Properties

Label 32-1050e16-1.1-c2e16-0-1
Degree $32$
Conductor $2.183\times 10^{48}$
Sign $1$
Analytic cond. $2.01553\times 10^{23}$
Root an. cond. $5.34887$
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  + 16·4-s − 24·9-s + 96·11-s + 144·16-s − 64·29-s − 384·36-s + 1.53e3·44-s − 112·49-s + 960·64-s − 384·71-s + 608·79-s + 324·81-s − 2.30e3·99-s + 112·109-s − 1.02e3·116-s + 3.71e3·121-s + 127-s + 131-s + 137-s + 139-s − 3.45e3·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 616·169-s + ⋯
L(s)  = 1  + 4·4-s − 8/3·9-s + 8.72·11-s + 9·16-s − 2.20·29-s − 10.6·36-s + 34.9·44-s − 2.28·49-s + 15·64-s − 5.40·71-s + 7.69·79-s + 4·81-s − 23.2·99-s + 1.02·109-s − 8.82·116-s + 30.6·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s − 24·144-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 3.64·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{32} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{32} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(2^{16} \cdot 3^{16} \cdot 5^{32} \cdot 7^{16}\)
Sign: $1$
Analytic conductor: \(2.01553\times 10^{23}\)
Root analytic conductor: \(5.34887\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1050} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 2^{16} \cdot 3^{16} \cdot 5^{32} \cdot 7^{16} ,\ ( \ : [1]^{16} ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.069693365\)
\(L(\frac12)\) \(\approx\) \(3.069693365\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 - p T^{2} )^{8} \)
3 \( ( 1 + p T^{2} )^{8} \)
5 \( 1 \)
7 \( 1 + 16 p T^{2} + 3932 T^{4} + 24720 T^{6} - 24154 p^{2} T^{8} + 24720 p^{4} T^{10} + 3932 p^{8} T^{12} + 16 p^{13} T^{14} + p^{16} T^{16} \)
good11 \( ( 1 - 24 T + 512 T^{2} - 6936 T^{3} + 89778 T^{4} - 6936 p^{2} T^{5} + 512 p^{4} T^{6} - 24 p^{6} T^{7} + p^{8} T^{8} )^{4} \)
13 \( ( 1 - 308 T^{2} + 54824 T^{4} - 3243420 T^{6} - 106725490 T^{8} - 3243420 p^{4} T^{10} + 54824 p^{8} T^{12} - 308 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
17 \( ( 1 - 1316 T^{2} + 51880 p T^{4} - 398466540 T^{6} + 132906424526 T^{8} - 398466540 p^{4} T^{10} + 51880 p^{9} T^{12} - 1316 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
19 \( ( 1 - 1092 T^{2} + 576680 T^{4} - 222586380 T^{6} + 81331407246 T^{8} - 222586380 p^{4} T^{10} + 576680 p^{8} T^{12} - 1092 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
23 \( ( 1 + 1300 T^{2} + 1274456 T^{4} + 947494620 T^{6} + 528617435630 T^{8} + 947494620 p^{4} T^{10} + 1274456 p^{8} T^{12} + 1300 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
29 \( ( 1 + 16 T + 44 p T^{2} + 55408 T^{3} + 751270 T^{4} + 55408 p^{2} T^{5} + 44 p^{5} T^{6} + 16 p^{6} T^{7} + p^{8} T^{8} )^{4} \)
31 \( ( 1 - 3948 T^{2} + 5903432 T^{4} - 3942646212 T^{6} + 1976573418510 T^{8} - 3942646212 p^{4} T^{10} + 5903432 p^{8} T^{12} - 3948 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
37 \( ( 1 + 2416 T^{2} + 5835932 T^{4} + 9959371920 T^{6} + 14384225705606 T^{8} + 9959371920 p^{4} T^{10} + 5835932 p^{8} T^{12} + 2416 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
41 \( ( 1 - 12516 T^{2} + 70001768 T^{4} - 228356671020 T^{6} + 475054496880078 T^{8} - 228356671020 p^{4} T^{10} + 70001768 p^{8} T^{12} - 12516 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
43 \( ( 1 + 1912 T^{2} + 2635964 T^{4} + 3280241736 T^{6} + 4054697588678 T^{8} + 3280241736 p^{4} T^{10} + 2635964 p^{8} T^{12} + 1912 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
47 \( ( 1 - 144 p T^{2} + 28283420 T^{4} - 81813152400 T^{6} + 198252999971526 T^{8} - 81813152400 p^{4} T^{10} + 28283420 p^{8} T^{12} - 144 p^{13} T^{14} + p^{16} T^{16} )^{2} \)
53 \( ( 1 + 6868 T^{2} + 39779480 T^{4} + 162177493980 T^{6} + 178595339294 p^{2} T^{8} + 162177493980 p^{4} T^{10} + 39779480 p^{8} T^{12} + 6868 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
59 \( ( 1 - 11736 T^{2} + 79787516 T^{4} - 412369170408 T^{6} + 1633889740528710 T^{8} - 412369170408 p^{4} T^{10} + 79787516 p^{8} T^{12} - 11736 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
61 \( ( 1 - 15944 T^{2} + 96425692 T^{4} - 269693317496 T^{6} + 596407807846918 T^{8} - 269693317496 p^{4} T^{10} + 96425692 p^{8} T^{12} - 15944 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
67 \( ( 1 + 18688 T^{2} + 141351740 T^{4} + 579001624320 T^{6} + 2070725817126086 T^{8} + 579001624320 p^{4} T^{10} + 141351740 p^{8} T^{12} + 18688 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
71 \( ( 1 + 96 T + 14048 T^{2} + 646080 T^{3} + 72093138 T^{4} + 646080 p^{2} T^{5} + 14048 p^{4} T^{6} + 96 p^{6} T^{7} + p^{8} T^{8} )^{4} \)
73 \( ( 1 - 18612 T^{2} + 177429992 T^{4} - 1152928560348 T^{6} + 6336699932034510 T^{8} - 1152928560348 p^{4} T^{10} + 177429992 p^{8} T^{12} - 18612 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
79 \( ( 1 - 152 T + 28916 T^{2} - 2541000 T^{3} + 270417254 T^{4} - 2541000 p^{2} T^{5} + 28916 p^{4} T^{6} - 152 p^{6} T^{7} + p^{8} T^{8} )^{4} \)
83 \( ( 1 - 39632 T^{2} + 756012572 T^{4} - 9074582537520 T^{6} + 74694570513655814 T^{8} - 9074582537520 p^{4} T^{10} + 756012572 p^{8} T^{12} - 39632 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
89 \( ( 1 - 38508 T^{2} + 778838024 T^{4} - 10318651040964 T^{6} + 96273467949842958 T^{8} - 10318651040964 p^{4} T^{10} + 778838024 p^{8} T^{12} - 38508 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
97 \( ( 1 - 54804 T^{2} + 1439825768 T^{4} - 23691050976252 T^{6} + 266516143722806094 T^{8} - 23691050976252 p^{4} T^{10} + 1439825768 p^{8} T^{12} - 54804 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
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

