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Properties

Label 16-280e8-1.1-c1e8-0-2
Degree $16$
Conductor $3.778\times 10^{19}$
Sign $1$
Analytic cond. $624.426$
Root an. cond. $1.49526$
Motivic weight $1$
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  − 2·4-s − 8·7-s + 4·8-s + 12·9-s + 2·16-s − 8·17-s − 8·23-s − 4·25-s + 16·28-s − 8·31-s − 8·32-s − 24·36-s + 24·41-s + 36·49-s − 32·56-s − 96·63-s + 8·64-s + 16·68-s + 16·71-s + 48·72-s + 32·73-s + 48·79-s + 70·81-s + 56·89-s + 16·92-s + 24·97-s + 8·100-s + ⋯
L(s)  = 1  − 4-s − 3.02·7-s + 1.41·8-s + 4·9-s + 1/2·16-s − 1.94·17-s − 1.66·23-s − 4/5·25-s + 3.02·28-s − 1.43·31-s − 1.41·32-s − 4·36-s + 3.74·41-s + 36/7·49-s − 4.27·56-s − 12.0·63-s + 64-s + 1.94·68-s + 1.89·71-s + 5.65·72-s + 3.74·73-s + 5.40·79-s + 70/9·81-s + 5.93·89-s + 1.66·92-s + 2.43·97-s + 4/5·100-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{24} \cdot 5^{8} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(624.426\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{24} \cdot 5^{8} \cdot 7^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.356143816\)
\(L(\frac12)\) \(\approx\) \(2.356143816\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + p T^{2} - p^{2} T^{3} + p T^{4} - p^{3} T^{5} + p^{3} T^{6} + p^{4} T^{8} \)
5 \( ( 1 + T^{2} )^{4} \)
7 \( ( 1 + T )^{8} \)
good3 \( ( 1 - 2 p T^{2} + 19 T^{4} - 2 p^{3} T^{6} + p^{4} T^{8} )^{2} \)
11 \( 1 - 36 T^{2} + 458 T^{4} - 624 T^{6} - 28877 T^{8} - 624 p^{2} T^{10} + 458 p^{4} T^{12} - 36 p^{6} T^{14} + p^{8} T^{16} \)
13 \( 1 - 4 p T^{2} + 842 T^{4} + 16 p T^{6} - 118637 T^{8} + 16 p^{3} T^{10} + 842 p^{4} T^{12} - 4 p^{7} T^{14} + p^{8} T^{16} \)
17 \( ( 1 + 4 T + 46 T^{2} + 152 T^{3} + 1027 T^{4} + 152 p T^{5} + 46 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
19 \( 1 - 32 T^{2} + 1580 T^{4} - 32608 T^{6} + 878854 T^{8} - 32608 p^{2} T^{10} + 1580 p^{4} T^{12} - 32 p^{6} T^{14} + p^{8} T^{16} \)
23 \( ( 1 + 4 T + 72 T^{2} + 196 T^{3} + 2198 T^{4} + 196 p T^{5} + 72 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
29 \( 1 - 76 T^{2} + 3290 T^{4} - 117680 T^{6} + 3745027 T^{8} - 117680 p^{2} T^{10} + 3290 p^{4} T^{12} - 76 p^{6} T^{14} + p^{8} T^{16} \)
31 \( ( 1 + 4 T + 80 T^{2} + 180 T^{3} + 2982 T^{4} + 180 p T^{5} + 80 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
37 \( 1 - 208 T^{2} + 21004 T^{4} - 1335984 T^{6} + 58738694 T^{8} - 1335984 p^{2} T^{10} + 21004 p^{4} T^{12} - 208 p^{6} T^{14} + p^{8} T^{16} \)
41 \( ( 1 - 12 T + 160 T^{2} - 1236 T^{3} + 10150 T^{4} - 1236 p T^{5} + 160 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
43 \( 1 - 240 T^{2} + 27148 T^{4} - 1935760 T^{6} + 97426310 T^{8} - 1935760 p^{2} T^{10} + 27148 p^{4} T^{12} - 240 p^{6} T^{14} + p^{8} T^{16} \)
47 \( ( 1 + 114 T^{2} + 64 T^{3} + 7019 T^{4} + 64 p T^{5} + 114 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
53 \( 1 - 296 T^{2} + 42172 T^{4} - 3798360 T^{6} + 238062182 T^{8} - 3798360 p^{2} T^{10} + 42172 p^{4} T^{12} - 296 p^{6} T^{14} + p^{8} T^{16} \)
59 \( 1 - 56 T^{2} + 9980 T^{4} - 452424 T^{6} + 48737382 T^{8} - 452424 p^{2} T^{10} + 9980 p^{4} T^{12} - 56 p^{6} T^{14} + p^{8} T^{16} \)
61 \( 1 - 64 T^{2} + 8428 T^{4} - 441408 T^{6} + 39497222 T^{8} - 441408 p^{2} T^{10} + 8428 p^{4} T^{12} - 64 p^{6} T^{14} + p^{8} T^{16} \)
67 \( 1 - 408 T^{2} + 78460 T^{4} - 9314920 T^{6} + 748909862 T^{8} - 9314920 p^{2} T^{10} + 78460 p^{4} T^{12} - 408 p^{6} T^{14} + p^{8} T^{16} \)
71 \( ( 1 - 8 T + 148 T^{2} - 1096 T^{3} + 14662 T^{4} - 1096 p T^{5} + 148 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
73 \( ( 1 - 16 T + 228 T^{2} - 2480 T^{3} + 24678 T^{4} - 2480 p T^{5} + 228 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
79 \( ( 1 - 24 T + 438 T^{2} - 5280 T^{3} + 54971 T^{4} - 5280 p T^{5} + 438 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
83 \( 1 - 488 T^{2} + 108668 T^{4} - 14956824 T^{6} + 1448765094 T^{8} - 14956824 p^{2} T^{10} + 108668 p^{4} T^{12} - 488 p^{6} T^{14} + p^{8} T^{16} \)
89 \( ( 1 - 28 T + 512 T^{2} - 6308 T^{3} + 66598 T^{4} - 6308 p T^{5} + 512 p^{2} T^{6} - 28 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
97 \( ( 1 - 12 T + 118 T^{2} + 72 T^{3} - 6213 T^{4} + 72 p T^{5} + 118 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
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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

