Properties

Label 32-300e16-1.1-c2e16-0-1
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 about

Normalization:  

Dirichlet series

L(s)  = 1  + 8·4-s + 24·9-s + 52·16-s + 64·29-s + 192·36-s − 32·41-s − 408·49-s + 272·61-s + 240·64-s + 324·81-s − 256·89-s + 224·101-s + 1.04e3·109-s + 512·116-s + 528·121-s + 127-s + 131-s + 137-s + 139-s + 1.24e3·144-s + 149-s + 151-s + 157-s + 163-s − 256·164-s + 167-s + 1.00e3·169-s + ⋯
L(s)  = 1  + 2·4-s + 8/3·9-s + 13/4·16-s + 2.20·29-s + 16/3·36-s − 0.780·41-s − 8.32·49-s + 4.45·61-s + 15/4·64-s + 4·81-s − 2.87·89-s + 2.21·101-s + 9.54·109-s + 4.41·116-s + 4.36·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 26/3·144-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s − 1.56·164-s + 0.00598·167-s + 5.91·169-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

\(L(\frac{3}{2})\) \(\approx\) \(53.65795110\)
\(L(\frac12)\) \(\approx\) \(53.65795110\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - p^{3} T^{2} + 3 p^{2} T^{4} + 5 p^{4} T^{6} - 31 p^{4} T^{8} + 5 p^{8} T^{10} + 3 p^{10} T^{12} - p^{15} T^{14} + p^{16} T^{16} \)
3 \( ( 1 - p T^{2} )^{8} \)
5 \( 1 \)
good7 \( ( 1 + 204 T^{2} + 18426 T^{4} + 1076112 T^{6} + 53449475 T^{8} + 1076112 p^{4} T^{10} + 18426 p^{8} T^{12} + 204 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
11 \( ( 1 - 24 p T^{2} + 33852 T^{4} - 3268920 T^{6} + 337315526 T^{8} - 3268920 p^{4} T^{10} + 33852 p^{8} T^{12} - 24 p^{13} T^{14} + p^{16} T^{16} )^{2} \)
13 \( ( 1 - 500 T^{2} + 138714 T^{4} - 168304 p^{2} T^{6} + 4888018019 T^{8} - 168304 p^{6} T^{10} + 138714 p^{8} T^{12} - 500 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
17 \( ( 1 - 648 T^{2} + 410236 T^{4} - 149339448 T^{6} + 52652093574 T^{8} - 149339448 p^{4} T^{10} + 410236 p^{8} T^{12} - 648 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
19 \( ( 1 - 1756 T^{2} + 1510026 T^{4} - 44288912 p T^{6} + 346336758035 T^{8} - 44288912 p^{5} T^{10} + 1510026 p^{8} T^{12} - 1756 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
23 \( ( 1 + 2248 T^{2} + 2472828 T^{4} + 1852675064 T^{6} + 1086691824134 T^{8} + 1852675064 p^{4} T^{10} + 2472828 p^{8} T^{12} + 2248 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
29 \( ( 1 - 16 T + 1764 T^{2} - 10544 T^{3} + 1646102 T^{4} - 10544 p^{2} T^{5} + 1764 p^{4} T^{6} - 16 p^{6} T^{7} + p^{8} T^{8} )^{4} \)
31 \( ( 1 - 2028 T^{2} + 2805402 T^{4} - 2884712400 T^{6} + 3076391251619 T^{8} - 2884712400 p^{4} T^{10} + 2805402 p^{8} T^{12} - 2028 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
37 \( ( 1 - 1784 T^{2} + 7604700 T^{4} - 9917137480 T^{6} + 21466335785606 T^{8} - 9917137480 p^{4} T^{10} + 7604700 p^{8} T^{12} - 1784 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
41 \( ( 1 + 8 T + 1756 T^{2} + 78008 T^{3} + 3756598 T^{4} + 78008 p^{2} T^{5} + 1756 p^{4} T^{6} + 8 p^{6} T^{7} + p^{8} T^{8} )^{4} \)
43 \( ( 1 + 7900 T^{2} + 33132810 T^{4} + 96255124016 T^{6} + 206878197918227 T^{8} + 96255124016 p^{4} T^{10} + 33132810 p^{8} T^{12} + 7900 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
47 \( ( 1 + 5656 T^{2} + 20883420 T^{4} + 53920616360 T^{6} + 122334534618566 T^{8} + 53920616360 p^{4} T^{10} + 20883420 p^{8} T^{12} + 5656 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
53 \( ( 1 - 8504 T^{2} + 37649340 T^{4} - 110228557960 T^{6} + 302042397010886 T^{8} - 110228557960 p^{4} T^{10} + 37649340 p^{8} T^{12} - 8504 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
59 \( ( 1 - 21656 T^{2} + 220430236 T^{4} - 1377870886568 T^{6} + 5784129582348550 T^{8} - 1377870886568 p^{4} T^{10} + 220430236 p^{8} T^{12} - 21656 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
61 \( ( 1 - 68 T + 6786 T^{2} - 434848 T^{3} + 32554859 T^{4} - 434848 p^{2} T^{5} + 6786 p^{4} T^{6} - 68 p^{6} T^{7} + p^{8} T^{8} )^{4} \)
67 \( ( 1 + 14364 T^{2} + 117868746 T^{4} + 700523495472 T^{6} + 3348544470504275 T^{8} + 700523495472 p^{4} T^{10} + 117868746 p^{8} T^{12} + 14364 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
71 \( ( 1 - 31512 T^{2} + 469696188 T^{4} - 4285342561320 T^{6} + 26126405445829766 T^{8} - 4285342561320 p^{4} T^{10} + 469696188 p^{8} T^{12} - 31512 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
73 \( ( 1 - 11064 T^{2} + 131124508 T^{4} - 901551032712 T^{6} + 5798260918327494 T^{8} - 901551032712 p^{4} T^{10} + 131124508 p^{8} T^{12} - 11064 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
79 \( ( 1 - 35400 T^{2} + 591951516 T^{4} - 6232687244280 T^{6} + 45861936410406470 T^{8} - 6232687244280 p^{4} T^{10} + 591951516 p^{8} T^{12} - 35400 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
83 \( ( 1 + 6424 T^{2} + 64588572 T^{4} + 770632625960 T^{6} + 4303554072892550 T^{8} + 770632625960 p^{4} T^{10} + 64588572 p^{8} T^{12} + 6424 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
89 \( ( 1 + 64 T + 28356 T^{2} + 1283264 T^{3} + 322223942 T^{4} + 1283264 p^{2} T^{5} + 28356 p^{4} T^{6} + 64 p^{6} T^{7} + p^{8} T^{8} )^{4} \)
97 \( ( 1 - 41636 T^{2} + 941790058 T^{4} - 14310982927568 T^{6} + 156484383916080883 T^{8} - 14310982927568 p^{4} T^{10} + 941790058 p^{8} T^{12} - 41636 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
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

