Properties

Label 32-275e16-1.1-c1e16-0-1
Degree $32$
Conductor $1.070\times 10^{39}$
Sign $1$
Analytic cond. $292251.$
Root an. cond. $1.48185$
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  − 2·4-s − 9-s + 6·11-s + 16-s − 30·19-s + 18·29-s − 20·31-s + 2·36-s + 16·41-s − 12·44-s − 15·49-s + 54·59-s + 12·61-s + 12·64-s − 40·71-s + 60·76-s − 74·79-s − 3·81-s + 32·89-s − 6·99-s − 2·101-s + 92·109-s − 36·116-s − 9·121-s + 40·124-s + 127-s + 131-s + ⋯
L(s)  = 1  − 4-s − 1/3·9-s + 1.80·11-s + 1/4·16-s − 6.88·19-s + 3.34·29-s − 3.59·31-s + 1/3·36-s + 2.49·41-s − 1.80·44-s − 2.14·49-s + 7.03·59-s + 1.53·61-s + 3/2·64-s − 4.74·71-s + 6.88·76-s − 8.32·79-s − 1/3·81-s + 3.39·89-s − 0.603·99-s − 0.199·101-s + 8.81·109-s − 3.34·116-s − 0.818·121-s + 3.59·124-s + 0.0887·127-s + 0.0873·131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{32} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{32} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(5^{32} \cdot 11^{16}\)
Sign: $1$
Analytic conductor: \(292251.\)
Root analytic conductor: \(1.48185\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 5^{32} \cdot 11^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.3674619277\)
\(L(\frac12)\) \(\approx\) \(0.3674619277\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( ( 1 - 3 T + 18 T^{2} + 9 T^{3} + 75 T^{4} + 9 p T^{5} + 18 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
good2 \( 1 + p T^{2} + 3 T^{4} - p^{3} T^{6} + T^{8} + 5 p^{3} T^{10} + 107 T^{12} - 35 p T^{14} - 463 T^{16} - 35 p^{3} T^{18} + 107 p^{4} T^{20} + 5 p^{9} T^{22} + p^{8} T^{24} - p^{13} T^{26} + 3 p^{12} T^{28} + p^{15} T^{30} + p^{16} T^{32} \)
3 \( 1 + T^{2} + 4 T^{4} - 14 T^{6} + 2 p T^{8} - 127 T^{10} - 38 p T^{12} - 812 T^{14} + 7 T^{16} - 812 p^{2} T^{18} - 38 p^{5} T^{20} - 127 p^{6} T^{22} + 2 p^{9} T^{24} - 14 p^{10} T^{26} + 4 p^{12} T^{28} + p^{14} T^{30} + p^{16} T^{32} \)
7 \( 1 + 15 T^{2} + 17 T^{4} - 1585 T^{6} - 12737 T^{8} + 20590 T^{10} + 667104 T^{12} + 930150 T^{14} - 18281245 T^{16} + 930150 p^{2} T^{18} + 667104 p^{4} T^{20} + 20590 p^{6} T^{22} - 12737 p^{8} T^{24} - 1585 p^{10} T^{26} + 17 p^{12} T^{28} + 15 p^{14} T^{30} + p^{16} T^{32} \)
13 \( 1 + 14 T^{2} + 7 p T^{4} - 2555 T^{6} - 28537 T^{8} - 114505 T^{10} + 2873459 T^{12} + 54468034 T^{14} + 687202213 T^{16} + 54468034 p^{2} T^{18} + 2873459 p^{4} T^{20} - 114505 p^{6} T^{22} - 28537 p^{8} T^{24} - 2555 p^{10} T^{26} + 7 p^{13} T^{28} + 14 p^{14} T^{30} + p^{16} T^{32} \)
17 \( 1 + 45 T^{2} + 1014 T^{4} + 9165 T^{6} + 2750 T^{8} - 757080 T^{10} + 24885636 T^{12} + 903283470 T^{14} + 21029277439 T^{16} + 903283470 p^{2} T^{18} + 24885636 p^{4} T^{20} - 757080 p^{6} T^{22} + 2750 p^{8} T^{24} + 9165 p^{10} T^{26} + 1014 p^{12} T^{28} + 45 p^{14} T^{30} + p^{16} T^{32} \)
19 \( ( 1 + 15 T + 97 T^{2} + 360 T^{3} + 673 T^{4} - 610 T^{5} - 686 T^{6} + 56575 T^{7} + 375305 T^{8} + 56575 p T^{9} - 686 p^{2} T^{10} - 610 p^{3} T^{11} + 673 p^{4} T^{12} + 360 p^{5} T^{13} + 97 p^{6} T^{14} + 15 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
23 \( ( 1 - 167 T^{2} + 12560 T^{4} - 554917 T^{6} + 15741919 T^{8} - 554917 p^{2} T^{10} + 12560 p^{4} T^{12} - 167 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
