Properties

Label 32-42e32-1.1-c3e16-0-2
Degree $32$
Conductor $8.790\times 10^{51}$
Sign $1$
Analytic cond. $1.89598\times 10^{32}$
Root an. cond. $10.2019$
Motivic weight $3$
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  + 72·19-s + 394·25-s + 708·31-s + 76·37-s + 1.40e3·43-s + 1.63e3·61-s − 1.52e3·67-s + 2.70e3·73-s − 364·79-s + 4.99e3·103-s + 772·109-s − 5.84e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2.95e4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  + 0.869·19-s + 3.15·25-s + 4.10·31-s + 0.337·37-s + 4.99·43-s + 3.42·61-s − 2.78·67-s + 4.32·73-s − 0.518·79-s + 4.77·103-s + 0.678·109-s − 4.38·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 13.4·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{32} \cdot 7^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{32} \cdot 7^{32}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(2^{32} \cdot 3^{32} \cdot 7^{32}\)
Sign: $1$
Analytic conductor: \(1.89598\times 10^{32}\)
Root analytic conductor: \(10.2019\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1764} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 2^{32} \cdot 3^{32} \cdot 7^{32} ,\ ( \ : [3/2]^{16} ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(1917.067563\)
\(L(\frac12)\) \(\approx\) \(1917.067563\)
\(L(\frac{5}{2})\) 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 \( 1 - 394 T^{2} + 79383 T^{4} - 13135862 T^{6} + 1573353173 T^{8} - 72279808932 T^{10} - 11170130994038 T^{12} + 735694168015888 p T^{14} - 600595446685632534 T^{16} + 735694168015888 p^{7} T^{18} - 11170130994038 p^{12} T^{20} - 72279808932 p^{18} T^{22} + 1573353173 p^{24} T^{24} - 13135862 p^{30} T^{26} + 79383 p^{36} T^{28} - 394 p^{42} T^{30} + p^{48} T^{32} \)
11 \( 1 + 5842 T^{2} + 17711847 T^{4} + 34861793198 T^{6} + 46760130210965 T^{8} + 35906625944396532 T^{10} - 10800538135038597014 T^{12} - \)\(85\!\cdots\!92\)\( T^{14} - \)\(15\!\cdots\!02\)\( T^{16} - \)\(85\!\cdots\!92\)\( p^{6} T^{18} - 10800538135038597014 p^{12} T^{20} + 35906625944396532 p^{18} T^{22} + 46760130210965 p^{24} T^{24} + 34861793198 p^{30} T^{26} + 17711847 p^{36} T^{28} + 5842 p^{42} T^{30} + p^{48} T^{32} \)
13 \( ( 1 - 14750 T^{2} + 100165933 T^{4} - 31441942490 p T^{6} + 1095322120237240 T^{8} - 31441942490 p^{7} T^{10} + 100165933 p^{12} T^{12} - 14750 p^{18} T^{14} + p^{24} T^{16} )^{2} \)
17 \( 1 - 17908 T^{2} + 183721800 T^{4} - 1337837599208 T^{6} + 6758666240315906 T^{8} - 17525131900492504332 T^{10} - \)\(47\!\cdots\!28\)\( T^{12} + \)\(88\!\cdots\!12\)\( T^{14} - \)\(56\!\cdots\!25\)\( T^{16} + \)\(88\!\cdots\!12\)\( p^{6} T^{18} - \)\(47\!\cdots\!28\)\( p^{12} T^{20} - 17525131900492504332 p^{18} T^{22} + 6758666240315906 p^{24} T^{24} - 1337837599208 p^{30} T^{26} + 183721800 p^{36} T^{28} - 17908 p^{42} T^{30} + p^{48} T^{32} \)
