Properties

Label 16-300e8-1.1-c2e8-0-1
Degree $16$
Conductor $6.561\times 10^{19}$
Sign $1$
Analytic cond. $1.99365\times 10^{7}$
Root an. cond. $2.85909$
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  − 4·2-s + 13·4-s − 24·8-s − 12·9-s − 16·13-s + 45·16-s + 48·18-s + 64·26-s + 64·29-s − 44·32-s − 156·36-s + 112·37-s − 16·41-s + 168·49-s − 208·52-s − 352·53-s − 256·58-s − 176·61-s + 29·64-s + 288·72-s + 240·73-s − 448·74-s + 90·81-s + 64·82-s + 80·89-s − 432·97-s − 672·98-s + ⋯
L(s)  = 1  − 2·2-s + 13/4·4-s − 3·8-s − 4/3·9-s − 1.23·13-s + 2.81·16-s + 8/3·18-s + 2.46·26-s + 2.20·29-s − 1.37·32-s − 4.33·36-s + 3.02·37-s − 0.390·41-s + 24/7·49-s − 4·52-s − 6.64·53-s − 4.41·58-s − 2.88·61-s + 0.453·64-s + 4·72-s + 3.28·73-s − 6.05·74-s + 10/9·81-s + 0.780·82-s + 0.898·89-s − 4.45·97-s − 6.85·98-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{16} \cdot 3^{8} \cdot 5^{16}\)
Sign: $1$
Analytic conductor: \(1.99365\times 10^{7}\)
Root analytic conductor: \(2.85909\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{300} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{16} \cdot 3^{8} \cdot 5^{16} ,\ ( \ : [1]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.130216062\)
\(L(\frac12)\) \(\approx\) \(1.130216062\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + p^{2} T + 3 T^{2} - p^{4} T^{3} - 13 p^{2} T^{4} - p^{6} T^{5} + 3 p^{4} T^{6} + p^{8} T^{7} + p^{8} T^{8} \)
3 \( ( 1 + p T^{2} )^{4} \)
5 \( 1 \)
good7 \( 1 - 24 p T^{2} + 13404 T^{4} - 690840 T^{6} + 31402310 T^{8} - 690840 p^{4} T^{10} + 13404 p^{8} T^{12} - 24 p^{13} T^{14} + p^{16} T^{16} \)
11 \( ( 1 - 276 T^{2} + 48006 T^{4} - 276 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
13 \( ( 1 + 8 T + 204 T^{2} - 1736 T^{3} - 634 T^{4} - 1736 p^{2} T^{5} + 204 p^{4} T^{6} + 8 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
17 \( ( 1 + 732 T^{2} - 3840 T^{3} + 247238 T^{4} - 3840 p^{2} T^{5} + 732 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
19 \( 1 - 1192 T^{2} + 901404 T^{4} - 482591000 T^{6} + 198221377670 T^{8} - 482591000 p^{4} T^{10} + 901404 p^{8} T^{12} - 1192 p^{12} T^{14} + p^{16} T^{16} \)
23 \( 1 - 616 T^{2} + 755676 T^{4} - 531969752 T^{6} + 267063306566 T^{8} - 531969752 p^{4} T^{10} + 755676 p^{8} T^{12} - 616 p^{12} T^{14} + p^{16} T^{16} \)
29 \( ( 1 - 32 T + 1212 T^{2} - 46048 T^{3} + 1958438 T^{4} - 46048 p^{2} T^{5} + 1212 p^{4} T^{6} - 32 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
31 \( 1 - 2280 T^{2} + 2108508 T^{4} - 623021400 T^{6} - 295925282362 T^{8} - 623021400 p^{4} T^{10} + 2108508 p^{8} T^{12} - 2280 p^{12} T^{14} + p^{16} T^{16} \)
37 \( ( 1 - 56 T + 3948 T^{2} - 174856 T^{3} + 6816518 T^{4} - 174856 p^{2} T^{5} + 3948 p^{4} T^{6} - 56 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
41 \( ( 1 + 8 T + 4924 T^{2} + 33080 T^{3} + 10990150 T^{4} + 33080 p^{2} T^{5} + 4924 p^{4} T^{6} + 8 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
43 \( 1 - 3976 T^{2} + 15992796 T^{4} - 34977997112 T^{6} + 80511749221766 T^{8} - 34977997112 p^{4} T^{10} + 15992796 p^{8} T^{12} - 3976 p^{12} T^{14} + p^{16} T^{16} \)
47 \( 1 - 9640 T^{2} + 45901788 T^{4} - 144992767640 T^{6} + 353863561499078 T^{8} - 144992767640 p^{4} T^{10} + 45901788 p^{8} T^{12} - 9640 p^{12} T^{14} + p^{16} T^{16} \)
53 \( ( 1 + 176 T + 396 p T^{2} + 1644496 T^{3} + 101651558 T^{4} + 1644496 p^{2} T^{5} + 396 p^{5} T^{6} + 176 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
59 \( 1 - 22952 T^{2} + 243302044 T^{4} - 1557392711960 T^{6} + 6588978912358150 T^{8} - 1557392711960 p^{4} T^{10} + 243302044 p^{8} T^{12} - 22952 p^{12} T^{14} + p^{16} T^{16} \)
61 \( ( 1 + 88 T + 12348 T^{2} + 708776 T^{3} + 62059430 T^{4} + 708776 p^{2} T^{5} + 12348 p^{4} T^{6} + 88 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
67 \( 1 - 19848 T^{2} + 189118044 T^{4} - 1228457910840 T^{6} + 6186195617725190 T^{8} - 1228457910840 p^{4} T^{10} + 189118044 p^{8} T^{12} - 19848 p^{12} T^{14} + p^{16} T^{16} \)
71 \( 1 - 26376 T^{2} + 353772444 T^{4} - 3049343603256 T^{6} + 18245696307579590 T^{8} - 3049343603256 p^{4} T^{10} + 353772444 p^{8} T^{12} - 26376 p^{12} T^{14} + p^{16} T^{16} \)
73 \( ( 1 - 120 T + 19740 T^{2} - 1592520 T^{3} + 158554502 T^{4} - 1592520 p^{2} T^{5} + 19740 p^{4} T^{6} - 120 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
79 \( 1 - 8040 T^{2} + 72964188 T^{4} - 471847486680 T^{6} + 4278949597529798 T^{8} - 471847486680 p^{4} T^{10} + 72964188 p^{8} T^{12} - 8040 p^{12} T^{14} + p^{16} T^{16} \)
83 \( 1 - 18184 T^{2} + 266916444 T^{4} - 2437903506104 T^{6} + 19925145362261510 T^{8} - 2437903506104 p^{4} T^{10} + 266916444 p^{8} T^{12} - 18184 p^{12} T^{14} + p^{16} T^{16} \)
89 \( ( 1 - 40 T + 11100 T^{2} - 192920 T^{3} + 121014662 T^{4} - 192920 p^{2} T^{5} + 11100 p^{4} T^{6} - 40 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
97 \( ( 1 + 216 T + 43516 T^{2} + 4942440 T^{3} + 582543750 T^{4} + 4942440 p^{2} T^{5} + 43516 p^{4} T^{6} + 216 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
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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

