Properties

Label 32-300e16-1.1-c2e16-0-0
Degree $32$
Conductor $4.305\times 10^{39}$
Sign $1$
Analytic cond. $3.97466\times 10^{14}$
Root an. cond. $2.85909$
Motivic weight $2$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 10·4-s + 24·9-s + 67·16-s − 128·29-s − 240·36-s − 32·41-s − 336·49-s − 352·61-s − 300·64-s + 324·81-s − 160·89-s − 448·101-s − 736·109-s + 1.28e3·116-s + 1.10e3·121-s + 127-s + 131-s + 137-s + 139-s + 1.60e3·144-s + 149-s + 151-s + 157-s + 163-s + 320·164-s + 167-s + 688·169-s + ⋯
L(s)  = 1  − 5/2·4-s + 8/3·9-s + 4.18·16-s − 4.41·29-s − 6.66·36-s − 0.780·41-s − 6.85·49-s − 5.77·61-s − 4.68·64-s + 4·81-s − 1.79·89-s − 4.43·101-s − 6.75·109-s + 11.0·116-s + 9.12·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 67/6·144-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 1.95·164-s + 0.00598·167-s + 4.07·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16} \cdot 5^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16} \cdot 5^{32}\right)^{s/2} \, \Gamma_{\C}(s+1)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(2^{32} \cdot 3^{16} \cdot 5^{32}\)
Sign: $1$
Analytic conductor: \(3.97466\times 10^{14}\)
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: \((32,\ 2^{32} \cdot 3^{16} \cdot 5^{32} ,\ ( \ : [1]^{16} ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.007978605\)
\(L(\frac12)\) \(\approx\) \(1.007978605\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 5 p T^{2} + 33 T^{4} - 5 p^{3} T^{6} - 37 p^{4} T^{8} - 5 p^{7} T^{10} + 33 p^{8} T^{12} + 5 p^{13} T^{14} + p^{16} T^{16} \)
3 \( ( 1 - p T^{2} )^{8} \)
5 \( 1 \)
good7 \( ( 1 + 24 p T^{2} + 13404 T^{4} + 690840 T^{6} + 31402310 T^{8} + 690840 p^{4} T^{10} + 13404 p^{8} T^{12} + 24 p^{13} T^{14} + p^{16} T^{16} )^{2} \)
11 \( ( 1 - 276 T^{2} + 48006 T^{4} - 276 p^{4} T^{6} + p^{8} T^{8} )^{4} \)
13 \( ( 1 - 344 T^{2} + 68124 T^{4} - 13074664 T^{6} + 2372591750 T^{8} - 13074664 p^{4} T^{10} + 68124 p^{8} T^{12} - 344 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
17 \( ( 1 - 1464 T^{2} + 1030300 T^{4} - 469485576 T^{6} + 156060299334 T^{8} - 469485576 p^{4} T^{10} + 1030300 p^{8} T^{12} - 1464 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
19 \( ( 1 - 1192 T^{2} + 901404 T^{4} - 482591000 T^{6} + 198221377670 T^{8} - 482591000 p^{4} T^{10} + 901404 p^{8} T^{12} - 1192 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
23 \( ( 1 + 616 T^{2} + 755676 T^{4} + 531969752 T^{6} + 267063306566 T^{8} + 531969752 p^{4} T^{10} + 755676 p^{8} T^{12} + 616 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
29 \( ( 1 + 32 T + 1212 T^{2} + 46048 T^{3} + 1958438 T^{4} + 46048 p^{2} T^{5} + 1212 p^{4} T^{6} + 32 p^{6} T^{7} + p^{8} T^{8} )^{4} \)
31 \( ( 1 - 2280 T^{2} + 2108508 T^{4} - 623021400 T^{6} - 295925282362 T^{8} - 623021400 p^{4} T^{10} + 2108508 p^{8} T^{12} - 2280 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
37 \( ( 1 - 4760 T^{2} + 9635868 T^{4} - 11236659880 T^{6} + 12108314450438 T^{8} - 11236659880 p^{4} T^{10} + 9635868 p^{8} T^{12} - 4760 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
41 \( ( 1 + 8 T + 4924 T^{2} + 33080 T^{3} + 10990150 T^{4} + 33080 p^{2} T^{5} + 4924 p^{4} T^{6} + 8 p^{6} T^{7} + p^{8} T^{8} )^{4} \)
43 \( ( 1 + 3976 T^{2} + 15992796 T^{4} + 34977997112 T^{6} + 80511749221766 T^{8} + 34977997112 p^{4} T^{10} + 15992796 p^{8} T^{12} + 3976 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
47 \( ( 1 + 9640 T^{2} + 45901788 T^{4} + 144992767640 T^{6} + 353863561499078 T^{8} + 144992767640 p^{4} T^{10} + 45901788 p^{8} T^{12} + 9640 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
