Properties

Label 32-42e32-1.1-c2e16-0-0
Degree $32$
Conductor $8.790\times 10^{51}$
Sign $1$
Analytic cond. $8.11592\times 10^{26}$
Root an. cond. $6.93293$
Motivic weight $2$
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  − 88·25-s − 128·37-s + 352·43-s + 256·67-s − 432·79-s + 80·109-s + 240·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2.62e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯
L(s)  = 1  − 3.51·25-s − 3.45·37-s + 8.18·43-s + 3.82·67-s − 5.46·79-s + 0.733·109-s + 1.98·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 15.5·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + 0.00448·223-s + 0.00440·227-s + ⋯

Functional equation

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

Invariants

Degree: \(32\)
Conductor: \(2^{32} \cdot 3^{32} \cdot 7^{32}\)
Sign: $1$
Analytic conductor: \(8.11592\times 10^{26}\)
Root analytic conductor: \(6.93293\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1764} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 2^{32} \cdot 3^{32} \cdot 7^{32} ,\ ( \ : [1]^{16} ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.5324491398\)
\(L(\frac12)\) \(\approx\) \(0.5324491398\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( ( 1 + 44 T^{2} + 12 p^{2} T^{4} + 16984 T^{6} + 1075871 T^{8} + 16984 p^{4} T^{10} + 12 p^{10} T^{12} + 44 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
11 \( ( 1 - 120 T^{2} + 13270 T^{4} + 3378240 T^{6} - 423912381 T^{8} + 3378240 p^{4} T^{10} + 13270 p^{8} T^{12} - 120 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
13 \( ( 1 - 656 T^{2} + 164608 T^{4} - 656 p^{4} T^{6} + p^{8} T^{8} )^{4} \)
17 \( ( 1 + 652 T^{2} + 159724 T^{4} + 64116376 T^{6} + 27945675775 T^{8} + 64116376 p^{4} T^{10} + 159724 p^{8} T^{12} + 652 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
19 \( ( 1 + 1308 T^{2} + 1022898 T^{4} + 558939792 T^{6} + 233352146771 T^{8} + 558939792 p^{4} T^{10} + 1022898 p^{8} T^{12} + 1308 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
23 \( ( 1 - 464 T^{2} - 112442 T^{4} + 107622016 T^{6} - 31147436573 T^{8} + 107622016 p^{4} T^{10} - 112442 p^{8} T^{12} - 464 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
29 \( ( 1 + 1264 T^{2} + 1437274 T^{4} + 1264 p^{4} T^{6} + p^{8} T^{8} )^{4} \)
31 \( ( 1 + 2684 T^{2} + 3669138 T^{4} + 4529722384 T^{6} + 4988428005107 T^{8} + 4529722384 p^{4} T^{10} + 3669138 p^{8} T^{12} + 2684 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
37 \( ( 1 + 32 T - 1088 T^{2} - 20032 T^{3} + 1184527 T^{4} - 20032 p^{2} T^{5} - 1088 p^{4} T^{6} + 32 p^{6} T^{7} + p^{8} T^{8} )^{4} \)
41 \( ( 1 - 1796 T^{2} + 4042324 T^{4} - 1796 p^{4} T^{6} + p^{8} T^{8} )^{4} \)
43 \( ( 1 - 44 T + 3790 T^{2} - 44 p^{2} T^{3} + p^{4} T^{4} )^{8} \)
47 \( ( 1 + 3908 T^{2} + 1704786 T^{4} + 6737392 p^{2} T^{6} + 17572019 p^{4} T^{8} + 6737392 p^{6} T^{10} + 1704786 p^{8} T^{12} + 3908 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
53 \( ( 1 - 9780 T^{2} + 56463370 T^{4} - 228891785040 T^{6} + 718495002809139 T^{8} - 228891785040 p^{4} T^{10} + 56463370 p^{8} T^{12} - 9780 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
59 \( ( 1 + 8772 T^{2} + 37475058 T^{4} + 133669525488 T^{6} + 478874949046931 T^{8} + 133669525488 p^{4} T^{10} + 37475058 p^{8} T^{12} + 8772 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
61 \( ( 1 + 4496 T^{2} - 11143392 T^{4} + 16481104096 T^{6} + 511699671973727 T^{8} + 16481104096 p^{4} T^{10} - 11143392 p^{8} T^{12} + 4496 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
67 \( ( 1 - 64 T - 2378 T^{2} + 160256 T^{3} + 3374611 T^{4} + 160256 p^{2} T^{5} - 2378 p^{4} T^{6} - 64 p^{6} T^{7} + p^{8} T^{8} )^{4} \)
71 \( ( 1 + 200 p T^{2} + 93659530 T^{4} + 200 p^{5} T^{6} + p^{8} T^{8} )^{4} \)
73 \( ( 1 - 3168 T^{2} - 22051872 T^{4} + 78276166848 T^{6} - 61825294870849 T^{8} + 78276166848 p^{4} T^{10} - 22051872 p^{8} T^{12} - 3168 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
79 \( ( 1 + 108 T + 2538 T^{2} - 362448 T^{3} - 28461229 T^{4} - 362448 p^{2} T^{5} + 2538 p^{4} T^{6} + 108 p^{6} T^{7} + p^{8} T^{8} )^{4} \)
83 \( ( 1 - 8292 T^{2} + 106271430 T^{4} - 8292 p^{4} T^{6} + p^{8} T^{8} )^{4} \)
89 \( ( 1 + 8892 T^{2} - 41191284 T^{4} - 46465448328 T^{6} + 5564743338369599 T^{8} - 46465448328 p^{4} T^{10} - 41191284 p^{8} T^{12} + 8892 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
97 \( ( 1 - 23328 T^{2} + 262163040 T^{4} - 23328 p^{4} T^{6} + p^{8} T^{8} )^{4} \)
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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

−2.18472560445751988355693787269, −1.99733289569093566713678219165, −1.95161605937087017625750571421, −1.93192048069690662580774743953, −1.90934719582070784810169448610, −1.82037390561147449407984260758, −1.68194008276719495543073429219, −1.65056562032829027250255096528, −1.52516440917848865358829483119, −1.45574512381874933247843699707, −1.29606753567595151877026186505, −1.24783259573475214453485403375, −1.15766230166576115159465012807, −1.10211440781374723122658880146, −0.842028956300217045699322320526, −0.805589451248777025815256807873, −0.75934610540822007825974619690, −0.75174366534020824038764739627, −0.72973915066622894663495877102, −0.62129957476292596866499512338, −0.57154029106921619716052275855, −0.32228857828000684684825926814, −0.26271714759700679504529998547, −0.079158476858791322608381996156, −0.04309320904890919743821691544, 0.04309320904890919743821691544, 0.079158476858791322608381996156, 0.26271714759700679504529998547, 0.32228857828000684684825926814, 0.57154029106921619716052275855, 0.62129957476292596866499512338, 0.72973915066622894663495877102, 0.75174366534020824038764739627, 0.75934610540822007825974619690, 0.805589451248777025815256807873, 0.842028956300217045699322320526, 1.10211440781374723122658880146, 1.15766230166576115159465012807, 1.24783259573475214453485403375, 1.29606753567595151877026186505, 1.45574512381874933247843699707, 1.52516440917848865358829483119, 1.65056562032829027250255096528, 1.68194008276719495543073429219, 1.82037390561147449407984260758, 1.90934719582070784810169448610, 1.93192048069690662580774743953, 1.95161605937087017625750571421, 1.99733289569093566713678219165, 2.18472560445751988355693787269

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

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