−4.96220299339527526628964254346, −4.93646776777897967380199884982, −4.69244066526921682013663499190, −4.48679034289566324898923396244, −4.41440114105526530880037629608, −4.40801758241639367304943323333, −4.20537693342380536323840486745, −3.95728969234595751249929480031, −3.43467498262371742651199555536, −3.34667578868649304567467528669, −3.28019375946473253548931554382, −3.22034456893542871142340455196, −3.08939236476823506431859599245, −2.76052223612923344602076858706, −2.49715562258898346522697079458, −2.37189247350972248843127245704, −2.29731079493737612065341272348, −2.21806415804789340045358297050, −1.78888308364050783113450319094, −1.57939099203242462227164767656, −1.22106835065571447854603991767, −1.15391622641662921526962140105, −0.890583935153528348516972048149, −0.49484728645730302719364501048, −0.21645978723584261123595078948, 0.21645978723584261123595078948, 0.49484728645730302719364501048, 0.890583935153528348516972048149, 1.15391622641662921526962140105, 1.22106835065571447854603991767, 1.57939099203242462227164767656, 1.78888308364050783113450319094, 2.21806415804789340045358297050, 2.29731079493737612065341272348, 2.37189247350972248843127245704, 2.49715562258898346522697079458, 2.76052223612923344602076858706, 3.08939236476823506431859599245, 3.22034456893542871142340455196, 3.28019375946473253548931554382, 3.34667578868649304567467528669, 3.43467498262371742651199555536, 3.95728969234595751249929480031, 4.20537693342380536323840486745, 4.40801758241639367304943323333, 4.41440114105526530880037629608, 4.48679034289566324898923396244, 4.69244066526921682013663499190, 4.93646776777897967380199884982, 4.96220299339527526628964254346

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

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