−5.24350371540809625595153757818, −5.23816230905559311907710205476, −5.18726828341894769002853771930, −4.70211172533508292122972380454, −4.61688458028020883038745895956, −4.46932696750048817302174138200, −4.45698306896100415107889455091, −4.22970037971459393661806953558, −4.02274177861558375984377382676, −4.00038920947525857990052623047, −3.79776384714988445567939271618, −3.63559836197354319758346018895, −3.58167701314939812941250014243, −3.45561762209284237654994176120, −3.33386684072321117868862346840, −2.90236608869803659311701468669, −2.40258374641210483121512696510, −2.25161435901464933315574593362, −2.22411586414352674696707961894, −2.20692449009271089806317985302, −1.77739856185962153337138047029, −1.73220669511550803654225237727, −0.932880745943828545876067282295, −0.822670326486147477649656091538, −0.63019074116518646852955770486, 0.63019074116518646852955770486, 0.822670326486147477649656091538, 0.932880745943828545876067282295, 1.73220669511550803654225237727, 1.77739856185962153337138047029, 2.20692449009271089806317985302, 2.22411586414352674696707961894, 2.25161435901464933315574593362, 2.40258374641210483121512696510, 2.90236608869803659311701468669, 3.33386684072321117868862346840, 3.45561762209284237654994176120, 3.58167701314939812941250014243, 3.63559836197354319758346018895, 3.79776384714988445567939271618, 4.00038920947525857990052623047, 4.02274177861558375984377382676, 4.22970037971459393661806953558, 4.45698306896100415107889455091, 4.46932696750048817302174138200, 4.61688458028020883038745895956, 4.70211172533508292122972380454, 5.18726828341894769002853771930, 5.23816230905559311907710205476, 5.24350371540809625595153757818

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

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