Properties

Label 32-690e16-1.1-c1e16-0-0
Degree $32$
Conductor $2.640\times 10^{45}$
Sign $1$
Analytic cond. $7.21139\times 10^{11}$
Root an. cond. $2.34727$
Motivic weight $1$
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  − 8·4-s + 16·5-s + 9-s + 12·11-s + 36·16-s − 128·20-s − 4·23-s + 136·25-s + 8·27-s + 4·31-s − 8·36-s − 96·44-s + 16·45-s + 54·49-s + 8·53-s + 192·55-s − 120·64-s − 16·73-s + 576·80-s + 6·81-s + 40·83-s − 80·89-s + 32·92-s + 12·99-s − 1.08e3·100-s − 64·108-s − 32·113-s + ⋯
L(s)  = 1  − 4·4-s + 7.15·5-s + 1/3·9-s + 3.61·11-s + 9·16-s − 28.6·20-s − 0.834·23-s + 27.1·25-s + 1.53·27-s + 0.718·31-s − 4/3·36-s − 14.4·44-s + 2.38·45-s + 54/7·49-s + 1.09·53-s + 25.8·55-s − 15·64-s − 1.87·73-s + 64.3·80-s + 2/3·81-s + 4.39·83-s − 8.47·89-s + 3.33·92-s + 1.20·99-s − 108.·100-s − 6.15·108-s − 3.01·113-s + ⋯

