Properties

Label 20-6032e10-1.1-c1e10-0-2
Degree $20$
Conductor $6.377\times 10^{37}$
Sign $1$
Analytic cond. $6.72023\times 10^{16}$
Root an. cond. $6.94015$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $10$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s + 4·5-s + 3·7-s − 4·9-s − 14·11-s − 10·13-s − 12·15-s + 5·17-s − 11·19-s − 9·21-s − 7·23-s − 12·25-s + 17·27-s − 10·29-s − 5·31-s + 42·33-s + 12·35-s + 8·37-s + 30·39-s + 14·41-s − 35·43-s − 16·45-s − 34·49-s − 15·51-s − 11·53-s − 56·55-s + 33·57-s + ⋯
L(s)  = 1  − 1.73·3-s + 1.78·5-s + 1.13·7-s − 4/3·9-s − 4.22·11-s − 2.77·13-s − 3.09·15-s + 1.21·17-s − 2.52·19-s − 1.96·21-s − 1.45·23-s − 2.39·25-s + 3.27·27-s − 1.85·29-s − 0.898·31-s + 7.31·33-s + 2.02·35-s + 1.31·37-s + 4.80·39-s + 2.18·41-s − 5.33·43-s − 2.38·45-s − 4.85·49-s − 2.10·51-s − 1.51·53-s − 7.55·55-s + 4.37·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 13^{10} \cdot 29^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 13^{10} \cdot 29^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(20\)
Conductor: \(2^{40} \cdot 13^{10} \cdot 29^{10}\)
Sign: $1$
Analytic conductor: \(6.72023\times 10^{16}\)
Root analytic conductor: \(6.94015\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(10\)
Selberg data: \((20,\ 2^{40} \cdot 13^{10} \cdot 29^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( ( 1 + T )^{10} \)
29 \( ( 1 + T )^{10} \)
good3 \( 1 + p T + 13 T^{2} + 34 T^{3} + 101 T^{4} + 220 T^{5} + 545 T^{6} + 1082 T^{7} + 760 p T^{8} + 1355 p T^{9} + 7696 T^{10} + 1355 p^{2} T^{11} + 760 p^{3} T^{12} + 1082 p^{3} T^{13} + 545 p^{4} T^{14} + 220 p^{5} T^{15} + 101 p^{6} T^{16} + 34 p^{7} T^{17} + 13 p^{8} T^{18} + p^{10} T^{19} + p^{10} T^{20} \)
5 \( 1 - 4 T + 28 T^{2} - 92 T^{3} + 419 T^{4} - 1177 T^{5} + 4181 T^{6} - 10331 T^{7} + 30971 T^{8} - 67556 T^{9} + 175536 T^{10} - 67556 p T^{11} + 30971 p^{2} T^{12} - 10331 p^{3} T^{13} + 4181 p^{4} T^{14} - 1177 p^{5} T^{15} + 419 p^{6} T^{16} - 92 p^{7} T^{17} + 28 p^{8} T^{18} - 4 p^{9} T^{19} + p^{10} T^{20} \)
7 \( 1 - 3 T + 43 T^{2} - 18 p T^{3} + 947 T^{4} - 2559 T^{5} + 13784 T^{6} - 33570 T^{7} + 20777 p T^{8} - 315844 T^{9} + 1161702 T^{10} - 315844 p T^{11} + 20777 p^{3} T^{12} - 33570 p^{3} T^{13} + 13784 p^{4} T^{14} - 2559 p^{5} T^{15} + 947 p^{6} T^{16} - 18 p^{8} T^{17} + 43 p^{8} T^{18} - 3 p^{9} T^{19} + p^{10} T^{20} \)
11 \( 1 + 14 T + 134 T^{2} + 904 T^{3} + 467 p T^{4} + 24445 T^{5} + 105024 T^{6} + 403390 T^{7} + 133684 p T^{8} + 5041255 T^{9} + 17059432 T^{10} + 5041255 p T^{11} + 133684 p^{3} T^{12} + 403390 p^{3} T^{13} + 105024 p^{4} T^{14} + 24445 p^{5} T^{15} + 467 p^{7} T^{16} + 904 p^{7} T^{17} + 134 p^{8} T^{18} + 14 p^{9} T^{19} + p^{10} T^{20} \)
