Properties

Label 28-429e14-1.1-c1e14-0-0
Degree $28$
Conductor $7.152\times 10^{36}$
Sign $1$
Analytic cond. $3.06394\times 10^{7}$
Root an. cond. $1.85083$
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  + 14·3-s + 5·4-s + 105·9-s + 70·12-s + 12·16-s + 4·17-s − 8·23-s + 22·25-s + 560·27-s − 24·29-s + 525·36-s + 32·43-s + 168·48-s + 26·49-s + 56·51-s + 20·53-s − 20·61-s + 20·64-s + 20·68-s − 112·69-s + 308·75-s + 12·79-s + 2.38e3·81-s − 336·87-s − 40·92-s + 110·100-s + 20·101-s + ⋯
L(s)  = 1  + 8.08·3-s + 5/2·4-s + 35·9-s + 20.2·12-s + 3·16-s + 0.970·17-s − 1.66·23-s + 22/5·25-s + 107.·27-s − 4.45·29-s + 87.5·36-s + 4.87·43-s + 24.2·48-s + 26/7·49-s + 7.84·51-s + 2.74·53-s − 2.56·61-s + 5/2·64-s + 2.42·68-s − 13.4·69-s + 35.5·75-s + 1.35·79-s + 264.·81-s − 36.0·87-s − 4.17·92-s + 11·100-s + 1.99·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{14} \cdot 11^{14} \cdot 13^{14}\right)^{s/2} \, \Gamma_{\C}(s)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{14} \cdot 11^{14} \cdot 13^{14}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(28\)
Conductor: \(3^{14} \cdot 11^{14} \cdot 13^{14}\)
Sign: $1$
Analytic conductor: \(3.06394\times 10^{7}\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{429} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((28,\ 3^{14} \cdot 11^{14} \cdot 13^{14} ,\ ( \ : [1/2]^{14} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1345.342548\)
\(L(\frac12)\) \(\approx\) \(1345.342548\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( ( 1 - T )^{14} \)
11 \( ( 1 + T^{2} )^{7} \)
13 \( 1 + 7 T^{2} - 32 T^{3} + 73 T^{4} + 256 T^{5} - 1129 T^{6} - 2752 T^{7} - 1129 p T^{8} + 256 p^{2} T^{9} + 73 p^{3} T^{10} - 32 p^{4} T^{11} + 7 p^{5} T^{12} + p^{7} T^{14} \)
good2 \( 1 - 5 T^{2} + 13 T^{4} - 25 T^{6} + 51 T^{8} - 119 T^{10} + 251 T^{12} - 467 T^{14} + 251 p^{2} T^{16} - 119 p^{4} T^{18} + 51 p^{6} T^{20} - 25 p^{8} T^{22} + 13 p^{10} T^{24} - 5 p^{12} T^{26} + p^{14} T^{28} \)
5 \( 1 - 22 T^{2} + 51 p T^{4} - 424 p T^{6} + 14429 T^{8} - 88134 T^{10} + 498723 T^{12} - 2600216 T^{14} + 498723 p^{2} T^{16} - 88134 p^{4} T^{18} + 14429 p^{6} T^{20} - 424 p^{9} T^{22} + 51 p^{11} T^{24} - 22 p^{12} T^{26} + p^{14} T^{28} \)
7 \( 1 - 26 T^{2} + 335 T^{4} - 2796 T^{6} + 17949 T^{8} - 83862 T^{10} + 180659 T^{12} + 258616 T^{14} + 180659 p^{2} T^{16} - 83862 p^{4} T^{18} + 17949 p^{6} T^{20} - 2796 p^{8} T^{22} + 335 p^{10} T^{24} - 26 p^{12} T^{26} + p^{14} T^{28} \)
17 \( ( 1 - 2 T + 69 T^{2} - 172 T^{3} + 2517 T^{4} - 6628 T^{5} + 60381 T^{6} - 144332 T^{7} + 60381 p T^{8} - 6628 p^{2} T^{9} + 2517 p^{3} T^{10} - 172 p^{4} T^{11} + 69 p^{5} T^{12} - 2 p^{6} T^{13} + p^{7} T^{14} )^{2} \)
19 \( 1 - 158 T^{2} + 11887 T^{4} - 568740 T^{6} + 19612501 T^{8} - 528839314 T^{10} + 11937378699 T^{12} - 238393179800 T^{14} + 11937378699 p^{2} T^{16} - 528839314 p^{4} T^{18} + 19612501 p^{6} T^{20} - 568740 p^{8} T^{22} + 11887 p^{10} T^{24} - 158 p^{12} T^{26} + p^{14} T^{28} \)
23 \( ( 1 + 4 T + 85 T^{2} + 504 T^{3} + 3817 T^{4} + 25280 T^{5} + 123013 T^{6} + 728008 T^{7} + 123013 p T^{8} + 25280 p^{2} T^{9} + 3817 p^{3} T^{10} + 504 p^{4} T^{11} + 85 p^{5} T^{12} + 4 p^{6} T^{13} + p^{7} T^{14} )^{2} \)
