Properties

Degree $16$
Conductor $1.100\times 10^{20}$
Sign $1$
Motivic weight $5$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 480·9-s + 1.19e4·25-s + 9.11e4·41-s + 7.54e4·49-s − 9.21e4·81-s − 7.94e5·89-s + 9.33e5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2.02e6·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s − 5.71e6·225-s + ⋯
L(s)  = 1  − 1.97·9-s + 3.80·25-s + 8.46·41-s + 4.48·49-s − 1.56·81-s − 10.6·89-s + 5.79·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s + 3.23e−6·157-s + 2.94e−6·163-s + 2.77e−6·167-s − 5.45·169-s + 2.54e−6·173-s + 2.33e−6·179-s + 2.26e−6·181-s + 1.98e−6·191-s + 1.93e−6·193-s + 1.83e−6·197-s + 1.79e−6·199-s + 1.54e−6·211-s + 1.34e−6·223-s − 7.52·225-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{48} \cdot 5^{8}\)
Sign: $1$
Motivic weight: \(5\)
Character: induced by $\chi_{320} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{48} \cdot 5^{8} ,\ ( \ : [5/2]^{8} ),\ 1 )\)

Particular Values

\(L(3)\) \(\approx\) \(4.764931597\)
\(L(\frac12)\) \(\approx\) \(4.764931597\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( ( 1 - 238 p^{2} T^{2} + p^{10} T^{4} )^{2} \)
good3 \( ( 1 + 40 p T^{2} + p^{10} T^{4} )^{4} \)
7 \( ( 1 - 18852 T^{2} + p^{10} T^{4} )^{4} \)
11 \( ( 1 - 233298 T^{2} + p^{10} T^{4} )^{4} \)
13 \( ( 1 + 506394 T^{2} + p^{10} T^{4} )^{4} \)
17 \( ( 1 - 2662570 T^{2} + p^{10} T^{4} )^{4} \)
19 \( ( 1 + 1673278 T^{2} + p^{10} T^{4} )^{4} \)
23 \( ( 1 - 12357236 T^{2} + p^{10} T^{4} )^{4} \)
29 \( ( 1 - 27077290 T^{2} + p^{10} T^{4} )^{4} \)
31 \( ( 1 + 19214 p^{2} T^{2} + p^{10} T^{4} )^{4} \)
37 \( ( 1 + 77394 p^{2} T^{2} + p^{10} T^{4} )^{4} \)
41 \( ( 1 - 11396 T + p^{5} T^{2} )^{8} \)
43 \( ( 1 - 72715480 T^{2} + p^{10} T^{4} )^{4} \)
47 \( ( 1 - 453462436 T^{2} + p^{10} T^{4} )^{4} \)
53 \( ( 1 + 539491786 T^{2} + p^{10} T^{4} )^{4} \)
59 \( ( 1 - 1161609714 T^{2} + p^{10} T^{4} )^{4} \)
61 \( ( 1 + 686048530 T^{2} + p^{10} T^{4} )^{4} \)
67 \( ( 1 + 2629714328 T^{2} + p^{10} T^{4} )^{4} \)
71 \( ( 1 + 2762141854 T^{2} + p^{10} T^{4} )^{4} \)
73 \( ( 1 - 4145966042 T^{2} + p^{10} T^{4} )^{4} \)
79 \( ( 1 + 5182544750 T^{2} + p^{10} T^{4} )^{4} \)
83 \( ( 1 - 3415955864 T^{2} + p^{10} T^{4} )^{4} \)
89 \( ( 1 + 99362 T + p^{5} T^{2} )^{8} \)
97 \( ( 1 + 15344227702 T^{2} + p^{10} T^{4} )^{4} \)
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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

−4.32614870012673123857644889375, −4.20791619078746892690725871352, −4.10920041099009648703834475426, −3.82161294533934488260216034412, −3.60573693998182465514884394674, −3.55990286837408158080034984225, −3.26190406491093730694055155232, −3.01759783965403678094677442767, −2.83314352951830629074152949656, −2.78443030897026439812842180275, −2.70465077125751782100970246009, −2.55218677988979436514775638013, −2.53690636252064932335392209528, −2.36706606272904450104723339505, −2.24840760247956606389440022708, −1.76831791927789783016103672179, −1.55158919650606585430643920607, −1.34327937123262441550404762117, −1.16831888350469217130965480641, −1.01976210223580627933231750675, −0.832517453706098061643229596280, −0.75438382260686888714622540711, −0.67476676310681496254354079087, −0.21715298591256696383692215825, −0.17910271212036626436635804063, 0.17910271212036626436635804063, 0.21715298591256696383692215825, 0.67476676310681496254354079087, 0.75438382260686888714622540711, 0.832517453706098061643229596280, 1.01976210223580627933231750675, 1.16831888350469217130965480641, 1.34327937123262441550404762117, 1.55158919650606585430643920607, 1.76831791927789783016103672179, 2.24840760247956606389440022708, 2.36706606272904450104723339505, 2.53690636252064932335392209528, 2.55218677988979436514775638013, 2.70465077125751782100970246009, 2.78443030897026439812842180275, 2.83314352951830629074152949656, 3.01759783965403678094677442767, 3.26190406491093730694055155232, 3.55990286837408158080034984225, 3.60573693998182465514884394674, 3.82161294533934488260216034412, 4.10920041099009648703834475426, 4.20791619078746892690725871352, 4.32614870012673123857644889375

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

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