19 \( ( 1 - 36 T + 12655 T^{2} - 440028 T^{3} + 69273163 T^{4} - 128221056 T^{5} - 8891738156 p T^{6} + 30396011988456 T^{7} - 3052907831237222 T^{8} + 30396011988456 p^{3} T^{9} - 8891738156 p^{7} T^{10} - 128221056 p^{9} T^{11} + 69273163 p^{12} T^{12} - 440028 p^{15} T^{13} + 12655 p^{18} T^{14} - 36 p^{21} T^{15} + p^{24} T^{16} )^{2} \)
23 \( 1 + 25876 T^{2} + 243513432 T^{4} + 1681286205992 T^{6} + 1505945652861218 T^{8} - \)\(37\!\cdots\!76\)\( T^{10} - \)\(25\!\cdots\!88\)\( T^{12} + \)\(23\!\cdots\!32\)\( p^{2} T^{14} + \)\(15\!\cdots\!91\)\( T^{16} + \)\(23\!\cdots\!32\)\( p^{8} T^{18} - \)\(25\!\cdots\!88\)\( p^{12} T^{20} - \)\(37\!\cdots\!76\)\( p^{18} T^{22} + 1505945652861218 p^{24} T^{24} + 1681286205992 p^{30} T^{26} + 243513432 p^{36} T^{28} + 25876 p^{42} T^{30} + p^{48} T^{32} \)
29 \( ( 1 - 2138 p T^{2} + 2900130409 T^{4} - 97031213045314 T^{6} + 2790721647797586244 T^{8} - 97031213045314 p^{6} T^{10} + 2900130409 p^{12} T^{12} - 2138 p^{19} T^{14} + p^{24} T^{16} )^{2} \)
31 \( ( 1 - 354 T + 75094 T^{2} - 11795988 T^{3} + 883174573 T^{4} - 3505223700 T^{5} - 50834565509222 T^{6} + 19486802838799662 T^{7} - 3853637391311667068 T^{8} + 19486802838799662 p^{3} T^{9} - 50834565509222 p^{6} T^{10} - 3505223700 p^{9} T^{11} + 883174573 p^{12} T^{12} - 11795988 p^{15} T^{13} + 75094 p^{18} T^{14} - 354 p^{21} T^{15} + p^{24} T^{16} )^{2} \)
37 \( ( 1 - 38 T - 91851 T^{2} - 20169358 T^{3} + 5124013139 T^{4} + 1492587433200 T^{5} + 200369748531112 T^{6} - 60212649379130144 T^{7} - 16861961095425650754 T^{8} - 60212649379130144 p^{3} T^{9} + 200369748531112 p^{6} T^{10} + 1492587433200 p^{9} T^{11} + 5124013139 p^{12} T^{12} - 20169358 p^{15} T^{13} - 91851 p^{18} T^{14} - 38 p^{21} T^{15} + p^{24} T^{16} )^{2} \)
41 \( ( 1 + 311308 T^{2} + 49778800408 T^{4} + 5249339160024580 T^{6} + \)\(41\!\cdots\!42\)\( T^{8} + 5249339160024580 p^{6} T^{10} + 49778800408 p^{12} T^{12} + 311308 p^{18} T^{14} + p^{24} T^{16} )^{2} \)
43 \( ( 1 - 352 T + 171577 T^{2} - 15193816 T^{3} + 8385609238 T^{4} - 15193816 p^{3} T^{5} + 171577 p^{6} T^{6} - 352 p^{9} T^{7} + p^{12} T^{8} )^{4} \)
47 \( 1 - 307672 T^{2} + 35469631884 T^{4} - 1804225864402640 T^{6} + 27431766123470856938 T^{8} + \)\(11\!\cdots\!68\)\( T^{10} - \)\(25\!\cdots\!40\)\( T^{12} + \)\(15\!\cdots\!32\)\( T^{14} - \)\(22\!\cdots\!25\)\( T^{16} + \)\(15\!\cdots\!32\)\( p^{6} T^{18} - \)\(25\!\cdots\!40\)\( p^{12} T^{20} + \)\(11\!\cdots\!68\)\( p^{18} T^{22} + 27431766123470856938 p^{24} T^{24} - 1804225864402640 p^{30} T^{26} + 35469631884 p^{36} T^{28} - 307672 p^{42} T^{30} + p^{48} T^{32} \)
