Properties

Label 16-120e8-1.1-c2e8-0-0
Degree $16$
Conductor $4.300\times 10^{16}$
Sign $1$
Analytic cond. $13065.6$
Root an. cond. $1.80824$
Motivic weight $2$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·3-s + 16·7-s + 18·9-s − 8·13-s − 8·19-s − 64·21-s − 20·25-s − 44·27-s + 120·31-s + 8·37-s + 32·39-s − 328·43-s − 36·49-s + 32·57-s + 8·61-s + 288·63-s + 152·67-s + 32·73-s + 80·75-s + 88·79-s + 170·81-s − 128·91-s − 480·93-s + 144·97-s − 144·103-s − 216·109-s − 32·111-s + ⋯
L(s)  = 1  − 4/3·3-s + 16/7·7-s + 2·9-s − 0.615·13-s − 0.421·19-s − 3.04·21-s − 4/5·25-s − 1.62·27-s + 3.87·31-s + 8/37·37-s + 0.820·39-s − 7.62·43-s − 0.734·49-s + 0.561·57-s + 8/61·61-s + 32/7·63-s + 2.26·67-s + 0.438·73-s + 1.06·75-s + 1.11·79-s + 2.09·81-s − 1.40·91-s − 5.16·93-s + 1.48·97-s − 1.39·103-s − 1.98·109-s − 0.288·111-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{24} \cdot 3^{8} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(13065.6\)
Root analytic conductor: \(1.80824\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{120} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{24} \cdot 3^{8} \cdot 5^{8} ,\ ( \ : [1]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.5011053538\)
\(L(\frac12)\) \(\approx\) \(0.5011053538\)
\(L(2)\) 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 \( 1 + 4 T - 2 T^{2} - 4 p^{2} T^{3} - 34 p T^{4} - 4 p^{4} T^{5} - 2 p^{4} T^{6} + 4 p^{6} T^{7} + p^{8} T^{8} \)
5 \( ( 1 + p T^{2} )^{4} \)
good7 \( ( 1 - 8 T + 114 T^{2} - 304 T^{3} + 4426 T^{4} - 304 p^{2} T^{5} + 114 p^{4} T^{6} - 8 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
11 \( 1 - 80 T^{2} - 8612 T^{4} + 1315920 T^{6} + 21445638 T^{8} + 1315920 p^{4} T^{10} - 8612 p^{8} T^{12} - 80 p^{12} T^{14} + p^{16} T^{16} \)
13 \( ( 1 + 4 T + 296 T^{2} - 1620 T^{3} + 39470 T^{4} - 1620 p^{2} T^{5} + 296 p^{4} T^{6} + 4 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
17 \( 1 - 1208 T^{2} + 736924 T^{4} - 304675080 T^{6} + 97385661510 T^{8} - 304675080 p^{4} T^{10} + 736924 p^{8} T^{12} - 1208 p^{12} T^{14} + p^{16} T^{16} \)
19 \( ( 1 + 4 T + 736 T^{2} + 908 T^{3} + 287486 T^{4} + 908 p^{2} T^{5} + 736 p^{4} T^{6} + 4 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
23 \( 1 - 1268 T^{2} + 1124824 T^{4} - 670011900 T^{6} + 391776576750 T^{8} - 670011900 p^{4} T^{10} + 1124824 p^{8} T^{12} - 1268 p^{12} T^{14} + p^{16} T^{16} \)
29 \( 1 - 4352 T^{2} + 8975164 T^{4} - 11861548800 T^{6} + 11435522898630 T^{8} - 11861548800 p^{4} T^{10} + 8975164 p^{8} T^{12} - 4352 p^{12} T^{14} + p^{16} T^{16} \)
31 \( ( 1 - 60 T + 2920 T^{2} - 101620 T^{3} + 3613902 T^{4} - 101620 p^{2} T^{5} + 2920 p^{4} T^{6} - 60 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
37 \( ( 1 - 4 T + 2248 T^{2} + 23316 T^{3} + 2375598 T^{4} + 23316 p^{2} T^{5} + 2248 p^{4} T^{6} - 4 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
41 \( 1 - 1920 T^{2} + 9436924 T^{4} - 13365676160 T^{6} + 37834324951686 T^{8} - 13365676160 p^{4} T^{10} + 9436924 p^{8} T^{12} - 1920 p^{12} T^{14} + p^{16} T^{16} \)
43 \( ( 1 + 164 T + 16878 T^{2} + 1130284 T^{3} + 57159898 T^{4} + 1130284 p^{2} T^{5} + 16878 p^{4} T^{6} + 164 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
47 \( 1 - 15060 T^{2} + 103552984 T^{4} - 425803156380 T^{6} + 1147648197554286 T^{8} - 425803156380 p^{4} T^{10} + 103552984 p^{8} T^{12} - 15060 p^{12} T^{14} + p^{16} T^{16} \)
53 \( 1 - 5496 T^{2} + 16907740 T^{4} - 66116431304 T^{6} + 230534565871494 T^{8} - 66116431304 p^{4} T^{10} + 16907740 p^{8} T^{12} - 5496 p^{12} T^{14} + p^{16} T^{16} \)
59 \( 1 - 15232 T^{2} + 125199548 T^{4} - 692185732224 T^{6} + 2791710943429190 T^{8} - 692185732224 p^{4} T^{10} + 125199548 p^{8} T^{12} - 15232 p^{12} T^{14} + p^{16} T^{16} \)
61 \( ( 1 - 4 T + 2376 T^{2} - 32108 T^{3} + 20621806 T^{4} - 32108 p^{2} T^{5} + 2376 p^{4} T^{6} - 4 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
67 \( ( 1 - 76 T + 13102 T^{2} - 787268 T^{3} + 76659290 T^{4} - 787268 p^{2} T^{5} + 13102 p^{4} T^{6} - 76 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
71 \( 1 - 24024 T^{2} + 302202556 T^{4} - 2503345638248 T^{6} + 14778735683172486 T^{8} - 2503345638248 p^{4} T^{10} + 302202556 p^{8} T^{12} - 24024 p^{12} T^{14} + p^{16} T^{16} \)
73 \( ( 1 - 16 T + 17788 T^{2} - 226416 T^{3} + 135726918 T^{4} - 226416 p^{2} T^{5} + 17788 p^{4} T^{6} - 16 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
79 \( ( 1 - 44 T + 23696 T^{2} - 832068 T^{3} + 217860926 T^{4} - 832068 p^{2} T^{5} + 23696 p^{4} T^{6} - 44 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
83 \( 1 - 13396 T^{2} + 137438936 T^{4} - 1239330480732 T^{6} + 8621185666547246 T^{8} - 1239330480732 p^{4} T^{10} + 137438936 p^{8} T^{12} - 13396 p^{12} T^{14} + p^{16} T^{16} \)
89 \( 1 - 14056 T^{2} + 62829020 T^{4} + 214903235496 T^{6} - 3856000255517626 T^{8} + 214903235496 p^{4} T^{10} + 62829020 p^{8} T^{12} - 14056 p^{12} T^{14} + p^{16} T^{16} \)
97 \( ( 1 - 72 T + 27260 T^{2} - 1496952 T^{3} + 346190214 T^{4} - 1496952 p^{2} T^{5} + 27260 p^{4} T^{6} - 72 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
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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

