Properties

Label 18-1856e9-1.1-c3e9-0-1
Degree $18$
Conductor $2.613\times 10^{29}$
Sign $-1$
Analytic cond. $2.26462\times 10^{18}$
Root an. cond. $10.4645$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $9$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 4·3-s − 10·5-s − 12·7-s − 89·9-s + 64·11-s − 70·13-s − 40·15-s − 66·17-s + 42·19-s − 48·21-s − 40·23-s − 457·25-s − 306·27-s + 261·29-s + 64·31-s + 256·33-s + 120·35-s + 54·37-s − 280·39-s − 378·41-s − 32·43-s + 890·45-s − 1.16e3·47-s − 1.64e3·49-s − 264·51-s − 278·53-s − 640·55-s + ⋯
L(s)  = 1  + 0.769·3-s − 0.894·5-s − 0.647·7-s − 3.29·9-s + 1.75·11-s − 1.49·13-s − 0.688·15-s − 0.941·17-s + 0.507·19-s − 0.498·21-s − 0.362·23-s − 3.65·25-s − 2.18·27-s + 1.67·29-s + 0.370·31-s + 1.35·33-s + 0.579·35-s + 0.239·37-s − 1.14·39-s − 1.43·41-s − 0.113·43-s + 2.94·45-s − 3.61·47-s − 4.80·49-s − 0.724·51-s − 0.720·53-s − 1.56·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{54} \cdot 29^{9}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & -\,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{54} \cdot 29^{9}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{9} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(18\)
Conductor: \(2^{54} \cdot 29^{9}\)
Sign: $-1$
Analytic conductor: \(2.26462\times 10^{18}\)
Root analytic conductor: \(10.4645\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1856} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(9\)
Selberg data: \((18,\ 2^{54} \cdot 29^{9} ,\ ( \ : [3/2]^{9} ),\ -1 )\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
29 \( ( 1 - p T )^{9} \)
good3 \( 1 - 4 T + 35 p T^{2} - 470 T^{3} + 6034 T^{4} - 28642 T^{5} + 9196 p^{3} T^{6} - 137636 p^{2} T^{7} + 8241757 T^{8} - 38984750 T^{9} + 8241757 p^{3} T^{10} - 137636 p^{8} T^{11} + 9196 p^{12} T^{12} - 28642 p^{12} T^{13} + 6034 p^{15} T^{14} - 470 p^{18} T^{15} + 35 p^{22} T^{16} - 4 p^{24} T^{17} + p^{27} T^{18} \)
5 \( 1 + 2 p T + 557 T^{2} + 5468 T^{3} + 150926 T^{4} + 49296 p^{2} T^{5} + 26414558 T^{6} + 166916184 T^{7} + 3553674671 T^{8} + 19655273498 T^{9} + 3553674671 p^{3} T^{10} + 166916184 p^{6} T^{11} + 26414558 p^{9} T^{12} + 49296 p^{14} T^{13} + 150926 p^{15} T^{14} + 5468 p^{18} T^{15} + 557 p^{21} T^{16} + 2 p^{25} T^{17} + p^{27} T^{18} \)
7 \( 1 + 12 T + 1791 T^{2} + 14992 T^{3} + 1531652 T^{4} + 9480048 T^{5} + 889838092 T^{6} + 4546988400 T^{7} + 394510750414 T^{8} + 252104386488 p T^{9} + 394510750414 p^{3} T^{10} + 4546988400 p^{6} T^{11} + 889838092 p^{9} T^{12} + 9480048 p^{12} T^{13} + 1531652 p^{15} T^{14} + 14992 p^{18} T^{15} + 1791 p^{21} T^{16} + 12 p^{24} T^{17} + p^{27} T^{18} \)
11 \( 1 - 64 T + 9633 T^{2} - 515106 T^{3} + 41565250 T^{4} - 1903070478 T^{5} + 108633045572 T^{6} - 4328865590444 T^{7} + 196340175103773 T^{8} - 6808183448600998 T^{9} + 196340175103773 p^{3} T^{10} - 4328865590444 p^{6} T^{11} + 108633045572 p^{9} T^{12} - 1903070478 p^{12} T^{13} + 41565250 p^{15} T^{14} - 515106 p^{18} T^{15} + 9633 p^{21} T^{16} - 64 p^{24} T^{17} + p^{27} T^{18} \)