−2.34622777897253617117993367122, −2.24051629699359034770510634399, −2.17278193079190356343473892131, −2.01122541492158013591872560269, −1.93802715423338939867490227815, −1.89131506436671914807147648220, −1.85579593192500491104401919015, −1.64433743770405973901230478588, −1.58605463298922192550122003364, −1.55068958920343672766935492373, −1.51728243168081129415651651456, −1.48306049619028936099992929209, −1.43983607357490877823393290149, −1.42650306753293439888644360054, −1.19985585380339539797477139174, −1.19860093363250882577272438286, −0.922608120316133728753286730212, −0.868863768381750448710895648200, −0.791681657583675627386292827360, −0.78585985437312698492956363601, −0.67656263667089504789955062196, −0.60756581594859515734202599802, −0.29634362754764275998668253146, −0.15544459701964950860229213540, −0.02436939212777101188929617006, 0.02436939212777101188929617006, 0.15544459701964950860229213540, 0.29634362754764275998668253146, 0.60756581594859515734202599802, 0.67656263667089504789955062196, 0.78585985437312698492956363601, 0.791681657583675627386292827360, 0.868863768381750448710895648200, 0.922608120316133728753286730212, 1.19860093363250882577272438286, 1.19985585380339539797477139174, 1.42650306753293439888644360054, 1.43983607357490877823393290149, 1.48306049619028936099992929209, 1.51728243168081129415651651456, 1.55068958920343672766935492373, 1.58605463298922192550122003364, 1.64433743770405973901230478588, 1.85579593192500491104401919015, 1.89131506436671914807147648220, 1.93802715423338939867490227815, 2.01122541492158013591872560269, 2.17278193079190356343473892131, 2.24051629699359034770510634399, 2.34622777897253617117993367122

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

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