−6.22002224173348558138485287582, −5.98881275739973653913262717555, −5.50722181782743425294737313468, −5.25014242162157746475944556534, −5.10010757774716832717239060328, −5.08756530531755948142658050746, −5.05001165922671806196940286036, −4.96969156684280485027445195783, −4.76590345747011203658606418902, −4.45800723735361511691008941740, −4.35807175105323344634527034128, −4.30956352670802815651470781337, −3.75758543140384269282713055404, −3.67969054727501125184306429972, −3.55885438397420757987690286948, −3.25976276116511647196398552708, −3.00178638915020503420753669697, −2.55159356346984180354721268432, −2.46338764434779017300378449339, −2.02205308251388566271748071326, −1.84567762268849174823633332001, −1.48675583626068797778415143476, −1.30057589121090105447609607889, −1.06723096432953235520736128538, −0.15687545200369335686533508709, 0.15687545200369335686533508709, 1.06723096432953235520736128538, 1.30057589121090105447609607889, 1.48675583626068797778415143476, 1.84567762268849174823633332001, 2.02205308251388566271748071326, 2.46338764434779017300378449339, 2.55159356346984180354721268432, 3.00178638915020503420753669697, 3.25976276116511647196398552708, 3.55885438397420757987690286948, 3.67969054727501125184306429972, 3.75758543140384269282713055404, 4.30956352670802815651470781337, 4.35807175105323344634527034128, 4.45800723735361511691008941740, 4.76590345747011203658606418902, 4.96969156684280485027445195783, 5.05001165922671806196940286036, 5.08756530531755948142658050746, 5.10010757774716832717239060328, 5.25014242162157746475944556534, 5.50722181782743425294737313468, 5.98881275739973653913262717555, 6.22002224173348558138485287582

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

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