29 \( ( 1 + 12 T + 205 T^{2} + 1730 T^{3} + 17161 T^{4} + 111022 T^{5} + 804745 T^{6} + 4114856 T^{7} + 804745 p T^{8} + 111022 p^{2} T^{9} + 17161 p^{3} T^{10} + 1730 p^{4} T^{11} + 205 p^{5} T^{12} + 12 p^{6} T^{13} + p^{7} T^{14} )^{2} \)
31 \( 1 - 210 T^{2} + 20767 T^{4} - 1286560 T^{6} + 55893861 T^{8} - 1825462602 T^{10} + 49686754603 T^{12} - 1397334021096 T^{14} + 49686754603 p^{2} T^{16} - 1825462602 p^{4} T^{18} + 55893861 p^{6} T^{20} - 1286560 p^{8} T^{22} + 20767 p^{10} T^{24} - 210 p^{12} T^{26} + p^{14} T^{28} \)
37 \( 1 - 210 T^{2} + 25331 T^{4} - 2212916 T^{6} + 152324457 T^{8} - 8585680686 T^{10} + 406102843635 T^{12} - 16290206957656 T^{14} + 406102843635 p^{2} T^{16} - 8585680686 p^{4} T^{18} + 152324457 p^{6} T^{20} - 2212916 p^{8} T^{22} + 25331 p^{10} T^{24} - 210 p^{12} T^{26} + p^{14} T^{28} \)
41 \( 1 - 310 T^{2} + 49543 T^{4} - 5370628 T^{6} + 440137925 T^{8} - 28795116890 T^{10} + 1548060413843 T^{12} - 69371631106008 T^{14} + 1548060413843 p^{2} T^{16} - 28795116890 p^{4} T^{18} + 440137925 p^{6} T^{20} - 5370628 p^{8} T^{22} + 49543 p^{10} T^{24} - 310 p^{12} T^{26} + p^{14} T^{28} \)
43 \( ( 1 - 16 T + 211 T^{2} - 2288 T^{3} + 22441 T^{4} - 191142 T^{5} + 1465639 T^{6} - 9828708 T^{7} + 1465639 p T^{8} - 191142 p^{2} T^{9} + 22441 p^{3} T^{10} - 2288 p^{4} T^{11} + 211 p^{5} T^{12} - 16 p^{6} T^{13} + p^{7} T^{14} )^{2} \)
47 \( 1 - 246 T^{2} + 28843 T^{4} - 2118652 T^{6} + 116717337 T^{8} - 6019751466 T^{10} + 327324863995 T^{12} - 16657366022856 T^{14} + 327324863995 p^{2} T^{16} - 6019751466 p^{4} T^{18} + 116717337 p^{6} T^{20} - 2118652 p^{8} T^{22} + 28843 p^{10} T^{24} - 246 p^{12} T^{26} + p^{14} T^{28} \)
53 \( ( 1 - 10 T + 283 T^{2} - 1932 T^{3} + 32981 T^{4} - 164822 T^{5} + 2330967 T^{6} - 9583240 T^{7} + 2330967 p T^{8} - 164822 p^{2} T^{9} + 32981 p^{3} T^{10} - 1932 p^{4} T^{11} + 283 p^{5} T^{12} - 10 p^{6} T^{13} + p^{7} T^{14} )^{2} \)
59 \( 1 - 502 T^{2} + 124851 T^{4} - 20334268 T^{6} + 2433427561 T^{8} - 228291315274 T^{10} + 17498707952051 T^{12} - 1123362838468680 T^{14} + 17498707952051 p^{2} T^{16} - 228291315274 p^{4} T^{18} + 2433427561 p^{6} T^{20} - 20334268 p^{8} T^{22} + 124851 p^{10} T^{24} - 502 p^{12} T^{26} + p^{14} T^{28} \)
61 \( ( 1 + 10 T + 439 T^{2} + 3544 T^{3} + 81553 T^{4} + 529902 T^{5} + 8345663 T^{6} + 42813248 T^{7} + 8345663 p T^{8} + 529902 p^{2} T^{9} + 81553 p^{3} T^{10} + 3544 p^{4} T^{11} + 439 p^{5} T^{12} + 10 p^{6} T^{13} + p^{7} T^{14} )^{2} \)
67 \( 1 - 430 T^{2} + 104143 T^{4} - 17808832 T^{6} + 2346863965 T^{8} - 249249928206 T^{10} + 21841953270611 T^{12} - 1598308795731944 T^{14} + 21841953270611 p^{2} T^{16} - 249249928206 p^{4} T^{18} + 2346863965 p^{6} T^{20} - 17808832 p^{8} T^{22} + 104143 p^{10} T^{24} - 430 p^{12} T^{26} + p^{14} T^{28} \)
71 \( 1 - 402 T^{2} + 92395 T^{4} - 15358932 T^{6} + 2003898441 T^{8} - 214998177934 T^{10} + 19407964695659 T^{12} - 1490410254549400 T^{14} + 19407964695659 p^{2} T^{16} - 214998177934 p^{4} T^{18} + 2003898441 p^{6} T^{20} - 15358932 p^{8} T^{22} + 92395 p^{10} T^{24} - 402 p^{12} T^{26} + p^{14} T^{28} \)
73 \( 1 - 394 T^{2} + 74319 T^{4} - 9097708 T^{6} + 884060477 T^{8} - 81504025254 T^{10} + 7365403140563 T^{12} - 588462013064648 T^{14} + 7365403140563 p^{2} T^{16} - 81504025254 p^{4} T^{18} + 884060477 p^{6} T^{20} - 9097708 p^{8} T^{22} + 74319 p^{10} T^{24} - 394 p^{12} T^{26} + p^{14} T^{28} \)
