Properties

Label 32-1568e16-1.1-c2e16-0-0
Degree $32$
Conductor $1.335\times 10^{51}$
Sign $1$
Analytic cond. $1.23281\times 10^{26}$
Root an. cond. $6.53642$
Motivic weight $2$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 32·9-s + 120·25-s − 16·29-s − 144·37-s − 80·53-s + 524·81-s − 224·109-s + 1.32e3·113-s − 864·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2.11e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
L(s)  = 1  + 32/9·9-s + 24/5·25-s − 0.551·29-s − 3.89·37-s − 1.50·53-s + 6.46·81-s − 2.05·109-s + 11.7·113-s − 7.14·121-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 + 12.4·169-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{80} \cdot 7^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{80} \cdot 7^{32}\right)^{s/2} \, \Gamma_{\C}(s+1)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(2^{80} \cdot 7^{32}\)
Sign: $1$
Analytic conductor: \(1.23281\times 10^{26}\)
Root analytic conductor: \(6.53642\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 2^{80} \cdot 7^{32} ,\ ( \ : [1]^{16} ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.2159265702\)
\(L(\frac12)\) \(\approx\) \(0.2159265702\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( ( 1 - 16 T^{2} + 122 T^{4} - 416 T^{6} + 331 T^{8} - 416 p^{4} T^{10} + 122 p^{8} T^{12} - 16 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
5 \( ( 1 - 12 p T^{2} + 2058 T^{4} - 10608 p T^{6} + 1281491 T^{8} - 10608 p^{5} T^{10} + 2058 p^{8} T^{12} - 12 p^{13} T^{14} + p^{16} T^{16} )^{2} \)
11 \( ( 1 + 432 T^{2} + 101770 T^{4} + 18091872 T^{6} + 2519133147 T^{8} + 18091872 p^{4} T^{10} + 101770 p^{8} T^{12} + 432 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
13 \( ( 1 - 1056 T^{2} + 510492 T^{4} - 150812640 T^{6} + 30377120006 T^{8} - 150812640 p^{4} T^{10} + 510492 p^{8} T^{12} - 1056 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
17 \( ( 1 - 1468 T^{2} + 1106730 T^{4} - 537928880 T^{6} + 183685895411 T^{8} - 537928880 p^{4} T^{10} + 1106730 p^{8} T^{12} - 1468 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
19 \( ( 1 - 1440 T^{2} + 1137690 T^{4} - 599664000 T^{6} + 244339668779 T^{8} - 599664000 p^{4} T^{10} + 1137690 p^{8} T^{12} - 1440 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
23 \( ( 1 + 1648 T^{2} + 1698202 T^{4} + 1274662432 T^{6} + 740326951915 T^{8} + 1274662432 p^{4} T^{10} + 1698202 p^{8} T^{12} + 1648 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
29 \( ( 1 + 4 T + 2608 T^{2} + 8572 T^{3} + 2977102 T^{4} + 8572 p^{2} T^{5} + 2608 p^{4} T^{6} + 4 p^{6} T^{7} + p^{8} T^{8} )^{4} \)
31 \( ( 1 - 4944 T^{2} + 12041994 T^{4} - 18914445024 T^{6} + 21218074483547 T^{8} - 18914445024 p^{4} T^{10} + 12041994 p^{8} T^{12} - 4944 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
37 \( ( 1 + 36 T + 3954 T^{2} + 161088 T^{3} + 7052651 T^{4} + 161088 p^{2} T^{5} + 3954 p^{4} T^{6} + 36 p^{6} T^{7} + p^{8} T^{8} )^{4} \)
41 \( ( 1 - 10816 T^{2} + 53826588 T^{4} - 162742362560 T^{6} + 329722113042758 T^{8} - 162742362560 p^{4} T^{10} + 53826588 p^{8} T^{12} - 10816 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
43 \( ( 1 + 5992 T^{2} + 25433596 T^{4} + 70773684952 T^{6} + 153071479203142 T^{8} + 70773684952 p^{4} T^{10} + 25433596 p^{8} T^{12} + 5992 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
47 \( ( 1 - 7200 T^{2} + 30753210 T^{4} - 100217929920 T^{6} + 252870771077579 T^{8} - 100217929920 p^{4} T^{10} + 30753210 p^{8} T^{12} - 7200 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
53 \( ( 1 + 20 T + 6498 T^{2} + 242656 T^{3} + 21504539 T^{4} + 242656 p^{2} T^{5} + 6498 p^{4} T^{6} + 20 p^{6} T^{7} + p^{8} T^{8} )^{4} \)
59 \( ( 1 - 13056 T^{2} + 103559946 T^{4} - 567820281792 T^{6} + 2269772666794523 T^{8} - 567820281792 p^{4} T^{10} + 103559946 p^{8} T^{12} - 13056 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
61 \( ( 1 - 19692 T^{2} + 191982810 T^{4} - 1196247954960 T^{6} + 5243298235324451 T^{8} - 1196247954960 p^{4} T^{10} + 191982810 p^{8} T^{12} - 19692 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
67 \( ( 1 + 19248 T^{2} + 198660442 T^{4} + 1359268530912 T^{6} + 6980138907036075 T^{8} + 1359268530912 p^{4} T^{10} + 198660442 p^{8} T^{12} + 19248 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
71 \( ( 1 + 24232 T^{2} + 314425660 T^{4} + 2642217676312 T^{6} + 15735155452974982 T^{8} + 2642217676312 p^{4} T^{10} + 314425660 p^{8} T^{12} + 24232 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
73 \( ( 1 - 20204 T^{2} + 194066938 T^{4} - 1215991652240 T^{6} + 6486410485261123 T^{8} - 1215991652240 p^{4} T^{10} + 194066938 p^{8} T^{12} - 20204 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
79 \( ( 1 + 34640 T^{2} + 596941050 T^{4} + 6494320935520 T^{6} + 48452484640984139 T^{8} + 6494320935520 p^{4} T^{10} + 596941050 p^{8} T^{12} + 34640 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
83 \( ( 1 - 35368 T^{2} + 588987420 T^{4} - 6301011254168 T^{6} + 49538235115611782 T^{8} - 6301011254168 p^{4} T^{10} + 588987420 p^{8} T^{12} - 35368 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
89 \( ( 1 - 17212 T^{2} + 211999818 T^{4} - 2253472306160 T^{6} + 21544216480380371 T^{8} - 2253472306160 p^{4} T^{10} + 211999818 p^{8} T^{12} - 17212 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
97 \( ( 1 - 39520 T^{2} + 789394140 T^{4} - 11378905321376 T^{6} + 124539045704549702 T^{8} - 11378905321376 p^{4} T^{10} + 789394140 p^{8} T^{12} - 39520 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
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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

