Properties

Label 32-280e16-1.1-c1e16-0-1
Degree $32$
Conductor $1.427\times 10^{39}$
Sign $1$
Analytic cond. $389908.$
Root an. cond. $1.49526$
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  + 2-s + 4-s + 16·5-s + 8-s + 16·9-s + 16·10-s − 4·11-s + 3·16-s + 16·18-s + 16·20-s − 4·22-s + 136·25-s − 16·31-s − 32-s + 16·36-s + 16·40-s − 4·43-s − 4·44-s + 256·45-s − 4·49-s + 136·50-s − 64·55-s − 8·61-s − 16·62-s − 5·64-s + 20·67-s + 16·72-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 7.15·5-s + 0.353·8-s + 16/3·9-s + 5.05·10-s − 1.20·11-s + 3/4·16-s + 3.77·18-s + 3.57·20-s − 0.852·22-s + 27.1·25-s − 2.87·31-s − 0.176·32-s + 8/3·36-s + 2.52·40-s − 0.609·43-s − 0.603·44-s + 38.1·45-s − 4/7·49-s + 19.2·50-s − 8.62·55-s − 1.02·61-s − 2.03·62-s − 5/8·64-s + 2.44·67-s + 1.88·72-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 5^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 5^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(2^{48} \cdot 5^{16} \cdot 7^{16}\)
Sign: $1$
Analytic conductor: \(389908.\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{280} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 2^{48} \cdot 5^{16} \cdot 7^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(120.5125120\)
\(L(\frac12)\) \(\approx\) \(120.5125120\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T - p T^{4} + 3 p T^{5} - 3 p^{2} T^{7} + p^{3} T^{8} - 3 p^{3} T^{9} + 3 p^{4} T^{11} - p^{5} T^{12} - p^{7} T^{15} + p^{8} T^{16} \)
5 \( ( 1 - T )^{16} \)
7 \( 1 + 4 T^{2} + 36 T^{3} + 4 T^{4} + 204 T^{5} + 100 p T^{6} + 904 T^{7} + 5814 T^{8} + 904 p T^{9} + 100 p^{3} T^{10} + 204 p^{3} T^{11} + 4 p^{4} T^{12} + 36 p^{5} T^{13} + 4 p^{6} T^{14} + p^{8} T^{16} \)
good3 \( 1 - 16 T^{2} + 134 T^{4} - 772 T^{6} + 1171 p T^{8} - 4648 p T^{10} + 51310 T^{12} - 176516 T^{14} + 185548 p T^{16} - 176516 p^{2} T^{18} + 51310 p^{4} T^{20} - 4648 p^{7} T^{22} + 1171 p^{9} T^{24} - 772 p^{10} T^{26} + 134 p^{12} T^{28} - 16 p^{14} T^{30} + p^{16} T^{32} \)
11 \( ( 1 + 2 T + 38 T^{2} + 102 T^{3} + 849 T^{4} + 2360 T^{5} + 13338 T^{6} + 36764 T^{7} + 162252 T^{8} + 36764 p T^{9} + 13338 p^{2} T^{10} + 2360 p^{3} T^{11} + 849 p^{4} T^{12} + 102 p^{5} T^{13} + 38 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
13 \( ( 1 + 42 T^{2} + 12 T^{3} + 1109 T^{4} + 552 T^{5} + 21146 T^{6} + 10156 T^{7} + 312412 T^{8} + 10156 p T^{9} + 21146 p^{2} T^{10} + 552 p^{3} T^{11} + 1109 p^{4} T^{12} + 12 p^{5} T^{13} + 42 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
17 \( 1 - 8 p T^{2} + 9398 T^{4} - 441684 T^{6} + 15867977 T^{8} - 462761176 T^{10} + 11338206366 T^{12} - 238218042700 T^{14} + 4340632328516 T^{16} - 238218042700 p^{2} T^{18} + 11338206366 p^{4} T^{20} - 462761176 p^{6} T^{22} + 15867977 p^{8} T^{24} - 441684 p^{10} T^{26} + 9398 p^{12} T^{28} - 8 p^{15} T^{30} + p^{16} T^{32} \)