53 \( 1 + 938962 T^{2} + 467576763291 T^{4} + 165402918375811934 T^{6} + \)\(46\!\cdots\!33\)\( T^{8} + \)\(10\!\cdots\!96\)\( T^{10} + \)\(22\!\cdots\!82\)\( T^{12} + \)\(39\!\cdots\!84\)\( T^{14} + \)\(62\!\cdots\!74\)\( T^{16} + \)\(39\!\cdots\!84\)\( p^{6} T^{18} + \)\(22\!\cdots\!82\)\( p^{12} T^{20} + \)\(10\!\cdots\!96\)\( p^{18} T^{22} + \)\(46\!\cdots\!33\)\( p^{24} T^{24} + 165402918375811934 p^{30} T^{26} + 467576763291 p^{36} T^{28} + 938962 p^{42} T^{30} + p^{48} T^{32} \)
59 \( 1 - 937618 T^{2} + 432339337947 T^{4} - 128957067202564334 T^{6} + \)\(29\!\cdots\!01\)\( T^{8} - \)\(61\!\cdots\!08\)\( T^{10} + \)\(14\!\cdots\!42\)\( T^{12} - \)\(10\!\cdots\!40\)\( p^{2} T^{14} + \)\(66\!\cdots\!62\)\( p^{4} T^{16} - \)\(10\!\cdots\!40\)\( p^{8} T^{18} + \)\(14\!\cdots\!42\)\( p^{12} T^{20} - \)\(61\!\cdots\!08\)\( p^{18} T^{22} + \)\(29\!\cdots\!01\)\( p^{24} T^{24} - 128957067202564334 p^{30} T^{26} + 432339337947 p^{36} T^{28} - 937618 p^{42} T^{30} + p^{48} T^{32} \)
61 \( ( 1 - 816 T + 1193926 T^{2} - 13002144 p T^{3} + 758515460050 T^{4} - 6819529429512 p T^{5} + 297926214028110592 T^{6} - \)\(13\!\cdots\!16\)\( T^{7} + \)\(81\!\cdots\!47\)\( T^{8} - \)\(13\!\cdots\!16\)\( p^{3} T^{9} + 297926214028110592 p^{6} T^{10} - 6819529429512 p^{10} T^{11} + 758515460050 p^{12} T^{12} - 13002144 p^{16} T^{13} + 1193926 p^{18} T^{14} - 816 p^{21} T^{15} + p^{24} T^{16} )^{2} \)
67 \( ( 1 + 764 T - 273663 T^{2} - 344243228 T^{3} - 11559707059 T^{4} + 21061240190208 T^{5} - 22917938128164926 T^{6} + 4683940411980034808 T^{7} + \)\(15\!\cdots\!66\)\( T^{8} + 4683940411980034808 p^{3} T^{9} - 22917938128164926 p^{6} T^{10} + 21061240190208 p^{9} T^{11} - 11559707059 p^{12} T^{12} - 344243228 p^{15} T^{13} - 273663 p^{18} T^{14} + 764 p^{21} T^{15} + p^{24} T^{16} )^{2} \)
71 \( ( 1 - 1856440 T^{2} + 1761671174068 T^{4} - 1073069689503907240 T^{6} + \)\(45\!\cdots\!66\)\( T^{8} - 1073069689503907240 p^{6} T^{10} + 1761671174068 p^{12} T^{12} - 1856440 p^{18} T^{14} + p^{24} T^{16} )^{2} \)
73 \( ( 1 - 1350 T + 1649863 T^{2} - 1407190050 T^{3} + 1214635307077 T^{4} - 919047647125260 T^{5} + 624120328609647082 T^{6} - \)\(41\!\cdots\!20\)\( T^{7} + \)\(25\!\cdots\!02\)\( T^{8} - \)\(41\!\cdots\!20\)\( p^{3} T^{9} + 624120328609647082 p^{6} T^{10} - 919047647125260 p^{9} T^{11} + 1214635307077 p^{12} T^{12} - 1407190050 p^{15} T^{13} + 1649863 p^{18} T^{14} - 1350 p^{21} T^{15} + p^{24} T^{16} )^{2} \)
79 \( ( 1 + 182 T - 1303458 T^{2} + 51410212 T^{3} + 935956948277 T^{4} - 127720931027844 T^{5} - 463014917975063918 T^{6} + 34428841086426043166 T^{7} + \)\(20\!\cdots\!88\)\( T^{8} + 34428841086426043166 p^{3} T^{9} - 463014917975063918 p^{6} T^{10} - 127720931027844 p^{9} T^{11} + 935956948277 p^{12} T^{12} + 51410212 p^{15} T^{13} - 1303458 p^{18} T^{14} + 182 p^{21} T^{15} + p^{24} T^{16} )^{2} \)