13 \( 1 + 70 T + 15349 T^{2} + 766236 T^{3} + 97697758 T^{4} + 3506309832 T^{5} + 358391986126 T^{6} + 9407381523608 T^{7} + 933789157523503 T^{8} + 20476944450242190 T^{9} + 933789157523503 p^{3} T^{10} + 9407381523608 p^{6} T^{11} + 358391986126 p^{9} T^{12} + 3506309832 p^{12} T^{13} + 97697758 p^{15} T^{14} + 766236 p^{18} T^{15} + 15349 p^{21} T^{16} + 70 p^{24} T^{17} + p^{27} T^{18} \)
17 \( 1 + 66 T + 23969 T^{2} + 929440 T^{3} + 232881740 T^{4} + 103821688 p T^{5} + 1172809828780 T^{6} - 2769603281824 p T^{7} + 222480045475062 p T^{8} - 404070385504592244 T^{9} + 222480045475062 p^{4} T^{10} - 2769603281824 p^{7} T^{11} + 1172809828780 p^{9} T^{12} + 103821688 p^{13} T^{13} + 232881740 p^{15} T^{14} + 929440 p^{18} T^{15} + 23969 p^{21} T^{16} + 66 p^{24} T^{17} + p^{27} T^{18} \)
19 \( 1 - 42 T + 39763 T^{2} - 1621328 T^{3} + 789320436 T^{4} - 31149559832 T^{5} + 10185547585708 T^{6} - 375868168173744 T^{7} + 94105339300264910 T^{8} - 3092874563550590524 T^{9} + 94105339300264910 p^{3} T^{10} - 375868168173744 p^{6} T^{11} + 10185547585708 p^{9} T^{12} - 31149559832 p^{12} T^{13} + 789320436 p^{15} T^{14} - 1621328 p^{18} T^{15} + 39763 p^{21} T^{16} - 42 p^{24} T^{17} + p^{27} T^{18} \)
23 \( 1 + 40 T + 51567 T^{2} + 972976 T^{3} + 1370943876 T^{4} + 2150324032 T^{5} + 25565920473164 T^{6} - 233225796078512 T^{7} + 377230103292645646 T^{8} - 4554081455272913104 T^{9} + 377230103292645646 p^{3} T^{10} - 233225796078512 p^{6} T^{11} + 25565920473164 p^{9} T^{12} + 2150324032 p^{12} T^{13} + 1370943876 p^{15} T^{14} + 972976 p^{18} T^{15} + 51567 p^{21} T^{16} + 40 p^{24} T^{17} + p^{27} T^{18} \)
31 \( 1 - 64 T + 137969 T^{2} - 4078514 T^{3} + 9627118922 T^{4} + 33430241786 T^{5} + 444509647662712 T^{6} + 13465773868128308 T^{7} + 15749005543521924729 T^{8} + \)\(61\!\cdots\!10\)\( T^{9} + 15749005543521924729 p^{3} T^{10} + 13465773868128308 p^{6} T^{11} + 444509647662712 p^{9} T^{12} + 33430241786 p^{12} T^{13} + 9627118922 p^{15} T^{14} - 4078514 p^{18} T^{15} + 137969 p^{21} T^{16} - 64 p^{24} T^{17} + p^{27} T^{18} \)
37 \( 1 - 54 T + 370965 T^{2} - 16465264 T^{3} + 64411695028 T^{4} - 2373244926504 T^{5} + 6872008237009588 T^{6} - 212283557107486352 T^{7} + \)\(49\!\cdots\!78\)\( T^{8} - \)\(12\!\cdots\!60\)\( T^{9} + \)\(49\!\cdots\!78\)\( p^{3} T^{10} - 212283557107486352 p^{6} T^{11} + 6872008237009588 p^{9} T^{12} - 2373244926504 p^{12} T^{13} + 64411695028 p^{15} T^{14} - 16465264 p^{18} T^{15} + 370965 p^{21} T^{16} - 54 p^{24} T^{17} + p^{27} T^{18} \)
41 \( 1 + 378 T + 463393 T^{2} + 158174400 T^{3} + 104201595028 T^{4} + 31149069757016 T^{5} + 14703694144864164 T^{6} + 3797291296902623296 T^{7} + \)\(14\!\cdots\!98\)\( T^{8} + \)\(31\!\cdots\!88\)\( T^{9} + \)\(14\!\cdots\!98\)\( p^{3} T^{10} + 3797291296902623296 p^{6} T^{11} + 14703694144864164 p^{9} T^{12} + 31149069757016 p^{12} T^{13} + 104201595028 p^{15} T^{14} + 158174400 p^{18} T^{15} + 463393 p^{21} T^{16} + 378 p^{24} T^{17} + p^{27} T^{18} \)