29 \( ( 1 - 9 T + 41 T^{2} - 450 T^{3} + 4023 T^{4} - 21060 T^{5} + 127504 T^{6} - 821169 T^{7} + 4563203 T^{8} - 821169 p T^{9} + 127504 p^{2} T^{10} - 21060 p^{3} T^{11} + 4023 p^{4} T^{12} - 450 p^{5} T^{13} + 41 p^{6} T^{14} - 9 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
31 \( ( 1 + 10 T + 63 T^{2} + 95 T^{3} - 1617 T^{4} - 16015 T^{5} - 48789 T^{6} + 115000 T^{7} + 1966355 T^{8} + 115000 p T^{9} - 48789 p^{2} T^{10} - 16015 p^{3} T^{11} - 1617 p^{4} T^{12} + 95 p^{5} T^{13} + 63 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
37 \( 1 + 6 T^{2} + 1941 T^{4} + 103250 T^{6} + 2185908 T^{8} + 237211460 T^{10} + 7708333819 T^{12} + 205953843036 T^{14} + 15787782567103 T^{16} + 205953843036 p^{2} T^{18} + 7708333819 p^{4} T^{20} + 237211460 p^{6} T^{22} + 2185908 p^{8} T^{24} + 103250 p^{10} T^{26} + 1941 p^{12} T^{28} + 6 p^{14} T^{30} + p^{16} T^{32} \)
41 \( ( 1 - 8 T + 11 T^{2} - 52 T^{3} + 2396 T^{4} - 11404 T^{5} - 17571 T^{6} - 488156 T^{7} + 6637407 T^{8} - 488156 p T^{9} - 17571 p^{2} T^{10} - 11404 p^{3} T^{11} + 2396 p^{4} T^{12} - 52 p^{5} T^{13} + 11 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
43 \( ( 1 - 225 T^{2} + 25151 T^{4} - 1821400 T^{6} + 92617071 T^{8} - 1821400 p^{2} T^{10} + 25151 p^{4} T^{12} - 225 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
47 \( 1 + 142 T^{2} + 12515 T^{4} + 824025 T^{6} + 43551035 T^{8} + 2283582491 T^{10} + 127668320367 T^{12} + 7127565880410 T^{14} + 365773054528725 T^{16} + 7127565880410 p^{2} T^{18} + 127668320367 p^{4} T^{20} + 2283582491 p^{6} T^{22} + 43551035 p^{8} T^{24} + 824025 p^{10} T^{26} + 12515 p^{12} T^{28} + 142 p^{14} T^{30} + p^{16} T^{32} \)
53 \( 1 + 211 T^{2} + 21457 T^{4} + 1648647 T^{6} + 107779487 T^{8} + 6382656310 T^{10} + 407907578108 T^{12} + 25684823282314 T^{14} + 1429524896577715 T^{16} + 25684823282314 p^{2} T^{18} + 407907578108 p^{4} T^{20} + 6382656310 p^{6} T^{22} + 107779487 p^{8} T^{24} + 1648647 p^{10} T^{26} + 21457 p^{12} T^{28} + 211 p^{14} T^{30} + p^{16} T^{32} \)
59 \( ( 1 - 27 T + 267 T^{2} - 1053 T^{3} + 1707 T^{4} - 10170 T^{5} - 256612 T^{6} + 7292916 T^{7} - 76795335 T^{8} + 7292916 p T^{9} - 256612 p^{2} T^{10} - 10170 p^{3} T^{11} + 1707 p^{4} T^{12} - 1053 p^{5} T^{13} + 267 p^{6} T^{14} - 27 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
61 \( ( 1 - 6 T - 112 T^{2} + 1194 T^{3} + 2297 T^{4} - 63510 T^{5} + 270712 T^{6} + 1244412 T^{7} - 19660775 T^{8} + 1244412 p T^{9} + 270712 p^{2} T^{10} - 63510 p^{3} T^{11} + 2297 p^{4} T^{12} + 1194 p^{5} T^{13} - 112 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
67 \( ( 1 - 219 T^{2} + 29116 T^{4} - 2559805 T^{6} + 189680615 T^{8} - 2559805 p^{2} T^{10} + 29116 p^{4} T^{12} - 219 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
71 \( ( 1 + 20 T + p T^{2} - 450 T^{3} + 5350 T^{4} + 64050 T^{5} - 426451 T^{6} - 1694530 T^{7} + 35725759 T^{8} - 1694530 p T^{9} - 426451 p^{2} T^{10} + 64050 p^{3} T^{11} + 5350 p^{4} T^{12} - 450 p^{5} T^{13} + p^{7} T^{14} + 20 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
73 \( 1 + 349 T^{2} + 56186 T^{4} + 5178845 T^{6} + 281523998 T^{8} + 3892182240 T^{10} - 1324918940276 T^{12} - 219342744316626 T^{14} - 20139724419895937 T^{16} - 219342744316626 p^{2} T^{18} - 1324918940276 p^{4} T^{20} + 3892182240 p^{6} T^{22} + 281523998 p^{8} T^{24} + 5178845 p^{10} T^{26} + 56186 p^{12} T^{28} + 349 p^{14} T^{30} + p^{16} T^{32} \)
