Properties

Label 28-684e14-1.1-c2e14-0-0
Degree $28$
Conductor $4.907\times 10^{39}$
Sign $1$
Analytic cond. $6.10205\times 10^{17}$
Root an. cond. $4.31713$
Motivic weight $2$
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  − 2·2-s + 3·4-s + 10·8-s + 54·13-s − 9·16-s − 34·17-s − 218·25-s − 108·26-s − 54·29-s − 6·32-s + 68·34-s + 100·37-s − 224·41-s + 233·49-s + 436·50-s + 162·52-s − 14·53-s + 108·58-s + 28·61-s + 99·64-s − 102·68-s + 70·73-s − 200·74-s + 448·82-s + 308·97-s − 466·98-s − 654·100-s + ⋯
L(s)  = 1  − 2-s + 3/4·4-s + 5/4·8-s + 4.15·13-s − 0.562·16-s − 2·17-s − 8.71·25-s − 4.15·26-s − 1.86·29-s − 0.187·32-s + 2·34-s + 2.70·37-s − 5.46·41-s + 4.75·49-s + 8.71·50-s + 3.11·52-s − 0.264·53-s + 1.86·58-s + 0.459·61-s + 1.54·64-s − 3/2·68-s + 0.958·73-s − 2.70·74-s + 5.46·82-s + 3.17·97-s − 4.75·98-s − 6.53·100-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{28} \cdot 19^{14}\right)^{s/2} \, \Gamma_{\C}(s)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{28} \cdot 19^{14}\right)^{s/2} \, \Gamma_{\C}(s+1)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(28\)
Conductor: \(2^{28} \cdot 3^{28} \cdot 19^{14}\)
Sign: $1$
Analytic conductor: \(6.10205\times 10^{17}\)
Root analytic conductor: \(4.31713\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((28,\ 2^{28} \cdot 3^{28} \cdot 19^{14} ,\ ( \ : [1]^{14} ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.082612259\)
\(L(\frac12)\) \(\approx\) \(3.082612259\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + p T + T^{2} - 7 p T^{3} - 21 p T^{4} - 7 p^{2} T^{5} + 33 p^{2} T^{6} + 55 p^{3} T^{7} + 33 p^{4} T^{8} - 7 p^{6} T^{9} - 21 p^{7} T^{10} - 7 p^{9} T^{11} + p^{10} T^{12} + p^{13} T^{13} + p^{14} T^{14} \)
3 \( 1 \)
19 \( ( 1 + p T^{2} )^{7} \)
good5 \( ( 1 + 109 T^{2} + 28 T^{3} + 6212 T^{4} + 1452 T^{5} + 362 p^{4} T^{6} + 50912 T^{7} + 362 p^{6} T^{8} + 1452 p^{4} T^{9} + 6212 p^{6} T^{10} + 28 p^{8} T^{11} + 109 p^{10} T^{12} + p^{14} T^{14} )^{2} \)
7 \( 1 - 233 T^{2} + 27852 T^{4} - 333685 p T^{6} + 163361841 T^{8} - 10452325750 T^{10} + 615602347125 T^{12} - 32240190830807 T^{14} + 615602347125 p^{4} T^{16} - 10452325750 p^{8} T^{18} + 163361841 p^{12} T^{20} - 333685 p^{17} T^{22} + 27852 p^{20} T^{24} - 233 p^{24} T^{26} + p^{28} T^{28} \)
11 \( 1 - 74 p T^{2} + 335393 T^{4} - 93186116 T^{6} + 1782447000 p T^{8} - 3345913060132 T^{10} + 486329171341590 T^{12} - 62284294191694644 T^{14} + 486329171341590 p^{4} T^{16} - 3345913060132 p^{8} T^{18} + 1782447000 p^{13} T^{20} - 93186116 p^{16} T^{22} + 335393 p^{20} T^{24} - 74 p^{25} T^{26} + p^{28} T^{28} \)
