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Properties

Label 16-2352e8-1.1-c2e8-0-4
Degree $16$
Conductor $9.365\times 10^{26}$
Sign $1$
Analytic cond. $2.84563\times 10^{14}$
Root an. cond. $8.00545$
Motivic weight $2$
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  − 12·9-s + 64·11-s + 144·23-s + 72·25-s + 16·29-s + 64·37-s − 16·43-s + 112·53-s − 192·67-s + 160·71-s + 176·79-s + 90·81-s − 768·99-s − 208·107-s − 176·109-s + 128·113-s + 1.58e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 448·169-s + ⋯
L(s)  = 1  − 4/3·9-s + 5.81·11-s + 6.26·23-s + 2.87·25-s + 0.551·29-s + 1.72·37-s − 0.372·43-s + 2.11·53-s − 2.86·67-s + 2.25·71-s + 2.22·79-s + 10/9·81-s − 7.75·99-s − 1.94·107-s − 1.61·109-s + 1.13·113-s + 13.0·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 2.65·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{32} \cdot 3^{8} \cdot 7^{16}\)
Sign: $1$
Analytic conductor: \(2.84563\times 10^{14}\)
Root analytic conductor: \(8.00545\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{32} \cdot 3^{8} \cdot 7^{16} ,\ ( \ : [1]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(90.95080856\)
\(L(\frac12)\) \(\approx\) \(90.95080856\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( ( 1 + p T^{2} )^{4} \)
7 \( 1 \)
good5 \( 1 - 72 T^{2} + 408 p T^{4} - 16296 T^{6} - 234766 T^{8} - 16296 p^{4} T^{10} + 408 p^{9} T^{12} - 72 p^{12} T^{14} + p^{16} T^{16} \)
11 \( ( 1 - 32 T + 744 T^{2} - 11632 T^{3} + 149834 T^{4} - 11632 p^{2} T^{5} + 744 p^{4} T^{6} - 32 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
13 \( 1 - 448 T^{2} + 80832 T^{4} - 12876224 T^{6} + 2438878274 T^{8} - 12876224 p^{4} T^{10} + 80832 p^{8} T^{12} - 448 p^{12} T^{14} + p^{16} T^{16} \)
17 \( 1 - 1288 T^{2} + 918072 T^{4} - 434650856 T^{6} + 147138021170 T^{8} - 434650856 p^{4} T^{10} + 918072 p^{8} T^{12} - 1288 p^{12} T^{14} + p^{16} T^{16} \)
19 \( 1 - 1624 T^{2} + 1175436 T^{4} - 535399208 T^{6} + 199113215846 T^{8} - 535399208 p^{4} T^{10} + 1175436 p^{8} T^{12} - 1624 p^{12} T^{14} + p^{16} T^{16} \)
23 \( ( 1 - 72 T + 3936 T^{2} - 133080 T^{3} + 3671834 T^{4} - 133080 p^{2} T^{5} + 3936 p^{4} T^{6} - 72 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
29 \( ( 1 - 8 T + 1248 T^{2} - 34936 T^{3} + 986042 T^{4} - 34936 p^{2} T^{5} + 1248 p^{4} T^{6} - 8 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
31 \( 1 - 4312 T^{2} + 9119052 T^{4} - 13306319912 T^{6} + 14680375735718 T^{8} - 13306319912 p^{4} T^{10} + 9119052 p^{8} T^{12} - 4312 p^{12} T^{14} + p^{16} T^{16} \)
37 \( ( 1 - 32 T + 4224 T^{2} - 111712 T^{3} + 7804898 T^{4} - 111712 p^{2} T^{5} + 4224 p^{4} T^{6} - 32 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
41 \( 1 - 6456 T^{2} + 23620056 T^{4} - 59716495320 T^{6} + 113784142705202 T^{8} - 59716495320 p^{4} T^{10} + 23620056 p^{8} T^{12} - 6456 p^{12} T^{14} + p^{16} T^{16} \)
43 \( ( 1 + 8 T + 5020 T^{2} + 39416 T^{3} + 12943654 T^{4} + 39416 p^{2} T^{5} + 5020 p^{4} T^{6} + 8 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
47 \( 1 - 4392 T^{2} + 16746060 T^{4} - 40867208664 T^{6} + 45687756518 p^{2} T^{8} - 40867208664 p^{4} T^{10} + 16746060 p^{8} T^{12} - 4392 p^{12} T^{14} + p^{16} T^{16} \)
53 \( ( 1 - 56 T + 10636 T^{2} - 459656 T^{3} + 43977478 T^{4} - 459656 p^{2} T^{5} + 10636 p^{4} T^{6} - 56 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
59 \( 1 - 20904 T^{2} + 206285580 T^{4} - 1265717294040 T^{6} + 5282541205354982 T^{8} - 1265717294040 p^{4} T^{10} + 206285580 p^{8} T^{12} - 20904 p^{12} T^{14} + p^{16} T^{16} \)
61 \( 1 - 12160 T^{2} + 47368704 T^{4} + 32287126912 T^{6} - 671662744204990 T^{8} + 32287126912 p^{4} T^{10} + 47368704 p^{8} T^{12} - 12160 p^{12} T^{14} + p^{16} T^{16} \)
67 \( ( 1 + 96 T + 11140 T^{2} + 350496 T^{3} + 35881254 T^{4} + 350496 p^{2} T^{5} + 11140 p^{4} T^{6} + 96 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
71 \( ( 1 - 80 T + 17544 T^{2} - 1141504 T^{3} + 127713770 T^{4} - 1141504 p^{2} T^{5} + 17544 p^{4} T^{6} - 80 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
73 \( 1 - 19392 T^{2} + 179810688 T^{4} - 1121770114752 T^{6} + 6024755745952898 T^{8} - 1121770114752 p^{4} T^{10} + 179810688 p^{8} T^{12} - 19392 p^{12} T^{14} + p^{16} T^{16} \)
79 \( ( 1 - 88 T + 13596 T^{2} - 991592 T^{3} + 123839942 T^{4} - 991592 p^{2} T^{5} + 13596 p^{4} T^{6} - 88 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
83 \( 1 - 38472 T^{2} + 704817564 T^{4} - 8155687703160 T^{6} + 66129062974945670 T^{8} - 8155687703160 p^{4} T^{10} + 704817564 p^{8} T^{12} - 38472 p^{12} T^{14} + p^{16} T^{16} \)
89 \( 1 - 25992 T^{2} + 341082360 T^{4} - 3264120127848 T^{6} + 26879492526931634 T^{8} - 3264120127848 p^{4} T^{10} + 341082360 p^{8} T^{12} - 25992 p^{12} T^{14} + p^{16} T^{16} \)
97 \( 1 - 12864 T^{2} + 271400256 T^{4} - 3007239316800 T^{6} + 34304436094172162 T^{8} - 3007239316800 p^{4} T^{10} + 271400256 p^{8} T^{12} - 12864 p^{12} T^{14} + p^{16} T^{16} \)
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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