17 \( 1 - 5 T + 112 T^{2} - 498 T^{3} + 6300 T^{4} - 25338 T^{5} + 231843 T^{6} - 834518 T^{7} + 359135 p T^{8} - 19432937 T^{9} + 119690082 T^{10} - 19432937 p T^{11} + 359135 p^{3} T^{12} - 834518 p^{3} T^{13} + 231843 p^{4} T^{14} - 25338 p^{5} T^{15} + 6300 p^{6} T^{16} - 498 p^{7} T^{17} + 112 p^{8} T^{18} - 5 p^{9} T^{19} + p^{10} T^{20} \)
19 \( 1 + 11 T + 170 T^{2} + 1241 T^{3} + 10798 T^{4} + 56565 T^{5} + 353190 T^{6} + 1371510 T^{7} + 7152375 T^{8} + 22753959 T^{9} + 125558756 T^{10} + 22753959 p T^{11} + 7152375 p^{2} T^{12} + 1371510 p^{3} T^{13} + 353190 p^{4} T^{14} + 56565 p^{5} T^{15} + 10798 p^{6} T^{16} + 1241 p^{7} T^{17} + 170 p^{8} T^{18} + 11 p^{9} T^{19} + p^{10} T^{20} \)
23 \( 1 + 7 T + 157 T^{2} + 928 T^{3} + 11663 T^{4} + 60280 T^{5} + 551638 T^{6} + 2539770 T^{7} + 18788248 T^{8} + 77346219 T^{9} + 489411626 T^{10} + 77346219 p T^{11} + 18788248 p^{2} T^{12} + 2539770 p^{3} T^{13} + 551638 p^{4} T^{14} + 60280 p^{5} T^{15} + 11663 p^{6} T^{16} + 928 p^{7} T^{17} + 157 p^{8} T^{18} + 7 p^{9} T^{19} + p^{10} T^{20} \)
31 \( 1 + 5 T + 181 T^{2} + 898 T^{3} + 17999 T^{4} + 82396 T^{5} + 1184032 T^{6} + 4929330 T^{7} + 56450384 T^{8} + 208915939 T^{9} + 2011211414 T^{10} + 208915939 p T^{11} + 56450384 p^{2} T^{12} + 4929330 p^{3} T^{13} + 1184032 p^{4} T^{14} + 82396 p^{5} T^{15} + 17999 p^{6} T^{16} + 898 p^{7} T^{17} + 181 p^{8} T^{18} + 5 p^{9} T^{19} + p^{10} T^{20} \)
37 \( 1 - 8 T + 167 T^{2} - 1017 T^{3} + 11943 T^{4} - 44941 T^{5} + 357677 T^{6} + 319009 T^{7} - 2718752 T^{8} + 110489181 T^{9} - 470218808 T^{10} + 110489181 p T^{11} - 2718752 p^{2} T^{12} + 319009 p^{3} T^{13} + 357677 p^{4} T^{14} - 44941 p^{5} T^{15} + 11943 p^{6} T^{16} - 1017 p^{7} T^{17} + 167 p^{8} T^{18} - 8 p^{9} T^{19} + p^{10} T^{20} \)
41 \( 1 - 14 T + 255 T^{2} - 2020 T^{3} + 23887 T^{4} - 157284 T^{5} + 1740012 T^{6} - 10727179 T^{7} + 100025972 T^{8} - 520487267 T^{9} + 4347974930 T^{10} - 520487267 p T^{11} + 100025972 p^{2} T^{12} - 10727179 p^{3} T^{13} + 1740012 p^{4} T^{14} - 157284 p^{5} T^{15} + 23887 p^{6} T^{16} - 2020 p^{7} T^{17} + 255 p^{8} T^{18} - 14 p^{9} T^{19} + p^{10} T^{20} \)
43 \( 1 + 35 T + 891 T^{2} + 16210 T^{3} + 246169 T^{4} + 3113896 T^{5} + 34475007 T^{6} + 333387848 T^{7} + 2881378156 T^{8} + 22163752175 T^{9} + 153655853008 T^{10} + 22163752175 p T^{11} + 2881378156 p^{2} T^{12} + 333387848 p^{3} T^{13} + 34475007 p^{4} T^{14} + 3113896 p^{5} T^{15} + 246169 p^{6} T^{16} + 16210 p^{7} T^{17} + 891 p^{8} T^{18} + 35 p^{9} T^{19} + p^{10} T^{20} \)
47 \( 1 + 198 T^{2} + 306 T^{3} + 20920 T^{4} + 65701 T^{5} + 1551573 T^{6} + 7040897 T^{7} + 91003583 T^{8} + 482886134 T^{9} + 4555687258 T^{10} + 482886134 p T^{11} + 91003583 p^{2} T^{12} + 7040897 p^{3} T^{13} + 1551573 p^{4} T^{14} + 65701 p^{5} T^{15} + 20920 p^{6} T^{16} + 306 p^{7} T^{17} + 198 p^{8} T^{18} + p^{10} T^{20} \)