−3.12983501676273750247927513841, −2.94903150571706995377093303137, −2.88231028383489034626104474656, −2.64141594936887003610508417274, −2.53944535675987756744664578954, −2.51982794840729700792226930622, −2.49620246120529837286403489245, −2.48382794445740902920922119458, −2.22725994310634659237363073724, −2.00743250199212259775634919645, −1.92258740815355644981542689314, −1.90804292484850650856039147502, −1.81974822531947962653226259124, −1.67952909197211848373730538296, −1.61360058510688238617256953206, −1.57214417903450694522291058764, −1.38846097723471396251476739114, −1.37197105480943763882842600594, −1.25532565454947277615486933516, −0.844756334688072721468892423462, −0.73344348095053977453551050497, −0.63382006096021536620316332427, −0.60404350698900908898287631610, −0.58471224703899958163219922539, −0.16912129534301495380833936648, 0.16912129534301495380833936648, 0.58471224703899958163219922539, 0.60404350698900908898287631610, 0.63382006096021536620316332427, 0.73344348095053977453551050497, 0.844756334688072721468892423462, 1.25532565454947277615486933516, 1.37197105480943763882842600594, 1.38846097723471396251476739114, 1.57214417903450694522291058764, 1.61360058510688238617256953206, 1.67952909197211848373730538296, 1.81974822531947962653226259124, 1.90804292484850650856039147502, 1.92258740815355644981542689314, 2.00743250199212259775634919645, 2.22725994310634659237363073724, 2.48382794445740902920922119458, 2.49620246120529837286403489245, 2.51982794840729700792226930622, 2.53944535675987756744664578954, 2.64141594936887003610508417274, 2.88231028383489034626104474656, 2.94903150571706995377093303137, 3.12983501676273750247927513841

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

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