43 \( 1 + 32 T + 429089 T^{2} + 18666174 T^{3} + 87172975202 T^{4} + 3973916110610 T^{5} + 11543912296163204 T^{6} + 466366406684044500 T^{7} + \)\(11\!\cdots\!05\)\( T^{8} + \)\(40\!\cdots\!78\)\( T^{9} + \)\(11\!\cdots\!05\)\( p^{3} T^{10} + 466366406684044500 p^{6} T^{11} + 11543912296163204 p^{9} T^{12} + 3973916110610 p^{12} T^{13} + 87172975202 p^{15} T^{14} + 18666174 p^{18} T^{15} + 429089 p^{21} T^{16} + 32 p^{24} T^{17} + p^{27} T^{18} \)
47 \( 1 + 1164 T + 1221825 T^{2} + 775560262 T^{3} + 449475197786 T^{4} + 189362851494538 T^{5} + 76010378830887304 T^{6} + 23718156786481864612 T^{7} + \)\(79\!\cdots\!69\)\( T^{8} + \)\(23\!\cdots\!86\)\( T^{9} + \)\(79\!\cdots\!69\)\( p^{3} T^{10} + 23718156786481864612 p^{6} T^{11} + 76010378830887304 p^{9} T^{12} + 189362851494538 p^{12} T^{13} + 449475197786 p^{15} T^{14} + 775560262 p^{18} T^{15} + 1221825 p^{21} T^{16} + 1164 p^{24} T^{17} + p^{27} T^{18} \)
53 \( 1 + 278 T + 15353 p T^{2} + 218008836 T^{3} + 337029734910 T^{4} + 84588258157888 T^{5} + 92593781811617374 T^{6} + 21222823142942272584 T^{7} + \)\(18\!\cdots\!19\)\( T^{8} + \)\(37\!\cdots\!98\)\( T^{9} + \)\(18\!\cdots\!19\)\( p^{3} T^{10} + 21222823142942272584 p^{6} T^{11} + 92593781811617374 p^{9} T^{12} + 84588258157888 p^{12} T^{13} + 337029734910 p^{15} T^{14} + 218008836 p^{18} T^{15} + 15353 p^{22} T^{16} + 278 p^{24} T^{17} + p^{27} T^{18} \)
59 \( 1 - 640 T + 1037723 T^{2} - 405974032 T^{3} + 464221237940 T^{4} - 129410581944608 T^{5} + 143719214140414572 T^{6} - 31515197364560689520 T^{7} + \)\(35\!\cdots\!70\)\( T^{8} - \)\(67\!\cdots\!56\)\( T^{9} + \)\(35\!\cdots\!70\)\( p^{3} T^{10} - 31515197364560689520 p^{6} T^{11} + 143719214140414572 p^{9} T^{12} - 129410581944608 p^{12} T^{13} + 464221237940 p^{15} T^{14} - 405974032 p^{18} T^{15} + 1037723 p^{21} T^{16} - 640 p^{24} T^{17} + p^{27} T^{18} \)
61 \( 1 + 1054 T + 1568037 T^{2} + 1162269824 T^{3} + 983362076188 T^{4} + 533559107827208 T^{5} + 334367270277065740 T^{6} + \)\(14\!\cdots\!88\)\( T^{7} + \)\(78\!\cdots\!10\)\( T^{8} + \)\(31\!\cdots\!08\)\( T^{9} + \)\(78\!\cdots\!10\)\( p^{3} T^{10} + \)\(14\!\cdots\!88\)\( p^{6} T^{11} + 334367270277065740 p^{9} T^{12} + 533559107827208 p^{12} T^{13} + 983362076188 p^{15} T^{14} + 1162269824 p^{18} T^{15} + 1568037 p^{21} T^{16} + 1054 p^{24} T^{17} + p^{27} T^{18} \)
67 \( 1 - 1184 T + 1288787 T^{2} - 1133540224 T^{3} + 964826408356 T^{4} - 707720778669696 T^{5} + 502817313454323868 T^{6} - \)\(31\!\cdots\!68\)\( T^{7} + \)\(19\!\cdots\!22\)\( T^{8} - \)\(10\!\cdots\!24\)\( T^{9} + \)\(19\!\cdots\!22\)\( p^{3} T^{10} - \)\(31\!\cdots\!68\)\( p^{6} T^{11} + 502817313454323868 p^{9} T^{12} - 707720778669696 p^{12} T^{13} + 964826408356 p^{15} T^{14} - 1133540224 p^{18} T^{15} + 1288787 p^{21} T^{16} - 1184 p^{24} T^{17} + p^{27} T^{18} \)
71 \( 1 + 28 p T + 3323031 T^{2} + 3702250096 T^{3} + 3671683881964 T^{4} + 2919164126927824 T^{5} + 2167371998542916596 T^{6} + \)\(13\!\cdots\!12\)\( T^{7} + \)\(89\!\cdots\!86\)\( T^{8} + \)\(52\!\cdots\!16\)\( T^{9} + \)\(89\!\cdots\!86\)\( p^{3} T^{10} + \)\(13\!\cdots\!12\)\( p^{6} T^{11} + 2167371998542916596 p^{9} T^{12} + 2919164126927824 p^{12} T^{13} + 3671683881964 p^{15} T^{14} + 3702250096 p^{18} T^{15} + 3323031 p^{21} T^{16} + 28 p^{25} T^{17} + p^{27} T^{18} \)