79 \( ( 1 + 37 T + 540 T^{2} + 3230 T^{3} - 14490 T^{4} - 396919 T^{5} - 1141518 T^{6} + 39718360 T^{7} + 582114095 T^{8} + 39718360 p T^{9} - 1141518 p^{2} T^{10} - 396919 p^{3} T^{11} - 14490 p^{4} T^{12} + 3230 p^{5} T^{13} + 540 p^{6} T^{14} + 37 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
83 \( 1 + 75 T^{2} + 10017 T^{4} + 1033995 T^{6} + 61414443 T^{8} - 4035522150 T^{10} - 277479475076 T^{12} - 74674877002950 T^{14} - 8228577548004045 T^{16} - 74674877002950 p^{2} T^{18} - 277479475076 p^{4} T^{20} - 4035522150 p^{6} T^{22} + 61414443 p^{8} T^{24} + 1033995 p^{10} T^{26} + 10017 p^{12} T^{28} + 75 p^{14} T^{30} + p^{16} T^{32} \)
89 \( ( 1 - 8 T + 254 T^{2} - 1664 T^{3} + 31231 T^{4} - 1664 p T^{5} + 254 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{4} \)
97 \( 1 - 534 T^{2} + 169761 T^{4} - 39345690 T^{6} + 7310468948 T^{8} - 1138531545060 T^{10} + 152541579319959 T^{12} - 17869715058596004 T^{14} + 1843745056436243143 T^{16} - 17869715058596004 p^{2} T^{18} + 152541579319959 p^{4} T^{20} - 1138531545060 p^{6} T^{22} + 7310468948 p^{8} T^{24} - 39345690 p^{10} T^{26} + 169761 p^{12} T^{28} - 534 p^{14} T^{30} + p^{16} T^{32} \)
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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

−3.39015195240389787076391756768, −3.37174242496646949265376430468, −3.31941564210228842123207691069, −3.20369092542583127844922053462, −3.13660324950560815314500896310, −2.71242486204635727632737602712, −2.65994058571815533326523097717, −2.61644220540105824637604222727, −2.60968400459934197277580531358, −2.59658801490981471964841031404, −2.36721587893701577865455391756, −2.35893837691115844547276659556, −2.34710766505392959803749112037, −2.14326458290675894342319222717, −2.07652702735642821483165631245, −1.86136518274029525010742001933, −1.62070469238405596816837411239, −1.56910256551485570150278275458, −1.56066121785546956610081794654, −1.42133109199588956277848707120, −1.18542951725274955585563570637, −1.05817599088596355671798679895, −0.66044677574741426923860480336, −0.48841012474344612065538495176, −0.13835601180493194171804971886, 0.13835601180493194171804971886, 0.48841012474344612065538495176, 0.66044677574741426923860480336, 1.05817599088596355671798679895, 1.18542951725274955585563570637, 1.42133109199588956277848707120, 1.56066121785546956610081794654, 1.56910256551485570150278275458, 1.62070469238405596816837411239, 1.86136518274029525010742001933, 2.07652702735642821483165631245, 2.14326458290675894342319222717, 2.34710766505392959803749112037, 2.35893837691115844547276659556, 2.36721587893701577865455391756, 2.59658801490981471964841031404, 2.60968400459934197277580531358, 2.61644220540105824637604222727, 2.65994058571815533326523097717, 2.71242486204635727632737602712, 3.13660324950560815314500896310, 3.20369092542583127844922053462, 3.31941564210228842123207691069, 3.37174242496646949265376430468, 3.39015195240389787076391756768

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

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