53 \( ( 1 - 11000 T^{2} + 64936668 T^{4} - 267744514120 T^{6} + 842750699414918 T^{8} - 267744514120 p^{4} T^{10} + 64936668 p^{8} T^{12} - 11000 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
59 \( ( 1 - 22952 T^{2} + 243302044 T^{4} - 1557392711960 T^{6} + 6588978912358150 T^{8} - 1557392711960 p^{4} T^{10} + 243302044 p^{8} T^{12} - 22952 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
61 \( ( 1 + 88 T + 12348 T^{2} + 708776 T^{3} + 62059430 T^{4} + 708776 p^{2} T^{5} + 12348 p^{4} T^{6} + 88 p^{6} T^{7} + p^{8} T^{8} )^{4} \)
67 \( ( 1 + 19848 T^{2} + 189118044 T^{4} + 1228457910840 T^{6} + 6186195617725190 T^{8} + 1228457910840 p^{4} T^{10} + 189118044 p^{8} T^{12} + 19848 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
71 \( ( 1 - 26376 T^{2} + 353772444 T^{4} - 3049343603256 T^{6} + 18245696307579590 T^{8} - 3049343603256 p^{4} T^{10} + 353772444 p^{8} T^{12} - 26376 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
73 \( ( 1 - 25080 T^{2} + 324571804 T^{4} - 2808004964040 T^{6} + 17495806969152966 T^{8} - 2808004964040 p^{4} T^{10} + 324571804 p^{8} T^{12} - 25080 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
79 \( ( 1 - 8040 T^{2} + 72964188 T^{4} - 471847486680 T^{6} + 4278949597529798 T^{8} - 471847486680 p^{4} T^{10} + 72964188 p^{8} T^{12} - 8040 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
83 \( ( 1 + 18184 T^{2} + 266916444 T^{4} + 2437903506104 T^{6} + 19925145362261510 T^{8} + 2437903506104 p^{4} T^{10} + 266916444 p^{8} T^{12} + 18184 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
89 \( ( 1 + 40 T + 11100 T^{2} + 192920 T^{3} + 121014662 T^{4} + 192920 p^{2} T^{5} + 11100 p^{4} T^{6} + 40 p^{6} T^{7} + p^{8} T^{8} )^{4} \)
97 \( ( 1 - 40376 T^{2} + 923595676 T^{4} - 13887638321672 T^{6} + 152910672794232646 T^{8} - 13887638321672 p^{4} T^{10} + 923595676 p^{8} T^{12} - 40376 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−2.99493642534246269967281113579, −2.98946791643939482727476248225, −2.95855493521653830754323066245, −2.93408774777415631668212880250, −2.76858669461678677633533233087, −2.60491794142069928976227159027, −2.46768337788018379203128083792, −2.22892526299122094064403736362, −2.22160864579171495833824578182, −1.98704329632274732566181784947, −1.89159837186016472968384981138, −1.72679125243687387077968248699, −1.67941239512989344135799976278, −1.67340402599160487299100163761, −1.62688537757731947559648723965, −1.55916413286521470525823595412, −1.47369844182490217849685811165, −1.31360433120439707739987049445, −1.05065975376765205110974217358, −0.911219626928576758199734300568, −0.797377112131599402338939739939, −0.49912882759719528029302456955, −0.27391662427501872479508343098, −0.25770002253266983709640297545, −0.15229773930998085731103407902, 0.15229773930998085731103407902, 0.25770002253266983709640297545, 0.27391662427501872479508343098, 0.49912882759719528029302456955, 0.797377112131599402338939739939, 0.911219626928576758199734300568, 1.05065975376765205110974217358, 1.31360433120439707739987049445, 1.47369844182490217849685811165, 1.55916413286521470525823595412, 1.62688537757731947559648723965, 1.67340402599160487299100163761, 1.67941239512989344135799976278, 1.72679125243687387077968248699, 1.89159837186016472968384981138, 1.98704329632274732566181784947, 2.22160864579171495833824578182, 2.22892526299122094064403736362, 2.46768337788018379203128083792, 2.60491794142069928976227159027, 2.76858669461678677633533233087, 2.93408774777415631668212880250, 2.95855493521653830754323066245, 2.98946791643939482727476248225, 2.99493642534246269967281113579

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

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