53 \( 1 + 11 T + 412 T^{2} + 66 p T^{3} + 75168 T^{4} + 522646 T^{5} + 8493208 T^{6} + 50592888 T^{7} + 684662055 T^{8} + 3567810117 T^{9} + 41575412808 T^{10} + 3567810117 p T^{11} + 684662055 p^{2} T^{12} + 50592888 p^{3} T^{13} + 8493208 p^{4} T^{14} + 522646 p^{5} T^{15} + 75168 p^{6} T^{16} + 66 p^{8} T^{17} + 412 p^{8} T^{18} + 11 p^{9} T^{19} + p^{10} T^{20} \)
59 \( 1 + 23 T + 568 T^{2} + 8622 T^{3} + 128106 T^{4} + 1484434 T^{5} + 16728074 T^{6} + 159548494 T^{7} + 1494192265 T^{8} + 12288692963 T^{9} + 100141178180 T^{10} + 12288692963 p T^{11} + 1494192265 p^{2} T^{12} + 159548494 p^{3} T^{13} + 16728074 p^{4} T^{14} + 1484434 p^{5} T^{15} + 128106 p^{6} T^{16} + 8622 p^{7} T^{17} + 568 p^{8} T^{18} + 23 p^{9} T^{19} + p^{10} T^{20} \)
61 \( 1 + 8 T + 417 T^{2} + 2698 T^{3} + 84049 T^{4} + 458650 T^{5} + 10933960 T^{6} + 51438837 T^{7} + 1020347998 T^{8} + 4180156915 T^{9} + 71406285838 T^{10} + 4180156915 p T^{11} + 1020347998 p^{2} T^{12} + 51438837 p^{3} T^{13} + 10933960 p^{4} T^{14} + 458650 p^{5} T^{15} + 84049 p^{6} T^{16} + 2698 p^{7} T^{17} + 417 p^{8} T^{18} + 8 p^{9} T^{19} + p^{10} T^{20} \)
67 \( 1 + 27 T + 427 T^{2} + 4542 T^{3} + 43209 T^{4} + 408414 T^{5} + 4210168 T^{6} + 40846430 T^{7} + 376162806 T^{8} + 3172940923 T^{9} + 26530748026 T^{10} + 3172940923 p T^{11} + 376162806 p^{2} T^{12} + 40846430 p^{3} T^{13} + 4210168 p^{4} T^{14} + 408414 p^{5} T^{15} + 43209 p^{6} T^{16} + 4542 p^{7} T^{17} + 427 p^{8} T^{18} + 27 p^{9} T^{19} + p^{10} T^{20} \)
71 \( 1 + 3 T + 339 T^{2} + 2438 T^{3} + 55875 T^{4} + 591768 T^{5} + 7071361 T^{6} + 74858184 T^{7} + 773733582 T^{8} + 6441590187 T^{9} + 64967003220 T^{10} + 6441590187 p T^{11} + 773733582 p^{2} T^{12} + 74858184 p^{3} T^{13} + 7071361 p^{4} T^{14} + 591768 p^{5} T^{15} + 55875 p^{6} T^{16} + 2438 p^{7} T^{17} + 339 p^{8} T^{18} + 3 p^{9} T^{19} + p^{10} T^{20} \)
73 \( 1 - 7 T + 149 T^{2} + 86 T^{3} + 19256 T^{4} - 62019 T^{5} + 2510809 T^{6} + 314926 T^{7} + 178368761 T^{8} - 208183770 T^{9} + 19040957664 T^{10} - 208183770 p T^{11} + 178368761 p^{2} T^{12} + 314926 p^{3} T^{13} + 2510809 p^{4} T^{14} - 62019 p^{5} T^{15} + 19256 p^{6} T^{16} + 86 p^{7} T^{17} + 149 p^{8} T^{18} - 7 p^{9} T^{19} + p^{10} T^{20} \)
79 \( 1 + 9 T + 430 T^{2} + 3943 T^{3} + 101090 T^{4} + 890355 T^{5} + 16018218 T^{6} + 132232226 T^{7} + 1875969073 T^{8} + 14092748579 T^{9} + 168293794632 T^{10} + 14092748579 p T^{11} + 1875969073 p^{2} T^{12} + 132232226 p^{3} T^{13} + 16018218 p^{4} T^{14} + 890355 p^{5} T^{15} + 101090 p^{6} T^{16} + 3943 p^{7} T^{17} + 430 p^{8} T^{18} + 9 p^{9} T^{19} + p^{10} T^{20} \)
83 \( 1 + 48 T + 1607 T^{2} + 39067 T^{3} + 791164 T^{4} + 13466480 T^{5} + 201308134 T^{6} + 2648331548 T^{7} + 31293429719 T^{8} + 331438073283 T^{9} + 3181189934558 T^{10} + 331438073283 p T^{11} + 31293429719 p^{2} T^{12} + 2648331548 p^{3} T^{13} + 201308134 p^{4} T^{14} + 13466480 p^{5} T^{15} + 791164 p^{6} T^{16} + 39067 p^{7} T^{17} + 1607 p^{8} T^{18} + 48 p^{9} T^{19} + p^{10} T^{20} \)