19 \( 1 - 120 T^{2} + 7112 T^{4} - 14584 p T^{6} + 7791548 T^{8} - 163541048 T^{10} + 2611826040 T^{12} - 33956352872 T^{14} + 500871514950 T^{16} - 33956352872 p^{2} T^{18} + 2611826040 p^{4} T^{20} - 163541048 p^{6} T^{22} + 7791548 p^{8} T^{24} - 14584 p^{11} T^{26} + 7112 p^{12} T^{28} - 120 p^{14} T^{30} + p^{16} T^{32} \)
23 \( 1 - 208 T^{2} + 22056 T^{4} - 1571904 T^{6} + 84053084 T^{8} - 3572959584 T^{10} + 124922536472 T^{12} - 3666816166032 T^{14} + 91371160875334 T^{16} - 3666816166032 p^{2} T^{18} + 124922536472 p^{4} T^{20} - 3572959584 p^{6} T^{22} + 84053084 p^{8} T^{24} - 1571904 p^{10} T^{26} + 22056 p^{12} T^{28} - 208 p^{14} T^{30} + p^{16} T^{32} \)
29 \( 1 - 292 T^{2} + 40806 T^{4} - 3658824 T^{6} + 238823825 T^{8} - 12229044600 T^{10} + 516201061814 T^{12} - 18541519099068 T^{14} + 576457234322116 T^{16} - 18541519099068 p^{2} T^{18} + 516201061814 p^{4} T^{20} - 12229044600 p^{6} T^{22} + 238823825 p^{8} T^{24} - 3658824 p^{10} T^{26} + 40806 p^{12} T^{28} - 292 p^{14} T^{30} + p^{16} T^{32} \)
31 \( ( 1 + 8 T + 156 T^{2} + 1296 T^{3} + 13292 T^{4} + 96288 T^{5} + 741700 T^{6} + 4392984 T^{7} + 27984230 T^{8} + 4392984 p T^{9} + 741700 p^{2} T^{10} + 96288 p^{3} T^{11} + 13292 p^{4} T^{12} + 1296 p^{5} T^{13} + 156 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
37 \( 1 - 340 T^{2} + 59936 T^{4} - 7152252 T^{6} + 641015212 T^{8} - 45497691444 T^{10} + 2635851611424 T^{12} - 126850111202108 T^{14} + 5118309159851494 T^{16} - 126850111202108 p^{2} T^{18} + 2635851611424 p^{4} T^{20} - 45497691444 p^{6} T^{22} + 641015212 p^{8} T^{24} - 7152252 p^{10} T^{26} + 59936 p^{12} T^{28} - 340 p^{14} T^{30} + p^{16} T^{32} \)
41 \( 1 - 232 T^{2} + 32104 T^{4} - 3202936 T^{6} + 255876156 T^{8} - 17043323304 T^{10} + 975399522136 T^{12} - 48558130743544 T^{14} + 2123654190398598 T^{16} - 48558130743544 p^{2} T^{18} + 975399522136 p^{4} T^{20} - 17043323304 p^{6} T^{22} + 255876156 p^{8} T^{24} - 3202936 p^{10} T^{26} + 32104 p^{12} T^{28} - 232 p^{14} T^{30} + p^{16} T^{32} \)
43 \( ( 1 + 2 T + 216 T^{2} + 270 T^{3} + 22980 T^{4} + 21190 T^{5} + 1612552 T^{6} + 1259098 T^{7} + 81139350 T^{8} + 1259098 p T^{9} + 1612552 p^{2} T^{10} + 21190 p^{3} T^{11} + 22980 p^{4} T^{12} + 270 p^{5} T^{13} + 216 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
47 \( ( 1 + 210 T^{2} + 424 T^{3} + 20501 T^{4} + 71348 T^{5} + 1376534 T^{6} + 5382808 T^{7} + 72656708 T^{8} + 5382808 p T^{9} + 1376534 p^{2} T^{10} + 71348 p^{3} T^{11} + 20501 p^{4} T^{12} + 424 p^{5} T^{13} + 210 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