−3.61493263661656128149769609941, −3.46096989718349077168794923149, −3.24962412218553508221406082825, −3.24922434203509853953542767923, −3.08633673591622673587439375367, −2.97230237885389768338588935212, −2.76133762354994258301262469023, −2.72691566954910755028253342698, −2.55152683173404425447092911709, −2.51439621636427001815606468409, −2.46675630964092868964667431273, −2.25402391394563751861741448663, −2.06211008920980923676012684119, −1.61342488970817002408373819622, −1.56235453229812654709214566959, −1.53106209228079096248091147572, −1.35519229601435408725179706194, −1.23067104313076942618980968646, −1.17395123441936549461967375759, −1.08994557906877489921440960084, −0.851579891362157082562194041358, −0.70999305161947280140676656261, −0.59986409414868874197202077096, −0.56231096812684730441431146874, −0.25654308720836781875442618497, 0.25654308720836781875442618497, 0.56231096812684730441431146874, 0.59986409414868874197202077096, 0.70999305161947280140676656261, 0.851579891362157082562194041358, 1.08994557906877489921440960084, 1.17395123441936549461967375759, 1.23067104313076942618980968646, 1.35519229601435408725179706194, 1.53106209228079096248091147572, 1.56235453229812654709214566959, 1.61342488970817002408373819622, 2.06211008920980923676012684119, 2.25402391394563751861741448663, 2.46675630964092868964667431273, 2.51439621636427001815606468409, 2.55152683173404425447092911709, 2.72691566954910755028253342698, 2.76133762354994258301262469023, 2.97230237885389768338588935212, 3.08633673591622673587439375367, 3.24922434203509853953542767923, 3.24962412218553508221406082825, 3.46096989718349077168794923149, 3.61493263661656128149769609941

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

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