Properties

Degree $16$
Conductor $1.431\times 10^{21}$
Sign $1$
Motivic weight $3$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·4-s + 67·16-s − 232·25-s + 64·37-s + 2.16e3·43-s + 324·64-s + 5.39e3·79-s − 464·100-s − 5.95e3·121-s + 127-s + 131-s + 137-s + 139-s + 128·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 7.04e3·169-s + 4.32e3·172-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯
L(s)  = 1  + 1/4·4-s + 1.04·16-s − 1.85·25-s + 0.284·37-s + 7.66·43-s + 0.632·64-s + 7.67·79-s − 0.463·100-s − 4.47·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.0710·148-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s − 3.20·169-s + 1.91·172-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + 0.000361·197-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(3^{16} \cdot 7^{16}\)
Sign: $1$
Motivic weight: \(3\)
Character: induced by $\chi_{441} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 3^{16} \cdot 7^{16} ,\ ( \ : [3/2]^{8} ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(19.43053551\)
\(L(\frac12)\) \(\approx\) \(19.43053551\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( ( 1 - T^{2} - p^{5} T^{4} - p^{6} T^{6} + p^{12} T^{8} )^{2} \)
5 \( ( 1 + 116 T^{2} + 1282 p^{2} T^{4} + 116 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
11 \( ( 1 + 2976 T^{2} + 47390 p^{2} T^{4} + 2976 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
13 \( ( 1 + 3520 T^{2} + 5818162 T^{4} + 3520 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
17 \( ( 1 + 9156 T^{2} + 49826306 T^{4} + 9156 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
19 \( ( 1 + 4356 T^{2} + 98743142 T^{4} + 4356 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
23 \( ( 1 + 42656 T^{2} + 750952798 T^{4} + 42656 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
29 \( ( 1 + 44468 T^{2} + 996014662 T^{4} + 44468 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
31 \( ( 1 + 99700 T^{2} + 4216698262 T^{4} + 99700 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
37 \( ( 1 - 16 T + 9066 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} )^{4} \)
41 \( ( 1 + 42420 T^{2} + 6386007458 T^{4} + 42420 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
43 \( ( 1 - 540 T + 167814 T^{2} - 540 p^{3} T^{3} + p^{6} T^{4} )^{4} \)
47 \( ( 1 + 80164 T^{2} + 14237262198 T^{4} + 80164 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
53 \( ( 1 + 462132 T^{2} + 97706926838 T^{4} + 462132 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
59 \( ( 1 + 590420 T^{2} + 168915151078 T^{4} + 590420 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
61 \( ( 1 + 162080 T^{2} - 25534056078 T^{4} + 162080 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
67 \( ( 1 + 345126 T^{2} + p^{6} T^{4} )^{4} \)
71 \( ( 1 + 472256 T^{2} + 269755485790 T^{4} + 472256 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
73 \( ( 1 + 1020400 T^{2} + 546326225122 T^{4} + 1020400 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
79 \( ( 1 - 1348 T + 16642 p T^{2} - 1348 p^{3} T^{3} + p^{6} T^{4} )^{4} \)
83 \( ( 1 + 367916 T^{2} + 118793240278 T^{4} + 367916 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
89 \( ( 1 + 2588100 T^{2} + 2668496393378 T^{4} + 2588100 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
97 \( ( 1 + 2268304 T^{2} + 2510084445826 T^{4} + 2268304 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
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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

−4.56414699938460105551127616082, −4.04260181612637535989656853116, −4.02082141889044503297319971960, −4.01298064682546215002871575029, −3.91584836810128479261098429934, −3.69913336318360698758648958574, −3.68546761829353287856535636339, −3.33177093107664872726295923103, −3.29723169708872725086202042218, −3.06271269733544288741566078447, −2.90157430893306751172173091893, −2.65821569817968930762030531205, −2.54341338814675595787958961351, −2.30209701323291969294128116791, −2.16981978401081169371352661618, −2.09357561229777712202459988683, −1.94408116719686640820178508438, −1.87472276351988175545453895981, −1.39002225116710743592304612509, −1.06808365019296098869595337861, −0.926171479124073196079099217059, −0.840820038756991393831167030671, −0.75355531249835741872345923194, −0.41954319055443436807373191201, −0.30800214825876499867451036834, 0.30800214825876499867451036834, 0.41954319055443436807373191201, 0.75355531249835741872345923194, 0.840820038756991393831167030671, 0.926171479124073196079099217059, 1.06808365019296098869595337861, 1.39002225116710743592304612509, 1.87472276351988175545453895981, 1.94408116719686640820178508438, 2.09357561229777712202459988683, 2.16981978401081169371352661618, 2.30209701323291969294128116791, 2.54341338814675595787958961351, 2.65821569817968930762030531205, 2.90157430893306751172173091893, 3.06271269733544288741566078447, 3.29723169708872725086202042218, 3.33177093107664872726295923103, 3.68546761829353287856535636339, 3.69913336318360698758648958574, 3.91584836810128479261098429934, 4.01298064682546215002871575029, 4.02082141889044503297319971960, 4.04260181612637535989656853116, 4.56414699938460105551127616082

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

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