Functional equation

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

Invariants

Degree: \(32\)
Conductor: \(2^{16} \cdot 3^{16} \cdot 5^{16} \cdot 23^{16}\)
Sign: $1$
Analytic conductor: \(7.21139\times 10^{11}\)
Root analytic conductor: \(2.34727\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 2^{16} \cdot 3^{16} \cdot 5^{16} \cdot 23^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.624614874\)
\(L(\frac12)\) \(\approx\) \(1.624614874\)
\(L(\frac{3}{2})\) 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 \( 1 - T^{2} - 8 T^{3} - 5 T^{4} + 4 p T^{5} + 17 T^{6} + 20 T^{7} - 56 T^{8} + 20 p T^{9} + 17 p^{2} T^{10} + 4 p^{4} T^{11} - 5 p^{4} T^{12} - 8 p^{5} T^{13} - p^{6} T^{14} + p^{8} T^{16} \)
5 \( ( 1 - T )^{16} \)
23 \( 1 + 4 T + 28 T^{2} + 4 T^{3} + 468 T^{4} + 1620 T^{5} + 16036 T^{6} - 26604 T^{7} - 81770 T^{8} - 26604 p T^{9} + 16036 p^{2} T^{10} + 1620 p^{3} T^{11} + 468 p^{4} T^{12} + 4 p^{5} T^{13} + 28 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \)
good7 \( 1 - 54 T^{2} + 1443 T^{4} - 3660 p T^{6} + 341787 T^{8} - 3680164 T^{10} + 33654325 T^{12} - 271748034 T^{14} + 1988748728 T^{16} - 271748034 p^{2} T^{18} + 33654325 p^{4} T^{20} - 3680164 p^{6} T^{22} + 341787 p^{8} T^{24} - 3660 p^{11} T^{26} + 1443 p^{12} T^{28} - 54 p^{14} T^{30} + p^{16} T^{32} \)
11 \( ( 1 - 6 T + 59 T^{2} - 260 T^{3} + 1567 T^{4} - 5840 T^{5} + 27553 T^{6} - 90402 T^{7} + 354796 T^{8} - 90402 p T^{9} + 27553 p^{2} T^{10} - 5840 p^{3} T^{11} + 1567 p^{4} T^{12} - 260 p^{5} T^{13} + 59 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
13 \( ( 1 + 25 T^{2} + 84 T^{3} + 633 T^{4} + 1028 T^{5} + 1065 p T^{6} + 26488 T^{7} + 155560 T^{8} + 26488 p T^{9} + 1065 p^{3} T^{10} + 1028 p^{3} T^{11} + 633 p^{4} T^{12} + 84 p^{5} T^{13} + 25 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
17 \( ( 1 + 45 T^{2} + 84 T^{3} + 1417 T^{4} + 3796 T^{5} + 31817 T^{6} + 95912 T^{7} + 628472 T^{8} + 95912 p T^{9} + 31817 p^{2} T^{10} + 3796 p^{3} T^{11} + 1417 p^{4} T^{12} + 84 p^{5} T^{13} + 45 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
19 \( 1 - 182 T^{2} + 16543 T^{4} - 995316 T^{6} + 44417439 T^{8} - 1563546664 T^{10} + 45082837165 T^{12} - 1090352697502 T^{14} + 22419124770792 T^{16} - 1090352697502 p^{2} T^{18} + 45082837165 p^{4} T^{20} - 1563546664 p^{6} T^{22} + 44417439 p^{8} T^{24} - 995316 p^{10} T^{26} + 16543 p^{12} T^{28} - 182 p^{14} T^{30} + p^{16} T^{32} \)
29 \( 1 - 252 T^{2} + 31652 T^{4} - 2635620 T^{6} + 163970996 T^{8} - 8155970316 T^{10} + 338634762524 T^{12} - 12059925888596 T^{14} + 373937267010262 T^{16} - 12059925888596 p^{2} T^{18} + 338634762524 p^{4} T^{20} - 8155970316 p^{6} T^{22} + 163970996 p^{8} T^{24} - 2635620 p^{10} T^{26} + 31652 p^{12} T^{28} - 252 p^{14} T^{30} + p^{16} T^{32} \)
31 \( ( 1 - 2 T + 87 T^{2} - 268 T^{3} + 3447 T^{4} - 13952 T^{5} + 96389 T^{6} - 414902 T^{7} + 2777880 T^{8} - 414902 p T^{9} + 96389 p^{2} T^{10} - 13952 p^{3} T^{11} + 3447 p^{4} T^{12} - 268 p^{5} T^{13} + 87 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
37 \( 1 - 324 T^{2} + 49844 T^{4} - 4751548 T^{6} + 306732372 T^{8} - 13682395604 T^{10} + 407712129932 T^{12} - 7490332746220 T^{14} + 132047410754582 T^{16} - 7490332746220 p^{2} T^{18} + 407712129932 p^{4} T^{20} - 13682395604 p^{6} T^{22} + 306732372 p^{8} T^{24} - 4751548 p^{10} T^{26} + 49844 p^{12} T^{28} - 324 p^{14} T^{30} + p^{16} T^{32} \)
41 \( 1 - 326 T^{2} + 54463 T^{4} - 6272404 T^{6} + 557053871 T^{8} - 40208669592 T^{10} + 2428724881757 T^{12} - 124909303313838 T^{14} + 5518425180737896 T^{16} - 124909303313838 p^{2} T^{18} + 2428724881757 p^{4} T^{20} - 40208669592 p^{6} T^{22} + 557053871 p^{8} T^{24} - 6272404 p^{10} T^{26} + 54463 p^{12} T^{28} - 326 p^{14} T^{30} + p^{16} T^{32} \)
43 \( 1 - 164 T^{2} + 18260 T^{4} - 1621980 T^{6} + 116941972 T^{8} - 7441797044 T^{10} + 415840226284 T^{12} - 20767843103372 T^{14} + 943845726667926 T^{16} - 20767843103372 p^{2} T^{18} + 415840226284 p^{4} T^{20} - 7441797044 p^{6} T^{22} + 116941972 p^{8} T^{24} - 1621980 p^{10} T^{26} + 18260 p^{12} T^{28} - 164 p^{14} T^{30} + p^{16} T^{32} \)
47 \( 1 - 392 T^{2} + 77656 T^{4} - 10400152 T^{6} + 22547812 p T^{8} - 87327299848 T^{10} + 6019029917288 T^{12} - 353621660643800 T^{14} + 17884944569496262 T^{16} - 353621660643800 p^{2} T^{18} + 6019029917288 p^{4} T^{20} - 87327299848 p^{6} T^{22} + 22547812 p^{9} T^{24} - 10400152 p^{10} T^{26} + 77656 p^{12} T^{28} - 392 p^{14} T^{30} + p^{16} T^{32} \)
53 \( ( 1 - 4 T + 224 T^{2} - 412 T^{3} + 22476 T^{4} + 14396 T^{5} + 1364000 T^{6} + 4181892 T^{7} + 69985990 T^{8} + 4181892 p T^{9} + 1364000 p^{2} T^{10} + 14396 p^{3} T^{11} + 22476 p^{4} T^{12} - 412 p^{5} T^{13} + 224 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
59 \( 1 - 608 T^{2} + 183128 T^{4} - 36356256 T^{6} + 5331002780 T^{8} - 612493377248 T^{10} + 57051853530600 T^{12} - 4395723361812512 T^{14} + 283159177125890310 T^{16} - 4395723361812512 p^{2} T^{18} + 57051853530600 p^{4} T^{20} - 612493377248 p^{6} T^{22} + 5331002780 p^{8} T^{24} - 36356256 p^{10} T^{26} + 183128 p^{12} T^{28} - 608 p^{14} T^{30} + p^{16} T^{32} \)
61 \( 1 - 322 T^{2} + 52979 T^{4} - 6397460 T^{6} + 653981107 T^{8} - 57930732628 T^{10} + 4514012542949 T^{12} - 318685089498582 T^{14} + 20468089760249352 T^{16} - 318685089498582 p^{2} T^{18} + 4514012542949 p^{4} T^{20} - 57930732628 p^{6} T^{22} + 653981107 p^{8} T^{24} - 6397460 p^{10} T^{26} + 52979 p^{12} T^{28} - 322 p^{14} T^{30} + p^{16} T^{32} \)
67 \( 1 - 588 T^{2} + 174196 T^{4} - 34597556 T^{6} + 5174985044 T^{8} - 619817996988 T^{10} + 61605612719180 T^{12} - 5189985489882628 T^{14} + 374861172443592598 T^{16} - 5189985489882628 p^{2} T^{18} + 61605612719180 p^{4} T^{20} - 619817996988 p^{6} T^{22} + 5174985044 p^{8} T^{24} - 34597556 p^{10} T^{26} + 174196 p^{12} T^{28} - 588 p^{14} T^{30} + p^{16} T^{32} \)
71 \( 1 - 522 T^{2} + 151603 T^{4} - 31090692 T^{6} + 4925561555 T^{8} - 632334679292 T^{10} + 67524594224213 T^{12} - 6091399824778054 T^{14} + 468058090950196456 T^{16} - 6091399824778054 p^{2} T^{18} + 67524594224213 p^{4} T^{20} - 632334679292 p^{6} T^{22} + 4925561555 p^{8} T^{24} - 31090692 p^{10} T^{26} + 151603 p^{12} T^{28} - 522 p^{14} T^{30} + p^{16} T^{32} \)
73 \( ( 1 + 8 T + 452 T^{2} + 2856 T^{3} + 93060 T^{4} + 472216 T^{5} + 11712252 T^{6} + 48943416 T^{7} + 1013081398 T^{8} + 48943416 p T^{9} + 11712252 p^{2} T^{10} + 472216 p^{3} T^{11} + 93060 p^{4} T^{12} + 2856 p^{5} T^{13} + 452 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
79 \( 1 - 756 T^{2} + 289604 T^{4} - 74013772 T^{6} + 14085566068 T^{8} - 2114668783076 T^{10} + 259096529587644 T^{12} - 26440183327775196 T^{14} + 2272179968891021846 T^{16} - 26440183327775196 p^{2} T^{18} + 259096529587644 p^{4} T^{20} - 2114668783076 p^{6} T^{22} + 14085566068 p^{8} T^{24} - 74013772 p^{10} T^{26} + 289604 p^{12} T^{28} - 756 p^{14} T^{30} + p^{16} T^{32} \)
83 \( ( 1 - 20 T + 398 T^{2} - 4724 T^{3} + 63576 T^{4} - 648020 T^{5} + 7357986 T^{6} - 65080276 T^{7} + 654158446 T^{8} - 65080276 p T^{9} + 7357986 p^{2} T^{10} - 648020 p^{3} T^{11} + 63576 p^{4} T^{12} - 4724 p^{5} T^{13} + 398 p^{6} T^{14} - 20 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
89 \( ( 1 + 40 T + 952 T^{2} + 14520 T^{3} + 173872 T^{4} + 1766024 T^{5} + 19130568 T^{6} + 207813272 T^{7} + 2146288414 T^{8} + 207813272 p T^{9} + 19130568 p^{2} T^{10} + 1766024 p^{3} T^{11} + 173872 p^{4} T^{12} + 14520 p^{5} T^{13} + 952 p^{6} T^{14} + 40 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
97 \( 1 - 814 T^{2} + 351339 T^{4} - 103868644 T^{6} + 23306738683 T^{8} - 4182848625620 T^{10} + 618947409793933 T^{12} - 76866067224491178 T^{14} + 8088105369841495608 T^{16} - 76866067224491178 p^{2} T^{18} + 618947409793933 p^{4} T^{20} - 4182848625620 p^{6} T^{22} + 23306738683 p^{8} T^{24} - 103868644 p^{10} T^{26} + 351339 p^{12} T^{28} - 814 p^{14} T^{30} + p^{16} T^{32} \)
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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

