Properties

Degree $32$
Conductor $4.890\times 10^{55}$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 2·7-s − 46·25-s − 8·37-s − 8·43-s + 3·49-s + 28·67-s + 44·79-s + 20·109-s + 74·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 98·169-s + 173-s − 92·175-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯
L(s)  = 1  + 0.755·7-s − 9.19·25-s − 1.31·37-s − 1.21·43-s + 3/7·49-s + 3.42·67-s + 4.95·79-s + 1.91·109-s + 6.72·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 7.53·169-s + 0.0760·173-s − 6.95·175-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{48} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{48} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(2^{64} \cdot 3^{48} \cdot 7^{16}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{3024} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 2^{64} \cdot 3^{48} \cdot 7^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(9.895527493\)
\(L(\frac12)\) \(\approx\) \(9.895527493\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( ( 1 - T - 5 T^{3} - 34 T^{4} - 5 p T^{5} - p^{3} T^{7} + p^{4} T^{8} )^{2} \)
good5 \( ( 1 + 23 T^{2} + 273 T^{4} + 2194 T^{6} + 12806 T^{8} + 2194 p^{2} T^{10} + 273 p^{4} T^{12} + 23 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
11 \( ( 1 - 37 T^{2} + 801 T^{4} - 12446 T^{6} + 149966 T^{8} - 12446 p^{2} T^{10} + 801 p^{4} T^{12} - 37 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
13 \( ( 1 - 49 T^{2} + 1398 T^{4} - 28055 T^{6} + 418226 T^{8} - 28055 p^{2} T^{10} + 1398 p^{4} T^{12} - 49 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
17 \( ( 1 + 72 T^{2} + 156 p T^{4} + 65784 T^{6} + 1248326 T^{8} + 65784 p^{2} T^{10} + 156 p^{5} T^{12} + 72 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
19 \( ( 1 - 100 T^{2} + 4914 T^{4} - 156608 T^{6} + 3515195 T^{8} - 156608 p^{2} T^{10} + 4914 p^{4} T^{12} - 100 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
23 \( ( 1 - 21 T^{2} + 697 T^{4} - 11718 T^{6} + 312438 T^{8} - 11718 p^{2} T^{10} + 697 p^{4} T^{12} - 21 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
29 \( ( 1 - 172 T^{2} + 14052 T^{4} - 714740 T^{6} + 24798806 T^{8} - 714740 p^{2} T^{10} + 14052 p^{4} T^{12} - 172 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
31 \( ( 1 - 187 T^{2} + 16353 T^{4} - 883634 T^{6} + 1053338 p T^{8} - 883634 p^{2} T^{10} + 16353 p^{4} T^{12} - 187 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
37 \( ( 1 + 2 T + 52 T^{2} - 100 T^{3} + 949 T^{4} - 100 p T^{5} + 52 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{4} \)
41 \( ( 1 + 135 T^{2} + 11289 T^{4} + 646218 T^{6} + 29821310 T^{8} + 646218 p^{2} T^{10} + 11289 p^{4} T^{12} + 135 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
43 \( ( 1 + 2 T + 40 T^{2} - 118 T^{3} - 194 T^{4} - 118 p T^{5} + 40 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{4} \)
47 \( ( 1 + 20 T^{2} + 4068 T^{4} + 4972 T^{6} + 7535798 T^{8} + 4972 p^{2} T^{10} + 4068 p^{4} T^{12} + 20 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
53 \( ( 1 - 184 T^{2} + 17436 T^{4} - 1193288 T^{6} + 68587430 T^{8} - 1193288 p^{2} T^{10} + 17436 p^{4} T^{12} - 184 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
59 \( ( 1 + 148 T^{2} + 10276 T^{4} + 457612 T^{6} + 23078422 T^{8} + 457612 p^{2} T^{10} + 10276 p^{4} T^{12} + 148 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
61 \( ( 1 - 317 T^{2} + 50098 T^{4} - 5135915 T^{6} + 370298362 T^{8} - 5135915 p^{2} T^{10} + 50098 p^{4} T^{12} - 317 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
67 \( ( 1 - 7 T + 114 T^{2} - 35 T^{3} + 3554 T^{4} - 35 p T^{5} + 114 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} )^{4} \)
71 \( ( 1 - 341 T^{2} + 63193 T^{4} - 7537622 T^{6} + 634982182 T^{8} - 7537622 p^{2} T^{10} + 63193 p^{4} T^{12} - 341 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
73 \( ( 1 - 321 T^{2} + 57726 T^{4} - 6821535 T^{6} + 583944098 T^{8} - 6821535 p^{2} T^{10} + 57726 p^{4} T^{12} - 321 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
79 \( ( 1 - 11 T + 246 T^{2} - 2515 T^{3} + 26570 T^{4} - 2515 p T^{5} + 246 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} )^{4} \)
83 \( ( 1 + 132 T^{2} + 12996 T^{4} + 1459932 T^{6} + 170705174 T^{8} + 1459932 p^{2} T^{10} + 12996 p^{4} T^{12} + 132 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
89 \( ( 1 + 351 T^{2} + 57993 T^{4} + 67866 p T^{6} + 534549758 T^{8} + 67866 p^{3} T^{10} + 57993 p^{4} T^{12} + 351 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
97 \( ( 1 - 285 T^{2} + 64986 T^{4} - 9080499 T^{6} + 1059084650 T^{8} - 9080499 p^{2} T^{10} + 64986 p^{4} T^{12} - 285 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
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

−2.06276403116188586314845361161, −2.02097922892590886727079751597, −1.99649355977867010398874918688, −1.95334404243282628314889666044, −1.94693940163061381889103323673, −1.89523129240823615708830330915, −1.70202169322486778217128795937, −1.64129680208098827805156591145, −1.57822780939865174197601192919, −1.54612959265250152506306548993, −1.50858773493573608121928363034, −1.36231360291101033906958035991, −1.28458130888012680358315141416, −1.27465244600209958627632312032, −1.10380185963872050585442237895, −0.968202698864195125912582872434, −0.73562319421716886607594677260, −0.73360907680076960474242873837, −0.65258723774628215767393936853, −0.59307647633699720773697149638, −0.53463040668040283487237287676, −0.48189226895012531539946381569, −0.30408497865373321582086250110, −0.22505942969715729428366499182, −0.12325388572855217142061574842, 0.12325388572855217142061574842, 0.22505942969715729428366499182, 0.30408497865373321582086250110, 0.48189226895012531539946381569, 0.53463040668040283487237287676, 0.59307647633699720773697149638, 0.65258723774628215767393936853, 0.73360907680076960474242873837, 0.73562319421716886607594677260, 0.968202698864195125912582872434, 1.10380185963872050585442237895, 1.27465244600209958627632312032, 1.28458130888012680358315141416, 1.36231360291101033906958035991, 1.50858773493573608121928363034, 1.54612959265250152506306548993, 1.57822780939865174197601192919, 1.64129680208098827805156591145, 1.70202169322486778217128795937, 1.89523129240823615708830330915, 1.94693940163061381889103323673, 1.95334404243282628314889666044, 1.99649355977867010398874918688, 2.02097922892590886727079751597, 2.06276403116188586314845361161

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

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