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Properties

Label 16-1734e8-1.1-c1e8-0-5
Degree $16$
Conductor $8.173\times 10^{25}$
Sign $1$
Analytic cond. $1.35085\times 10^{9}$
Root an. cond. $3.72102$
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  + 8·2-s + 36·4-s + 120·8-s − 4·9-s + 330·16-s − 32·18-s − 16·19-s + 8·25-s + 792·32-s − 144·36-s − 128·38-s − 32·47-s + 16·49-s + 64·50-s + 32·53-s + 48·59-s + 1.71e3·64-s + 16·67-s − 480·72-s − 576·76-s + 10·81-s + 16·83-s − 32·89-s − 256·94-s + 128·98-s + 288·100-s + 256·106-s + ⋯
L(s)  = 1  + 5.65·2-s + 18·4-s + 42.4·8-s − 4/3·9-s + 82.5·16-s − 7.54·18-s − 3.67·19-s + 8/5·25-s + 140.·32-s − 24·36-s − 20.7·38-s − 4.66·47-s + 16/7·49-s + 9.05·50-s + 4.39·53-s + 6.24·59-s + 214.5·64-s + 1.95·67-s − 56.5·72-s − 66.0·76-s + 10/9·81-s + 1.75·83-s − 3.39·89-s − 26.4·94-s + 12.9·98-s + 28.7·100-s + 24.8·106-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 17^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 17^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{8} \cdot 3^{8} \cdot 17^{16}\)
Sign: $1$
Analytic conductor: \(1.35085\times 10^{9}\)
Root analytic conductor: \(3.72102\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{8} \cdot 3^{8} \cdot 17^{16} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(746.4011312\)
\(L(\frac12)\) \(\approx\) \(746.4011312\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 - T )^{8} \)
3 \( ( 1 + T^{2} )^{4} \)
17 \( 1 \)
good5 \( 1 - 8 T^{2} + 56 T^{4} - 168 T^{6} + 786 T^{8} - 168 p^{2} T^{10} + 56 p^{4} T^{12} - 8 p^{6} T^{14} + p^{8} T^{16} \)
7 \( 1 - 16 T^{2} + 100 T^{4} - 48 T^{6} - 2938 T^{8} - 48 p^{2} T^{10} + 100 p^{4} T^{12} - 16 p^{6} T^{14} + p^{8} T^{16} \)
11 \( 1 - 8 T^{2} + 188 T^{4} + 392 T^{6} + 11878 T^{8} + 392 p^{2} T^{10} + 188 p^{4} T^{12} - 8 p^{6} T^{14} + p^{8} T^{16} \)
13 \( ( 1 + 8 T^{2} - 16 T^{3} + 186 T^{4} - 16 p T^{5} + 8 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
19 \( ( 1 + 8 T + 4 p T^{2} + 424 T^{3} + 2158 T^{4} + 424 p T^{5} + 4 p^{3} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
23 \( ( 1 - 44 T^{2} + p^{2} T^{4} )^{4} \)
29 \( 1 - 88 T^{2} + 2840 T^{4} - 48184 T^{6} + 896850 T^{8} - 48184 p^{2} T^{10} + 2840 p^{4} T^{12} - 88 p^{6} T^{14} + p^{8} T^{16} \)
31 \( 1 - 192 T^{2} + 16772 T^{4} - 895552 T^{6} + 32881158 T^{8} - 895552 p^{2} T^{10} + 16772 p^{4} T^{12} - 192 p^{6} T^{14} + p^{8} T^{16} \)
37 \( 1 - 184 T^{2} + 16664 T^{4} - 992408 T^{6} + 42713746 T^{8} - 992408 p^{2} T^{10} + 16664 p^{4} T^{12} - 184 p^{6} T^{14} + p^{8} T^{16} \)
41 \( 1 - 272 T^{2} + 34208 T^{4} - 63248 p T^{6} + 129477826 T^{8} - 63248 p^{3} T^{10} + 34208 p^{4} T^{12} - 272 p^{6} T^{14} + p^{8} T^{16} \)
43 \( ( 1 + 140 T^{2} - 64 T^{3} + 8310 T^{4} - 64 p T^{5} + 140 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
47 \( ( 1 + 16 T + 244 T^{2} + 2192 T^{3} + 386 p T^{4} + 2192 p T^{5} + 244 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
53 \( ( 1 - 16 T + 200 T^{2} - 1856 T^{3} + 15898 T^{4} - 1856 p T^{5} + 200 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
59 \( ( 1 - 24 T + 396 T^{2} - 4472 T^{3} + 39374 T^{4} - 4472 p T^{5} + 396 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
61 \( 1 - 360 T^{2} + 58936 T^{4} - 5949448 T^{6} + 422974418 T^{8} - 5949448 p^{2} T^{10} + 58936 p^{4} T^{12} - 360 p^{6} T^{14} + p^{8} T^{16} \)
67 \( ( 1 - 8 T + 156 T^{2} - 1096 T^{3} + 11790 T^{4} - 1096 p T^{5} + 156 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
71 \( 1 - 48 T^{2} + 7460 T^{4} - 292496 T^{6} + 57271302 T^{8} - 292496 p^{2} T^{10} + 7460 p^{4} T^{12} - 48 p^{6} T^{14} + p^{8} T^{16} \)
73 \( 1 - 320 T^{2} + 50816 T^{4} - 5419840 T^{6} + 441882946 T^{8} - 5419840 p^{2} T^{10} + 50816 p^{4} T^{12} - 320 p^{6} T^{14} + p^{8} T^{16} \)
79 \( 1 - 336 T^{2} + 63524 T^{4} - 8079600 T^{6} + 743666374 T^{8} - 8079600 p^{2} T^{10} + 63524 p^{4} T^{12} - 336 p^{6} T^{14} + p^{8} T^{16} \)
83 \( ( 1 - 8 T + 204 T^{2} - 680 T^{3} + 17390 T^{4} - 680 p T^{5} + 204 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
89 \( ( 1 + 16 T + 240 T^{2} + 3056 T^{3} + 33474 T^{4} + 3056 p T^{5} + 240 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
97 \( 1 - 320 T^{2} + 68096 T^{4} - 9736256 T^{6} + 1095923970 T^{8} - 9736256 p^{2} T^{10} + 68096 p^{4} T^{12} - 320 p^{6} T^{14} + p^{8} T^{16} \)
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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