13 \( ( 1 - 27 T + 890 T^{2} - 17419 T^{3} + 360020 T^{4} - 5742409 T^{5} + 91406057 T^{6} - 1207839730 T^{7} + 91406057 p^{2} T^{8} - 5742409 p^{4} T^{9} + 360020 p^{6} T^{10} - 17419 p^{8} T^{11} + 890 p^{10} T^{12} - 27 p^{12} T^{13} + p^{14} T^{14} )^{2} \)
17 \( ( 1 + p T + 916 T^{2} + 19523 T^{3} + 510137 T^{4} + 9372294 T^{5} + 210313661 T^{6} + 3045803367 T^{7} + 210313661 p^{2} T^{8} + 9372294 p^{4} T^{9} + 510137 p^{6} T^{10} + 19523 p^{8} T^{11} + 916 p^{10} T^{12} + p^{13} T^{13} + p^{14} T^{14} )^{2} \)
23 \( 1 - 3525 T^{2} + 6033626 T^{4} - 6939636893 T^{6} + 6245527985252 T^{8} - 4738591413835359 T^{10} + 3105703116983336097 T^{12} - 3335624923997196062 p^{2} T^{14} + 3105703116983336097 p^{4} T^{16} - 4738591413835359 p^{8} T^{18} + 6245527985252 p^{12} T^{20} - 6939636893 p^{16} T^{22} + 6033626 p^{20} T^{24} - 3525 p^{24} T^{26} + p^{28} T^{28} \)
29 \( ( 1 + 27 T + 2986 T^{2} + 60451 T^{3} + 3494188 T^{4} + 41548281 T^{5} + 2328143537 T^{6} + 16605107986 T^{7} + 2328143537 p^{2} T^{8} + 41548281 p^{4} T^{9} + 3494188 p^{6} T^{10} + 60451 p^{8} T^{11} + 2986 p^{10} T^{12} + 27 p^{12} T^{13} + p^{14} T^{14} )^{2} \)
31 \( 1 - 9406 T^{2} + 43063003 T^{4} - 127561822796 T^{6} + 274012245953257 T^{8} - 452932384005531682 T^{10} + \)\(59\!\cdots\!87\)\( T^{12} - \)\(63\!\cdots\!48\)\( T^{14} + \)\(59\!\cdots\!87\)\( p^{4} T^{16} - 452932384005531682 p^{8} T^{18} + 274012245953257 p^{12} T^{20} - 127561822796 p^{16} T^{22} + 43063003 p^{20} T^{24} - 9406 p^{24} T^{26} + p^{28} T^{28} \)
37 \( ( 1 - 50 T + 8075 T^{2} - 310628 T^{3} + 28949409 T^{4} - 905769422 T^{5} + 61676132707 T^{6} - 1571320657400 T^{7} + 61676132707 p^{2} T^{8} - 905769422 p^{4} T^{9} + 28949409 p^{6} T^{10} - 310628 p^{8} T^{11} + 8075 p^{10} T^{12} - 50 p^{12} T^{13} + p^{14} T^{14} )^{2} \)
41 \( ( 1 + 112 T + 9991 T^{2} + 696160 T^{3} + 43222773 T^{4} + 2275515152 T^{5} + 111151836323 T^{6} + 4761059441728 T^{7} + 111151836323 p^{2} T^{8} + 2275515152 p^{4} T^{9} + 43222773 p^{6} T^{10} + 696160 p^{8} T^{11} + 9991 p^{10} T^{12} + 112 p^{12} T^{13} + p^{14} T^{14} )^{2} \)
43 \( 1 - 294 p T^{2} + 74616113 T^{4} - 260313791652 T^{6} + 545973209374408 T^{8} - 511190797106479716 T^{10} - \)\(60\!\cdots\!78\)\( T^{12} + \)\(25\!\cdots\!32\)\( T^{14} - \)\(60\!\cdots\!78\)\( p^{4} T^{16} - 511190797106479716 p^{8} T^{18} + 545973209374408 p^{12} T^{20} - 260313791652 p^{16} T^{22} + 74616113 p^{20} T^{24} - 294 p^{25} T^{26} + p^{28} T^{28} \)
47 \( 1 - 22722 T^{2} + 253418065 T^{4} - 1829883537364 T^{6} + 9526669878144136 T^{8} - 37747540463963425524 T^{10} + \)\(11\!\cdots\!62\)\( T^{12} - \)\(28\!\cdots\!08\)\( T^{14} + \)\(11\!\cdots\!62\)\( p^{4} T^{16} - 37747540463963425524 p^{8} T^{18} + 9526669878144136 p^{12} T^{20} - 1829883537364 p^{16} T^{22} + 253418065 p^{20} T^{24} - 22722 p^{24} T^{26} + p^{28} T^{28} \)
