 1 - 49 T^{2} + 1656 T^{4} - 49 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
43 \( ( 1 - 101 T^{2} + 5008 T^{4} - 101 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
47 \( ( 1 + 136 T^{2} + 8878 T^{4} + 136 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
53 \( ( 1 - 176 T^{2} + 13198 T^{4} - 176 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
59 \( ( 1 + p T^{2} )^{8} \)
61 \( ( 1 + 5 T + 118 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{4} \)
67 \( ( 1 - 234 T^{2} + 22503 T^{4} - 234 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
71 \( ( 1 + 9918 T^{4} + p^{4} T^{8} )^{2} \)
73 \( ( 1 + 9 T + 156 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{4} \)
79 \( ( 1 - 240 T^{2} + 26718 T^{4} - 240 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
83 \( ( 1 + 319 T^{2} + 39208 T^{4} + 319 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
89 \( ( 1 - 305 T^{2} + 39088 T^{4} - 305 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
97 \( ( 1 + 3 T + 104 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{4} \)
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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

−5.18846785511935601332943982090, −5.09611165470014568834411359405, −4.91097279564200273946532134827, −4.91035266434224874123427373673, −4.71987970962092421022380363894, −4.48230417326699697896028262777, −4.25208470034793145585596531315, −4.20442727113141712215161408166, −4.10958492030487539044182663372, −3.84986612335308103292857002092, −3.76368590506245872423999576078, −3.60875232184222763226009836933, −3.50947399904895256247682526073, −3.01421242577524037355828365221, −2.99347916384676939488650535405, −2.85289373973446634562174677562, −2.66495544682658220965434713518, −2.66126834058225395264992515093, −2.06213102508203619720255057633, −2.00401386698956217742619062919, −1.75944430066183494829233942261, −1.27566324601648472071922053583, −1.19060063888702337925324625850, −1.12929481694610625423474453754, −0.33049392558902920622752100175, 0.33049392558902920622752100175, 1.12929481694610625423474453754, 1.19060063888702337925324625850, 1.27566324601648472071922053583, 1.75944430066183494829233942261, 2.00401386698956217742619062919, 2.06213102508203619720255057633, 2.66126834058225395264992515093, 2.66495544682658220965434713518, 2.85289373973446634562174677562, 2.99347916384676939488650535405, 3.01421242577524037355828365221, 3.50947399904895256247682526073, 3.60875232184222763226009836933, 3.76368590506245872423999576078, 3.84986612335308103292857002092, 4.10958492030487539044182663372, 4.20442727113141712215161408166, 4.25208470034793145585596531315, 4.48230417326699697896028262777, 4.71987970962092421022380363894, 4.91035266434224874123427373673, 4.91097279564200273946532134827, 5.09611165470014568834411359405, 5.18846785511935601332943982090

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

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