−3.95519537529711478784641222692, −3.92860551600092075312725104913, −3.92077933203072950270611036933, −3.49772807649712763231243289663, −3.49126895293796040959375997592, −3.41305261633828909623166103284, −3.33206815920414108776692809843, −3.12476651447945603946324642738, −3.02414521977156926856967093405, −2.88586098766089532137228993878, −2.65403205966203678491222803930, −2.62197127426860465597306986502, −2.48724687341120675248062494602, −2.43326799407112778584728972957, −2.24213738908307120788235706433, −2.15250130287601622089932351916, −1.95035018813433856486319859024, −1.92923816171683050349452143258, −1.79474663568171014402348298939, −1.61565336009846811891817883578, −1.27230581412646768647729096848, −1.03499115869827527608898813539, −0.75891349169467150985521489828, −0.53711015005721979802564207471, −0.52186634132046378823888384189, 0.52186634132046378823888384189, 0.53711015005721979802564207471, 0.75891349169467150985521489828, 1.03499115869827527608898813539, 1.27230581412646768647729096848, 1.61565336009846811891817883578, 1.79474663568171014402348298939, 1.92923816171683050349452143258, 1.95035018813433856486319859024, 2.15250130287601622089932351916, 2.24213738908307120788235706433, 2.43326799407112778584728972957, 2.48724687341120675248062494602, 2.62197127426860465597306986502, 2.65403205966203678491222803930, 2.88586098766089532137228993878, 3.02414521977156926856967093405, 3.12476651447945603946324642738, 3.33206815920414108776692809843, 3.41305261633828909623166103284, 3.49126895293796040959375997592, 3.49772807649712763231243289663, 3.92077933203072950270611036933, 3.92860551600092075312725104913, 3.95519537529711478784641222692

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

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