−2.72929031795263146701171374069, −2.72928049917321481118621344933, −2.44469537870079846632513977789, −2.43802200237516426368025860865, −2.34043145335211852335077527609, −2.32720071861534730433796703249, −2.27580669699279282432018471717, −2.27117947128194734398869164652, −2.25172949288264614956751817579, −2.06670943907084468210127962646, −1.87747651163302666763170906581, −1.83350173762269505275834147506, −1.73771263389873205168025543536, −1.37041968194304937593246222976, −1.32236868441983736874280895923, −1.30487855337306275813699494218, −1.24011701488949410314518026161, −1.21207209704899208871012570858, −1.20111529164242070913323474352, −1.16128850104757334437537448283, −1.07696770045772550266224497628, −0.912370895745096603589070550176, −0.834920238449842380930053079015, −0.30689417752803153949179237453, −0.05243670130419116951299273653, 0.05243670130419116951299273653, 0.30689417752803153949179237453, 0.834920238449842380930053079015, 0.912370895745096603589070550176, 1.07696770045772550266224497628, 1.16128850104757334437537448283, 1.20111529164242070913323474352, 1.21207209704899208871012570858, 1.24011701488949410314518026161, 1.30487855337306275813699494218, 1.32236868441983736874280895923, 1.37041968194304937593246222976, 1.73771263389873205168025543536, 1.83350173762269505275834147506, 1.87747651163302666763170906581, 2.06670943907084468210127962646, 2.25172949288264614956751817579, 2.27117947128194734398869164652, 2.27580669699279282432018471717, 2.32720071861534730433796703249, 2.34043145335211852335077527609, 2.43802200237516426368025860865, 2.44469537870079846632513977789, 2.72928049917321481118621344933, 2.72929031795263146701171374069

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

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