79 \( ( 1 - 6 T + 399 T^{2} - 2284 T^{3} + 76961 T^{4} - 389888 T^{5} + 9195259 T^{6} - 38954460 T^{7} + 9195259 p T^{8} - 389888 p^{2} T^{9} + 76961 p^{3} T^{10} - 2284 p^{4} T^{11} + 399 p^{5} T^{12} - 6 p^{6} T^{13} + p^{7} T^{14} )^{2} \)
83 \( 1 - 726 T^{2} + 256707 T^{4} - 59071420 T^{6} + 9979471401 T^{8} - 1322910753450 T^{10} + 143254516457027 T^{12} - 12960500582214984 T^{14} + 143254516457027 p^{2} T^{16} - 1322910753450 p^{4} T^{18} + 9979471401 p^{6} T^{20} - 59071420 p^{8} T^{22} + 256707 p^{10} T^{24} - 726 p^{12} T^{26} + p^{14} T^{28} \)
89 \( 1 - 750 T^{2} + 268287 T^{4} - 61998156 T^{6} + 10627182589 T^{8} - 1462407219214 T^{10} + 167923263484019 T^{12} - 16265991933164368 T^{14} + 167923263484019 p^{2} T^{16} - 1462407219214 p^{4} T^{18} + 10627182589 p^{6} T^{20} - 61998156 p^{8} T^{22} + 268287 p^{10} T^{24} - 750 p^{12} T^{26} + p^{14} T^{28} \)
97 \( 1 - 678 T^{2} + 218283 T^{4} - 45292412 T^{6} + 7010177337 T^{8} - 892333575418 T^{10} + 99694475830171 T^{12} - 10106365268275656 T^{14} + 99694475830171 p^{2} T^{16} - 892333575418 p^{4} T^{18} + 7010177337 p^{6} T^{20} - 45292412 p^{8} T^{22} + 218283 p^{10} T^{24} - 678 p^{12} T^{26} + p^{14} T^{28} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{28} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−3.36548166472854992964484530046, −3.32304678488124777283055367981, −3.26024751275536021365570087787, −2.84064756745205897963106416112, −2.82316294649373214623309007894, −2.80288778278513741670823146783, −2.77427391614251054057964748137, −2.62580869438595163665181359938, −2.59497172826488486684952458292, −2.54996222963310855081426280688, −2.53552748306959786824467233407, −2.34953874392678266094555662051, −2.18894990468798976168821848605, −2.06132777468751451878608717052, −2.02670089233783380950797050614, −1.98628983682244623158151723082, −1.93787664591795539504671941840, −1.92720406763282712299682050200, −1.55963550244804990555737911146, −1.30745626498582484053715845528, −1.30153165306499893560337598840, −1.28293401292494700422053799570, −0.986421031472242657310553305087, −0.854368405444295335915957851674, −0.816930096140076660963423538988, 0.816930096140076660963423538988, 0.854368405444295335915957851674, 0.986421031472242657310553305087, 1.28293401292494700422053799570, 1.30153165306499893560337598840, 1.30745626498582484053715845528, 1.55963550244804990555737911146, 1.92720406763282712299682050200, 1.93787664591795539504671941840, 1.98628983682244623158151723082, 2.02670089233783380950797050614, 2.06132777468751451878608717052, 2.18894990468798976168821848605, 2.34953874392678266094555662051, 2.53552748306959786824467233407, 2.54996222963310855081426280688, 2.59497172826488486684952458292, 2.62580869438595163665181359938, 2.77427391614251054057964748137, 2.80288778278513741670823146783, 2.82316294649373214623309007894, 2.84064756745205897963106416112, 3.26024751275536021365570087787, 3.32304678488124777283055367981, 3.36548166472854992964484530046

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

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