83 \( ( 1 + 2984326 T^{2} + 4434375721357 T^{4} + 4272111895669322434 T^{6} + \)\(28\!\cdots\!68\)\( T^{8} + 4272111895669322434 p^{6} T^{10} + 4434375721357 p^{12} T^{12} + 2984326 p^{18} T^{14} + p^{24} T^{16} )^{2} \)
89 \( 1 - 2563264 T^{2} + 3275474914692 T^{4} - 2650752142645956224 T^{6} + \)\(12\!\cdots\!98\)\( T^{8} + \)\(65\!\cdots\!92\)\( T^{10} - \)\(73\!\cdots\!16\)\( T^{12} + \)\(87\!\cdots\!44\)\( T^{14} - \)\(70\!\cdots\!05\)\( T^{16} + \)\(87\!\cdots\!44\)\( p^{6} T^{18} - \)\(73\!\cdots\!16\)\( p^{12} T^{20} + \)\(65\!\cdots\!92\)\( p^{18} T^{22} + \)\(12\!\cdots\!98\)\( p^{24} T^{24} - 2650752142645956224 p^{30} T^{26} + 3275474914692 p^{36} T^{28} - 2563264 p^{42} T^{30} + p^{48} T^{32} \)
97 \( ( 1 - 2165930 T^{2} + 2100306931633 T^{4} - 316975242262618658 T^{6} - \)\(53\!\cdots\!72\)\( T^{8} - 316975242262618658 p^{6} T^{10} + 2100306931633 p^{12} T^{12} - 2165930 p^{18} T^{14} + p^{24} T^{16} )^{2} \)
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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

−1.84402629606082249532457925397, −1.81883608636843528159775702868, −1.81407780342717247008407190750, −1.72221519867921360528708385765, −1.66907998270467635003976233638, −1.66448126406684737381528481633, −1.58242251634078591076344351054, −1.43468066596024235559428212192, −1.41918850168401296130606183826, −1.11208551543194347190842860167, −1.03612885797259258776381543436, −0.999201223581665633324522601299, −0.909371453132971143041610003519, −0.872895992626310288291299726034, −0.857974027136214195056905800382, −0.70679213864043055150697903442, −0.61586817050689507837838573214, −0.61586191392768813900852756434, −0.61039354121528913961778899173, −0.56930978941114828214716432564, −0.54201512326477246297748584260, −0.53719540107026925174478833319, −0.44104839799206860300686691549, −0.21099719541794123411754625681, −0.20315737136938805627001332181, 0.20315737136938805627001332181, 0.21099719541794123411754625681, 0.44104839799206860300686691549, 0.53719540107026925174478833319, 0.54201512326477246297748584260, 0.56930978941114828214716432564, 0.61039354121528913961778899173, 0.61586191392768813900852756434, 0.61586817050689507837838573214, 0.70679213864043055150697903442, 0.857974027136214195056905800382, 0.872895992626310288291299726034, 0.909371453132971143041610003519, 0.999201223581665633324522601299, 1.03612885797259258776381543436, 1.11208551543194347190842860167, 1.41918850168401296130606183826, 1.43468066596024235559428212192, 1.58242251634078591076344351054, 1.66448126406684737381528481633, 1.66907998270467635003976233638, 1.72221519867921360528708385765, 1.81407780342717247008407190750, 1.81883608636843528159775702868, 1.84402629606082249532457925397

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

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