Properties

Label 16-384e8-1.1-c8e8-0-1
Degree $16$
Conductor $4.728\times 10^{20}$
Sign $1$
Analytic cond. $3.58620\times 10^{17}$
Root an. cond. $12.5073$
Motivic weight $8$
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  + 8.74e3·9-s + 2.76e5·17-s + 1.41e5·25-s + 5.82e6·41-s + 1.47e7·49-s − 5.77e7·73-s + 4.78e7·81-s − 2.58e8·89-s − 7.71e8·97-s − 3.27e8·113-s − 3.21e8·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 2.42e9·153-s + 157-s + 163-s + 167-s + 4.95e9·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  + 4/3·9-s + 3.31·17-s + 0.362·25-s + 2.06·41-s + 2.56·49-s − 2.03·73-s + 10/9·81-s − 4.12·89-s − 8.71·97-s − 2.00·113-s − 1.49·121-s + 4.41·153-s + 6.07·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(9-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+4)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{56} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(3.58620\times 10^{17}\)
Root analytic conductor: \(12.5073\)
Motivic weight: \(8\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{384} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{56} \cdot 3^{8} ,\ ( \ : [4]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(37.50938150\)
\(L(\frac12)\) \(\approx\) \(37.50938150\)
\(L(5)\) 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 \( ( 1 - p^{7} T^{2} )^{4} \)
good5 \( ( 1 - 70756 T^{2} - 7732564938 p^{2} T^{4} - 70756 p^{16} T^{6} + p^{32} T^{8} )^{2} \)
7 \( ( 1 - 7379620 T^{2} + 757368357414 p^{2} T^{4} - 7379620 p^{16} T^{6} + p^{32} T^{8} )^{2} \)
11 \( ( 1 + 160503844 T^{2} + 92432076593555142 T^{4} + 160503844 p^{16} T^{6} + p^{32} T^{8} )^{2} \)
13 \( ( 1 - 2475941380 T^{2} + 2790480367912748166 T^{4} - 2475941380 p^{16} T^{6} + p^{32} T^{8} )^{2} \)
17 \( ( 1 - 69212 T + 14116566342 T^{2} - 69212 p^{8} T^{3} + p^{16} T^{4} )^{4} \)
19 \( ( 1 + 25078256164 T^{2} + \)\(29\!\cdots\!86\)\( T^{4} + 25078256164 p^{16} T^{6} + p^{32} T^{8} )^{2} \)
23 \( ( 1 - 261444073348 T^{2} + \)\(28\!\cdots\!42\)\( T^{4} - 261444073348 p^{16} T^{6} + p^{32} T^{8} )^{2} \)
29 \( ( 1 - 1434224064100 T^{2} + \)\(95\!\cdots\!58\)\( T^{4} - 1434224064100 p^{16} T^{6} + p^{32} T^{8} )^{2} \)
31 \( ( 1 - 2042390161828 T^{2} + \)\(20\!\cdots\!22\)\( T^{4} - 2042390161828 p^{16} T^{6} + p^{32} T^{8} )^{2} \)
37 \( ( 1 + 999600928892 T^{2} + \)\(24\!\cdots\!82\)\( T^{4} + 999600928892 p^{16} T^{6} + p^{32} T^{8} )^{2} \)
41 \( ( 1 - 1456708 T + 15954144930054 T^{2} - 1456708 p^{8} T^{3} + p^{16} T^{4} )^{4} \)
43 \( ( 1 + 21554652987940 T^{2} + \)\(23\!\cdots\!02\)\( T^{4} + 21554652987940 p^{16} T^{6} + p^{32} T^{8} )^{2} \)
47 \( ( 1 - 42662657557636 T^{2} + \)\(11\!\cdots\!62\)\( T^{4} - 42662657557636 p^{16} T^{6} + p^{32} T^{8} )^{2} \)
53 \( ( 1 - 211751128952932 T^{2} + \)\(18\!\cdots\!62\)\( T^{4} - 211751128952932 p^{16} T^{6} + p^{32} T^{8} )^{2} \)
59 \( ( 1 + 49948669021732 T^{2} + \)\(22\!\cdots\!62\)\( T^{4} + 49948669021732 p^{16} T^{6} + p^{32} T^{8} )^{2} \)
61 \( ( 1 - 227345717386372 T^{2} + \)\(18\!\cdots\!42\)\( T^{4} - 227345717386372 p^{16} T^{6} + p^{32} T^{8} )^{2} \)
67 \( ( 1 + 1294221353339428 T^{2} + \)\(72\!\cdots\!58\)\( T^{4} + 1294221353339428 p^{16} T^{6} + p^{32} T^{8} )^{2} \)
71 \( ( 1 - 1067557951562116 T^{2} + \)\(10\!\cdots\!02\)\( T^{4} - 1067557951562116 p^{16} T^{6} + p^{32} T^{8} )^{2} \)
73 \( ( 1 + 14439940 T + 1545688402083462 T^{2} + 14439940 p^{8} T^{3} + p^{16} T^{4} )^{4} \)
79 \( ( 1 + 1334505648457820 T^{2} + \)\(50\!\cdots\!18\)\( T^{4} + 1334505648457820 p^{16} T^{6} + p^{32} T^{8} )^{2} \)
83 \( ( 1 + 5531442914217508 T^{2} + \)\(15\!\cdots\!78\)\( T^{4} + 5531442914217508 p^{16} T^{6} + p^{32} T^{8} )^{2} \)
89 \( ( 1 + 64694692 T + 6451676807249862 T^{2} + 64694692 p^{8} T^{3} + p^{16} T^{4} )^{4} \)
97 \( ( 1 + 192872356 T + 23512646696808006 T^{2} + 192872356 p^{8} T^{3} + p^{16} T^{4} )^{4} \)
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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

−3.62528278373514986238690820603, −3.61174935088914318415300712036, −3.27376713853987659802175948250, −3.17731081398473764804276824110, −3.08725411369295373274197492763, −2.95816161807724298190562531080, −2.81457050245772211507745701802, −2.62616504901386711267324018692, −2.51711550921339800617023228805, −2.48564460944353289764056423098, −2.22466645006836181890199104514, −2.18184147387089328856920231869, −1.74483705321450699127825914073, −1.55368855901808711303194061733, −1.50738330507970349546589034209, −1.44970219410906193815149110960, −1.34448366616894379798602868686, −1.26052534345362809637032345597, −1.12531646387901995197463506978, −0.78219401712516889175959458582, −0.76699947850529358089875820153, −0.63132826843217930861921557754, −0.30593048389631665556702727456, −0.30329793256587393163377211997, −0.27416043204052987201991267031, 0.27416043204052987201991267031, 0.30329793256587393163377211997, 0.30593048389631665556702727456, 0.63132826843217930861921557754, 0.76699947850529358089875820153, 0.78219401712516889175959458582, 1.12531646387901995197463506978, 1.26052534345362809637032345597, 1.34448366616894379798602868686, 1.44970219410906193815149110960, 1.50738330507970349546589034209, 1.55368855901808711303194061733, 1.74483705321450699127825914073, 2.18184147387089328856920231869, 2.22466645006836181890199104514, 2.48564460944353289764056423098, 2.51711550921339800617023228805, 2.62616504901386711267324018692, 2.81457050245772211507745701802, 2.95816161807724298190562531080, 3.08725411369295373274197492763, 3.17731081398473764804276824110, 3.27376713853987659802175948250, 3.61174935088914318415300712036, 3.62528278373514986238690820603

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

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