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Properties

Label 32-2070e16-1.1-c1e16-0-0
Degree $32$
Conductor $1.136\times 10^{53}$
Sign $1$
Analytic cond. $3.10426\times 10^{19}$
Root an. cond. $4.06559$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 8·4-s + 36·16-s − 32·19-s + 32·31-s + 8·49-s + 24·61-s − 120·64-s + 256·76-s − 24·79-s − 88·109-s − 24·121-s − 256·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 48·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  − 4·4-s + 9·16-s − 7.34·19-s + 5.74·31-s + 8/7·49-s + 3.07·61-s − 15·64-s + 29.3·76-s − 2.70·79-s − 8.42·109-s − 2.18·121-s − 22.9·124-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 + 3.69·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{32} \cdot 5^{16} \cdot 23^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{32} \cdot 5^{16} \cdot 23^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(2^{16} \cdot 3^{32} \cdot 5^{16} \cdot 23^{16}\)
Sign: $1$
Analytic conductor: \(3.10426\times 10^{19}\)
Root analytic conductor: \(4.06559\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 2^{16} \cdot 3^{32} \cdot 5^{16} \cdot 23^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.06746855142\)
\(L(\frac12)\) \(\approx\) \(0.06746855142\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + T^{2} )^{8} \)
3 \( 1 \)
5 \( 1 + 12 T^{4} + 144 T^{6} - 154 T^{8} + 144 p^{2} T^{10} + 12 p^{4} T^{12} + p^{8} T^{16} \)
23 \( ( 1 + T^{2} )^{8} \)
good7 \( ( 1 - 4 T^{2} + 4 T^{4} - 60 T^{6} + 2486 T^{8} - 60 p^{2} T^{10} + 4 p^{4} T^{12} - 4 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
11 \( ( 1 + 12 T^{2} + 28 p T^{4} + 2340 T^{6} + 41110 T^{8} + 2340 p^{2} T^{10} + 28 p^{5} T^{12} + 12 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
13 \( ( 1 - 24 T^{2} + 44 p T^{4} - 9960 T^{6} + 144934 T^{8} - 9960 p^{2} T^{10} + 44 p^{5} T^{12} - 24 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
17 \( ( 1 - 36 T^{2} + 1220 T^{4} - 23900 T^{6} + 494838 T^{8} - 23900 p^{2} T^{10} + 1220 p^{4} T^{12} - 36 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
19 \( ( 1 + 8 T + 80 T^{2} + 420 T^{3} + 2334 T^{4} + 420 p T^{5} + 80 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{4} \)
29 \( ( 1 + 120 T^{2} + 7484 T^{4} + 11304 p T^{6} + 10876582 T^{8} + 11304 p^{3} T^{10} + 7484 p^{4} T^{12} + 120 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
31 \( ( 1 - 8 T + 60 T^{2} - 280 T^{3} + 1670 T^{4} - 280 p T^{5} + 60 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{4} \)
37 \( ( 1 - 212 T^{2} + 21908 T^{4} - 1420508 T^{6} + 62893014 T^{8} - 1420508 p^{2} T^{10} + 21908 p^{4} T^{12} - 212 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
41 \( ( 1 + 200 T^{2} + 19420 T^{4} + 1226232 T^{6} + 57280262 T^{8} + 1226232 p^{2} T^{10} + 19420 p^{4} T^{12} + 200 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
43 \( ( 1 - 128 T^{2} + 10620 T^{4} - 634192 T^{6} + 29985158 T^{8} - 634192 p^{2} T^{10} + 10620 p^{4} T^{12} - 128 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
47 \( ( 1 - 168 T^{2} + 15548 T^{4} - 1059608 T^{6} + 56840070 T^{8} - 1059608 p^{2} T^{10} + 15548 p^{4} T^{12} - 168 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
53 \( ( 1 - 384 T^{2} + 66444 T^{4} - 6757872 T^{6} + 441611750 T^{8} - 6757872 p^{2} T^{10} + 66444 p^{4} T^{12} - 384 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
59 \( ( 1 + 344 T^{2} + 55996 T^{4} + 5688360 T^{6} + 398550758 T^{8} + 5688360 p^{2} T^{10} + 55996 p^{4} T^{12} + 344 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
61 \( ( 1 - 6 T - 48 T^{2} - 174 T^{3} + 9030 T^{4} - 174 p T^{5} - 48 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{4} \)
67 \( ( 1 - 256 T^{2} + 41852 T^{4} - 4414800 T^{6} + 349810374 T^{8} - 4414800 p^{2} T^{10} + 41852 p^{4} T^{12} - 256 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
71 \( ( 1 + 408 T^{2} + 76604 T^{4} + 8958504 T^{6} + 741024454 T^{8} + 8958504 p^{2} T^{10} + 76604 p^{4} T^{12} + 408 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
73 \( ( 1 - 344 T^{2} + 61820 T^{4} - 7402600 T^{6} + 632949958 T^{8} - 7402600 p^{2} T^{10} + 61820 p^{4} T^{12} - 344 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
79 \( ( 1 + 6 T + 224 T^{2} + 10 p T^{3} + 22686 T^{4} + 10 p^{2} T^{5} + 224 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{4} \)
83 \( ( 1 - 276 T^{2} + 46724 T^{4} - 5402876 T^{6} + 503942070 T^{8} - 5402876 p^{2} T^{10} + 46724 p^{4} T^{12} - 276 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
89 \( ( 1 + 188 T^{2} + 29908 T^{4} + 3331460 T^{6} + 317273110 T^{8} + 3331460 p^{2} T^{10} + 29908 p^{4} T^{12} + 188 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
97 \( ( 1 - 664 T^{2} + 201692 T^{4} - 36608808 T^{6} + 4341652422 T^{8} - 36608808 p^{2} T^{10} + 201692 p^{4} T^{12} - 664 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−2.26805429625517565043197811765, −2.20824943257459312686424058358, −2.10160736583688670145262108567, −2.09360834353526656453145188551, −2.09276275175339931591336517510, −1.95526639687463282041092325901, −1.81965035723145339479439550136, −1.76344046531708907635521370428, −1.66198859217601188231223712419, −1.57631487731505079599340330164, −1.32900318430169986595919816200, −1.25584721272114381216719318374, −1.24007128766631263607024937643, −1.19826604250555431885519332323, −1.09645371320016701070106366432, −1.08767646163180014639125849166, −1.00868185178319111143994973509, −0.960245080468753429415675382450, −0.879233960916650907810109387934, −0.60248585908754258151202273558, −0.35493645848826206364458258250, −0.32042010382552752930240868651, −0.25622371163545277420758028534, −0.16603478116718023143286911334, −0.06577697308603666265344261445, 0.06577697308603666265344261445, 0.16603478116718023143286911334, 0.25622371163545277420758028534, 0.32042010382552752930240868651, 0.35493645848826206364458258250, 0.60248585908754258151202273558, 0.879233960916650907810109387934, 0.960245080468753429415675382450, 1.00868185178319111143994973509, 1.08767646163180014639125849166, 1.09645371320016701070106366432, 1.19826604250555431885519332323, 1.24007128766631263607024937643, 1.25584721272114381216719318374, 1.32900318430169986595919816200, 1.57631487731505079599340330164, 1.66198859217601188231223712419, 1.76344046531708907635521370428, 1.81965035723145339479439550136, 1.95526639687463282041092325901, 2.09276275175339931591336517510, 2.09360834353526656453145188551, 2.10160736583688670145262108567, 2.20824943257459312686424058358, 2.26805429625517565043197811765

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

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