53 \( ( 1 + 7 T + 13018 T^{2} + 229959 T^{3} + 29420 p^{2} T^{4} + 1696096917 T^{5} + 342669958913 T^{6} + 6191770691754 T^{7} + 342669958913 p^{2} T^{8} + 1696096917 p^{4} T^{9} + 29420 p^{8} T^{10} + 229959 p^{8} T^{11} + 13018 p^{10} T^{12} + 7 p^{12} T^{13} + p^{14} T^{14} )^{2} \)
59 \( 1 - 18221 T^{2} + 164693114 T^{4} - 1004575517989 T^{6} + 4965431549381844 T^{8} - 22248054318742909623 T^{10} + \)\(92\!\cdots\!21\)\( T^{12} - \)\(34\!\cdots\!14\)\( T^{14} + \)\(92\!\cdots\!21\)\( p^{4} T^{16} - 22248054318742909623 p^{8} T^{18} + 4965431549381844 p^{12} T^{20} - 1004575517989 p^{16} T^{22} + 164693114 p^{20} T^{24} - 18221 p^{24} T^{26} + p^{28} T^{28} \)
61 \( ( 1 - 14 T + 11889 T^{2} + 172516 T^{3} + 56767944 T^{4} + 3312034548 T^{5} + 154999392078 T^{6} + 18950890770796 T^{7} + 154999392078 p^{2} T^{8} + 3312034548 p^{4} T^{9} + 56767944 p^{6} T^{10} + 172516 p^{8} T^{11} + 11889 p^{10} T^{12} - 14 p^{12} T^{13} + p^{14} T^{14} )^{2} \)
67 \( 1 - 34957 T^{2} + 624606042 T^{4} - 7492897631461 T^{6} + 67308310489788340 T^{8} - \)\(47\!\cdots\!55\)\( T^{10} + \)\(28\!\cdots\!73\)\( T^{12} - \)\(13\!\cdots\!46\)\( T^{14} + \)\(28\!\cdots\!73\)\( p^{4} T^{16} - \)\(47\!\cdots\!55\)\( p^{8} T^{18} + 67308310489788340 p^{12} T^{20} - 7492897631461 p^{16} T^{22} + 624606042 p^{20} T^{24} - 34957 p^{24} T^{26} + p^{28} T^{28} \)
71 \( 1 - 32762 T^{2} + 551205595 T^{4} - 6262515479204 T^{6} + 53721586965030281 T^{8} - \)\(37\!\cdots\!86\)\( T^{10} + \)\(22\!\cdots\!23\)\( T^{12} - \)\(11\!\cdots\!36\)\( T^{14} + \)\(22\!\cdots\!23\)\( p^{4} T^{16} - \)\(37\!\cdots\!86\)\( p^{8} T^{18} + 53721586965030281 p^{12} T^{20} - 6262515479204 p^{16} T^{22} + 551205595 p^{20} T^{24} - 32762 p^{24} T^{26} + p^{28} T^{28} \)
73 \( ( 1 - 35 T + 31396 T^{2} - 1023593 T^{3} + 450101977 T^{4} - 12866553058 T^{5} + 3790482347389 T^{6} - 89666608859493 T^{7} + 3790482347389 p^{2} T^{8} - 12866553058 p^{4} T^{9} + 450101977 p^{6} T^{10} - 1023593 p^{8} T^{11} + 31396 p^{10} T^{12} - 35 p^{12} T^{13} + p^{14} T^{14} )^{2} \)
79 \( 1 - 48634 T^{2} + 1141797723 T^{4} - 17270324260196 T^{6} + 190582392545703977 T^{8} - \)\(16\!\cdots\!38\)\( T^{10} + \)\(12\!\cdots\!59\)\( T^{12} - \)\(80\!\cdots\!24\)\( T^{14} + \)\(12\!\cdots\!59\)\( p^{4} T^{16} - \)\(16\!\cdots\!38\)\( p^{8} T^{18} + 190582392545703977 p^{12} T^{20} - 17270324260196 p^{16} T^{22} + 1141797723 p^{20} T^{24} - 48634 p^{24} T^{26} + p^{28} T^{28} \)