53 \( 1 - 444 T^{2} + 95440 T^{4} - 13625140 T^{6} + 1487417420 T^{8} - 133281679708 T^{10} + 10117851372400 T^{12} - 660669978323828 T^{14} + 37485039307099558 T^{16} - 660669978323828 p^{2} T^{18} + 10117851372400 p^{4} T^{20} - 133281679708 p^{6} T^{22} + 1487417420 p^{8} T^{24} - 13625140 p^{10} T^{26} + 95440 p^{12} T^{28} - 444 p^{14} T^{30} + p^{16} T^{32} \)
59 \( 1 - 544 T^{2} + 142872 T^{4} - 24347616 T^{6} + 3056211228 T^{8} - 304237741216 T^{10} + 25229659138856 T^{12} - 1801962119594848 T^{14} + 113058863038407174 T^{16} - 1801962119594848 p^{2} T^{18} + 25229659138856 p^{4} T^{20} - 304237741216 p^{6} T^{22} + 3056211228 p^{8} T^{24} - 24347616 p^{10} T^{26} + 142872 p^{12} T^{28} - 544 p^{14} T^{30} + p^{16} T^{32} \)
61 \( ( 1 + 4 T + 220 T^{2} + 452 T^{3} + 21276 T^{4} + 14700 T^{5} + 1481892 T^{6} + 584396 T^{7} + 94645158 T^{8} + 584396 p T^{9} + 1481892 p^{2} T^{10} + 14700 p^{3} T^{11} + 21276 p^{4} T^{12} + 452 p^{5} T^{13} + 220 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
67 \( ( 1 - 10 T + 356 T^{2} - 3486 T^{3} + 61044 T^{4} - 581590 T^{5} + 6733628 T^{6} - 59512546 T^{7} + 528259766 T^{8} - 59512546 p T^{9} + 6733628 p^{2} T^{10} - 581590 p^{3} T^{11} + 61044 p^{4} T^{12} - 3486 p^{5} T^{13} + 356 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
71 \( 1 - 644 T^{2} + 198624 T^{4} - 39221932 T^{6} + 5591888940 T^{8} - 617721349412 T^{10} + 55999222297184 T^{12} - 4424201931687980 T^{14} + 322698601587423398 T^{16} - 4424201931687980 p^{2} T^{18} + 55999222297184 p^{4} T^{20} - 617721349412 p^{6} T^{22} + 5591888940 p^{8} T^{24} - 39221932 p^{10} T^{26} + 198624 p^{12} T^{28} - 644 p^{14} T^{30} + p^{16} T^{32} \)
73 \( 1 - 556 T^{2} + 165360 T^{4} - 34160612 T^{6} + 5437568204 T^{8} - 702529893004 T^{10} + 75867773158224 T^{12} - 6964409971692516 T^{14} + 548248027302940262 T^{16} - 6964409971692516 p^{2} T^{18} + 75867773158224 p^{4} T^{20} - 702529893004 p^{6} T^{22} + 5437568204 p^{8} T^{24} - 34160612 p^{10} T^{26} + 165360 p^{12} T^{28} - 556 p^{14} T^{30} + p^{16} T^{32} \)
79 \( 1 - 672 T^{2} + 239646 T^{4} - 58660812 T^{6} + 10893265057 T^{8} - 1614943604520 T^{10} + 196775937800454 T^{12} - 20041769506403180 T^{14} + 1721626596797889012 T^{16} - 20041769506403180 p^{2} T^{18} + 196775937800454 p^{4} T^{20} - 1614943604520 p^{6} T^{22} + 10893265057 p^{8} T^{24} - 58660812 p^{10} T^{26} + 239646 p^{12} T^{28} - 672 p^{14} T^{30} + p^{16} T^{32} \)
83 \( 1 - 764 T^{2} + 280944 T^{4} - 66837700 T^{6} + 11715405100 T^{8} - 1636191670636 T^{10} + 191589412200016 T^{12} - 19378096949526164 T^{14} + 1715972865423799398 T^{16} - 19378096949526164 p^{2} T^{18} + 191589412200016 p^{4} T^{20} - 1636191670636 p^{6} T^{22} + 11715405100 p^{8} T^{24} - 66837700 p^{10} T^{26} + 280944 p^{12} T^{28} - 764 p^{14} T^{30} + p^{16} T^{32} \)
