Properties

Label 16-384e8-1.1-c6e8-0-1
Degree $16$
Conductor $4.728\times 10^{20}$
Sign $1$
Analytic cond. $3.70927\times 10^{15}$
Root an. cond. $9.39897$
Motivic weight $6$
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  − 72·3-s + 1.76e3·9-s − 1.64e3·11-s − 6.26e4·25-s − 9.72e3·27-s + 1.18e5·33-s − 4.17e5·49-s − 8.36e5·59-s − 1.96e6·73-s + 4.50e6·75-s − 1.28e5·81-s + 5.87e5·83-s − 1.47e6·97-s − 2.90e6·99-s + 7.65e5·107-s − 1.21e7·121-s + 127-s + 131-s + 137-s + 139-s + 3.00e7·147-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.38e7·169-s + ⋯
L(s)  = 1  − 8/3·3-s + 2.41·9-s − 1.23·11-s − 4.00·25-s − 0.493·27-s + 3.30·33-s − 3.55·49-s − 4.07·59-s − 5.05·73-s + 10.6·75-s − 0.242·81-s + 1.02·83-s − 1.61·97-s − 2.99·99-s + 0.624·107-s − 6.86·121-s + 9.47·147-s + 2.87·169-s + 10.8·177-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(7-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+3)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{56} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(3.70927\times 10^{15}\)
Root analytic conductor: \(9.39897\)
Motivic weight: \(6\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{384} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{56} \cdot 3^{8} ,\ ( \ : [3]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.002476320356\)
\(L(\frac12)\) \(\approx\) \(0.002476320356\)
\(L(4)\) 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 \( ( 1 + 4 p^{2} T + 118 p^{2} T^{2} + 4 p^{8} T^{3} + p^{12} T^{4} )^{2} \)
good5 \( ( 1 + 1252 p^{2} T^{2} + 5863326 p^{3} T^{4} + 1252 p^{14} T^{6} + p^{24} T^{8} )^{2} \)
7 \( ( 1 + 208996 T^{2} + 30837454086 T^{4} + 208996 p^{12} T^{6} + p^{24} T^{8} )^{2} \)
11 \( ( 1 + 412 T + 314898 p T^{2} + 412 p^{6} T^{3} + p^{12} T^{4} )^{4} \)
13 \( ( 1 - 6934756 T^{2} + 21911820809766 T^{4} - 6934756 p^{12} T^{6} + p^{24} T^{8} )^{2} \)
17 \( ( 1 - 64896772 T^{2} + 2198931719582598 T^{4} - 64896772 p^{12} T^{6} + p^{24} T^{8} )^{2} \)
19 \( ( 1 - 177090628 T^{2} + 12242150590475238 T^{4} - 177090628 p^{12} T^{6} + p^{24} T^{8} )^{2} \)
23 \( ( 1 - 1858756 T^{2} + 43746394215781446 T^{4} - 1858756 p^{12} T^{6} + p^{24} T^{8} )^{2} \)
29 \( ( 1 + 384071044 T^{2} + 262631660267478246 T^{4} + 384071044 p^{12} T^{6} + p^{24} T^{8} )^{2} \)
31 \( ( 1 + 2960240164 T^{2} + 3713060857169843526 T^{4} + 2960240164 p^{12} T^{6} + p^{24} T^{8} )^{2} \)
37 \( ( 1 - 3235129636 T^{2} + 7977254229970771686 T^{4} - 3235129636 p^{12} T^{6} + p^{24} T^{8} )^{2} \)
41 \( ( 1 - 13543268548 T^{2} + 89790600389496386118 T^{4} - 13543268548 p^{12} T^{6} + p^{24} T^{8} )^{2} \)
43 \( ( 1 - 6754505476 T^{2} + 39481734937898737446 T^{4} - 6754505476 p^{12} T^{6} + p^{24} T^{8} )^{2} \)
47 \( ( 1 - 13978081156 T^{2} + \)\(16\!\cdots\!86\)\( T^{4} - 13978081156 p^{12} T^{6} + p^{24} T^{8} )^{2} \)
53 \( ( 1 + 52183699396 T^{2} + \)\(13\!\cdots\!66\)\( T^{4} + 52183699396 p^{12} T^{6} + p^{24} T^{8} )^{2} \)
59 \( ( 1 + 209156 T + 54505500486 T^{2} + 209156 p^{6} T^{3} + p^{12} T^{4} )^{4} \)
61 \( ( 1 - 97087344484 T^{2} + \)\(75\!\cdots\!26\)\( T^{4} - 97087344484 p^{12} T^{6} + p^{24} T^{8} )^{2} \)
67 \( ( 1 - 172396628932 T^{2} + \)\(20\!\cdots\!98\)\( T^{4} - 172396628932 p^{12} T^{6} + p^{24} T^{8} )^{2} \)
71 \( ( 1 - 254155821124 T^{2} + \)\(48\!\cdots\!46\)\( T^{4} - 254155821124 p^{12} T^{6} + p^{24} T^{8} )^{2} \)
73 \( ( 1 + 491236 T + 352157486502 T^{2} + 491236 p^{6} T^{3} + p^{12} T^{4} )^{4} \)
79 \( ( 1 - 65015134556 T^{2} + \)\(11\!\cdots\!46\)\( T^{4} - 65015134556 p^{12} T^{6} + p^{24} T^{8} )^{2} \)
83 \( ( 1 - 146756 T + 654996035622 T^{2} - 146756 p^{6} T^{3} + p^{12} T^{4} )^{4} \)
89 \( ( 1 - 329765563588 T^{2} + \)\(84\!\cdots\!58\)\( T^{4} - 329765563588 p^{12} T^{6} + p^{24} T^{8} )^{2} \)
97 \( ( 1 + 369308 T - 106588984506 T^{2} + 369308 p^{6} T^{3} + p^{12} T^{4} )^{4} \)
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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

−3.99973174634618858058168892198, −3.94508009503308979327078664826, −3.81579865399070435919516897565, −3.44213424905554442978442141685, −3.26831870144606819490372044645, −3.14910692376095903566513908645, −3.10689056580321899390976604942, −3.03389736316164954936714612712, −2.77690179936766228526139723162, −2.56043296900980092392223334353, −2.45679572846379824932870091492, −2.18419831640837436865958123188, −1.98634057355883693160535028966, −1.85714352207533761366641014560, −1.83119211808523495219119848869, −1.66190650762518891453499278558, −1.35392587858220824903084327605, −1.18486296577278221393234427374, −1.04312006997351556909285542518, −1.04272332817705470681669316290, −0.819634484598732597117388441498, −0.15042249454059650353842047774, −0.13915291791865138686664786528, −0.11224284858795867195704544089, −0.10437270713263148148757655563, 0.10437270713263148148757655563, 0.11224284858795867195704544089, 0.13915291791865138686664786528, 0.15042249454059650353842047774, 0.819634484598732597117388441498, 1.04272332817705470681669316290, 1.04312006997351556909285542518, 1.18486296577278221393234427374, 1.35392587858220824903084327605, 1.66190650762518891453499278558, 1.83119211808523495219119848869, 1.85714352207533761366641014560, 1.98634057355883693160535028966, 2.18419831640837436865958123188, 2.45679572846379824932870091492, 2.56043296900980092392223334353, 2.77690179936766228526139723162, 3.03389736316164954936714612712, 3.10689056580321899390976604942, 3.14910692376095903566513908645, 3.26831870144606819490372044645, 3.44213424905554442978442141685, 3.81579865399070435919516897565, 3.94508009503308979327078664826, 3.99973174634618858058168892198

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

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