89 \( 1 - 20 T + 826 T^{2} - 13265 T^{3} + 309864 T^{4} - 4125997 T^{5} + 69930056 T^{6} - 784800296 T^{7} + 10554625333 T^{8} - 100327277122 T^{9} + 1115257123896 T^{10} - 100327277122 p T^{11} + 10554625333 p^{2} T^{12} - 784800296 p^{3} T^{13} + 69930056 p^{4} T^{14} - 4125997 p^{5} T^{15} + 309864 p^{6} T^{16} - 13265 p^{7} T^{17} + 826 p^{8} T^{18} - 20 p^{9} T^{19} + p^{10} T^{20} \)
97 \( 1 - T + 474 T^{2} - 1011 T^{3} + 124564 T^{4} - 289437 T^{5} + 22765700 T^{6} - 50389262 T^{7} + 3148321249 T^{8} - 6418058069 T^{9} + 341520385872 T^{10} - 6418058069 p T^{11} + 3148321249 p^{2} T^{12} - 50389262 p^{3} T^{13} + 22765700 p^{4} T^{14} - 289437 p^{5} T^{15} + 124564 p^{6} T^{16} - 1011 p^{7} T^{17} + 474 p^{8} T^{18} - p^{9} T^{19} + p^{10} T^{20} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{20} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−3.10555923748797888532277213388, −3.09991738146704475396624792261, −3.01441648757327375532802742651, −2.94710234800948422067273228789, −2.90881044551209002585123289456, −2.67984053317663685094997810118, −2.55711680761688534488709158408, −2.39217746254944566657226248601, −2.30524743659129641636960238985, −2.29287797820286960313750192855, −2.29285377416583059095219107776, −2.27400866679591937776133653694, −2.26093130050488857402466515823, −2.13394878472620363664103345547, −1.88455779583905307528172406163, −1.76568692246176394110572440888, −1.76114062282575106089906489027, −1.59680864412539880896145972978, −1.51016222367431392740855512969, −1.45410444103830657635910875293, −1.41063643149162553996671863000, −1.24371808111022863649447713971, −1.17890911071607434833805702009, −0.950031592526461356321205677635, −0.78232403494764473402700346996, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.78232403494764473402700346996, 0.950031592526461356321205677635, 1.17890911071607434833805702009, 1.24371808111022863649447713971, 1.41063643149162553996671863000, 1.45410444103830657635910875293, 1.51016222367431392740855512969, 1.59680864412539880896145972978, 1.76114062282575106089906489027, 1.76568692246176394110572440888, 1.88455779583905307528172406163, 2.13394878472620363664103345547, 2.26093130050488857402466515823, 2.27400866679591937776133653694, 2.29285377416583059095219107776, 2.29287797820286960313750192855, 2.30524743659129641636960238985, 2.39217746254944566657226248601, 2.55711680761688534488709158408, 2.67984053317663685094997810118, 2.90881044551209002585123289456, 2.94710234800948422067273228789, 3.01441648757327375532802742651, 3.09991738146704475396624792261, 3.10555923748797888532277213388

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

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