83 \( 1 - 33658 T^{2} + 594449419 T^{4} - 7466735526180 T^{6} + 76091481064390409 T^{8} - \)\(68\!\cdots\!82\)\( T^{10} + \)\(55\!\cdots\!67\)\( T^{12} - \)\(40\!\cdots\!32\)\( T^{14} + \)\(55\!\cdots\!67\)\( p^{4} T^{16} - \)\(68\!\cdots\!82\)\( p^{8} T^{18} + 76091481064390409 p^{12} T^{20} - 7466735526180 p^{16} T^{22} + 594449419 p^{20} T^{24} - 33658 p^{24} T^{26} + p^{28} T^{28} \)
89 \( ( 1 + 31807 T^{2} - 1100800 T^{3} + 405025533 T^{4} - 32794364416 T^{5} + 3158083087971 T^{6} - 381306692946944 T^{7} + 3158083087971 p^{2} T^{8} - 32794364416 p^{4} T^{9} + 405025533 p^{6} T^{10} - 1100800 p^{8} T^{11} + 31807 p^{10} T^{12} + p^{14} T^{14} )^{2} \)
97 \( ( 1 - 154 T + 38467 T^{2} - 4319284 T^{3} + 686082817 T^{4} - 60314185574 T^{5} + 7876032152891 T^{6} - 622370614755736 T^{7} + 7876032152891 p^{2} T^{8} - 60314185574 p^{4} T^{9} + 686082817 p^{6} T^{10} - 4319284 p^{8} T^{11} + 38467 p^{10} T^{12} - 154 p^{12} T^{13} + p^{14} T^{14} )^{2} \)
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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

−2.86905371698358830308076208245, −2.43614286409568021185739503722, −2.37805401329468711648689146697, −2.33645040958249264977419021739, −2.32191895913823019934081750359, −2.29769451324709876623063380395, −2.16095329940405999303563058815, −2.09250196310594999724413324499, −2.05016286237213814826404095016, −1.98095072851468236612814515811, −1.95433069607071373033709114464, −1.73479722321722610310385834901, −1.61864308978691479339411449070, −1.53827273205299578867579758316, −1.34371165528392402438612413412, −1.24016586777330062866961572871, −1.21769723287882998510702962190, −1.16151383336220639976469857680, −1.00987948362913085741370143414, −0.895753647867246518157378979753, −0.56733309982358012755780492601, −0.51411816592709628896973315380, −0.30224066522737491407661488163, −0.18301970378896940961949315850, −0.17233094655408973739994677113, 0.17233094655408973739994677113, 0.18301970378896940961949315850, 0.30224066522737491407661488163, 0.51411816592709628896973315380, 0.56733309982358012755780492601, 0.895753647867246518157378979753, 1.00987948362913085741370143414, 1.16151383336220639976469857680, 1.21769723287882998510702962190, 1.24016586777330062866961572871, 1.34371165528392402438612413412, 1.53827273205299578867579758316, 1.61864308978691479339411449070, 1.73479722321722610310385834901, 1.95433069607071373033709114464, 1.98095072851468236612814515811, 2.05016286237213814826404095016, 2.09250196310594999724413324499, 2.16095329940405999303563058815, 2.29769451324709876623063380395, 2.32191895913823019934081750359, 2.33645040958249264977419021739, 2.37805401329468711648689146697, 2.43614286409568021185739503722, 2.86905371698358830308076208245

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

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