89 \( 1 - 904 T^{2} + 398408 T^{4} - 114210008 T^{6} + 23987033852 T^{8} - 3943146167496 T^{10} + 529135284297848 T^{12} - 59592961731275736 T^{14} + 5726182845912077830 T^{16} - 59592961731275736 p^{2} T^{18} + 529135284297848 p^{4} T^{20} - 3943146167496 p^{6} T^{22} + 23987033852 p^{8} T^{24} - 114210008 p^{10} T^{26} + 398408 p^{12} T^{28} - 904 p^{14} T^{30} + p^{16} T^{32} \)
97 \( 1 - 664 T^{2} + 230726 T^{4} - 56036276 T^{6} + 10668809817 T^{8} - 1685656930712 T^{10} + 227831471257294 T^{12} - 26797828856727836 T^{14} + 2769020205902604324 T^{16} - 26797828856727836 p^{2} T^{18} + 227831471257294 p^{4} T^{20} - 1685656930712 p^{6} T^{22} + 10668809817 p^{8} T^{24} - 56036276 p^{10} T^{26} + 230726 p^{12} T^{28} - 664 p^{14} T^{30} + p^{16} T^{32} \)
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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

−3.40762692700717015302216192792, −3.32848696423407492408787933545, −3.15104473447429293623280408379, −3.08738565641852910157477153139, −3.02988721233242942861211599516, −2.67434687563554975762415297729, −2.64005542417613593550426469598, −2.57211465962240082266948683814, −2.49499067021003374299123461651, −2.42693316664145583197368751989, −2.40234878092203970455099851567, −2.37699733499221864306159171328, −2.25667426603866610563988639939, −1.94429898334409791617361001881, −1.91233050535756757058563382852, −1.88084607191702109434365472577, −1.65985738756740725443342994207, −1.54886346216681532228131749993, −1.50000635327445606719313504597, −1.45598503723112908233610765494, −1.45396513664675929370458057608, −1.44054566451780328956572257013, −1.07602007705532920140135810520, −0.954027433515987197822461561712, −0.61740747125339861430811793903, 0.61740747125339861430811793903, 0.954027433515987197822461561712, 1.07602007705532920140135810520, 1.44054566451780328956572257013, 1.45396513664675929370458057608, 1.45598503723112908233610765494, 1.50000635327445606719313504597, 1.54886346216681532228131749993, 1.65985738756740725443342994207, 1.88084607191702109434365472577, 1.91233050535756757058563382852, 1.94429898334409791617361001881, 2.25667426603866610563988639939, 2.37699733499221864306159171328, 2.40234878092203970455099851567, 2.42693316664145583197368751989, 2.49499067021003374299123461651, 2.57211465962240082266948683814, 2.64005542417613593550426469598, 2.67434687563554975762415297729, 3.02988721233242942861211599516, 3.08738565641852910157477153139, 3.15104473447429293623280408379, 3.32848696423407492408787933545, 3.40762692700717015302216192792

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

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