−1.98207758431763212161783481208, −1.95398265651525923556397903206, −1.91154138258348214407654904776, −1.88271995249614921805497621952, −1.86055250732722048237017097320, −1.85946168879494904914786568069, −1.85180229487372684623286014777, −1.55281406061478563229090674004, −1.55237215918478175295515354163, −1.42847262390158066641769089624, −1.37014103510169540218374347870, −1.30280271558532416468374912397, −1.18360770517959723015955367755, −1.14637478568764462971823678654, −1.13555320950782702843285079245, −0.899054034040930692890792492328, −0.842004208848577415037848250484, −0.70958132589998675203061038192, −0.70884615625316058728867861242, −0.66950895020887725540272765594, −0.60149080567802404059670583758, −0.46048140732699692132638642269, −0.21097670369069429745310531594, −0.13680693687915118089167008499, −0.01300852075436866857674574250, 0.01300852075436866857674574250, 0.13680693687915118089167008499, 0.21097670369069429745310531594, 0.46048140732699692132638642269, 0.60149080567802404059670583758, 0.66950895020887725540272765594, 0.70884615625316058728867861242, 0.70958132589998675203061038192, 0.842004208848577415037848250484, 0.899054034040930692890792492328, 1.13555320950782702843285079245, 1.14637478568764462971823678654, 1.18360770517959723015955367755, 1.30280271558532416468374912397, 1.37014103510169540218374347870, 1.42847262390158066641769089624, 1.55237215918478175295515354163, 1.55281406061478563229090674004, 1.85180229487372684623286014777, 1.85946168879494904914786568069, 1.86055250732722048237017097320, 1.88271995249614921805497621952, 1.91154138258348214407654904776, 1.95398265651525923556397903206, 1.98207758431763212161783481208

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

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