73 \( 1 + 750 T + 1278657 T^{2} + 455393264 T^{3} + 707537549700 T^{4} + 178907994243976 T^{5} + 365621467346670292 T^{6} + 57042805873060402832 T^{7} + \)\(13\!\cdots\!26\)\( T^{8} + \)\(33\!\cdots\!48\)\( T^{9} + \)\(13\!\cdots\!26\)\( p^{3} T^{10} + 57042805873060402832 p^{6} T^{11} + 365621467346670292 p^{9} T^{12} + 178907994243976 p^{12} T^{13} + 707537549700 p^{15} T^{14} + 455393264 p^{18} T^{15} + 1278657 p^{21} T^{16} + 750 p^{24} T^{17} + p^{27} T^{18} \)
79 \( 1 + 2916 T + 6921569 T^{2} + 11219603842 T^{3} + 15899392325866 T^{4} + 18421776994143678 T^{5} + 19289605175163710808 T^{6} + \)\(17\!\cdots\!08\)\( T^{7} + \)\(14\!\cdots\!33\)\( T^{8} + \)\(10\!\cdots\!18\)\( T^{9} + \)\(14\!\cdots\!33\)\( p^{3} T^{10} + \)\(17\!\cdots\!08\)\( p^{6} T^{11} + 19289605175163710808 p^{9} T^{12} + 18421776994143678 p^{12} T^{13} + 15899392325866 p^{15} T^{14} + 11219603842 p^{18} T^{15} + 6921569 p^{21} T^{16} + 2916 p^{24} T^{17} + p^{27} T^{18} \)
83 \( 1 - 2832 T + 6778531 T^{2} - 11761336688 T^{3} + 17540112874164 T^{4} - 22097573321356192 T^{5} + 24748748903174983564 T^{6} - \)\(24\!\cdots\!00\)\( T^{7} + \)\(21\!\cdots\!62\)\( T^{8} - \)\(17\!\cdots\!92\)\( T^{9} + \)\(21\!\cdots\!62\)\( p^{3} T^{10} - \)\(24\!\cdots\!00\)\( p^{6} T^{11} + 24748748903174983564 p^{9} T^{12} - 22097573321356192 p^{12} T^{13} + 17540112874164 p^{15} T^{14} - 11761336688 p^{18} T^{15} + 6778531 p^{21} T^{16} - 2832 p^{24} T^{17} + p^{27} T^{18} \)
89 \( 1 + 370 T + 3700321 T^{2} + 2119446432 T^{3} + 6789288951716 T^{4} + 4759281662686776 T^{5} + 8502181235036434196 T^{6} + \)\(60\!\cdots\!40\)\( T^{7} + \)\(79\!\cdots\!06\)\( T^{8} + \)\(50\!\cdots\!84\)\( T^{9} + \)\(79\!\cdots\!06\)\( p^{3} T^{10} + \)\(60\!\cdots\!40\)\( p^{6} T^{11} + 8502181235036434196 p^{9} T^{12} + 4759281662686776 p^{12} T^{13} + 6789288951716 p^{15} T^{14} + 2119446432 p^{18} T^{15} + 3700321 p^{21} T^{16} + 370 p^{24} T^{17} + p^{27} T^{18} \)
97 \( 1 + 2234 T + 7747865 T^{2} + 13656982720 T^{3} + 26987814014020 T^{4} + 38339711774685656 T^{5} + 55519290330743069780 T^{6} + \)\(64\!\cdots\!44\)\( T^{7} + \)\(74\!\cdots\!54\)\( T^{8} + \)\(71\!\cdots\!56\)\( T^{9} + \)\(74\!\cdots\!54\)\( p^{3} T^{10} + \)\(64\!\cdots\!44\)\( p^{6} T^{11} + 55519290330743069780 p^{9} T^{12} + 38339711774685656 p^{12} T^{13} + 26987814014020 p^{15} T^{14} + 13656982720 p^{18} T^{15} + 7747865 p^{21} T^{16} + 2234 p^{24} T^{17} + p^{27} T^{18} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{18} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−3.58351772455205149355166883734, −3.51758822375033546467664669720, −3.50972779520580033447092189958, −3.34062641581840989356124829938, −3.11902011848613314061275842880, −3.10803653125865008567545667265, −3.06602712297177920667608180478, −2.93624279897876652459736183688, −2.72629218893552971598967717870, −2.64230120125822972450369506668, −2.60493701803437284423164880060, −2.37997804740801584917375860951, −2.29884281728275968152478526153, −2.27707437038743652981665384999, −2.15485781443355369864955877017, −2.14753571347409760857434373270, −1.76325034318848125178284070203, −1.48862076186139928477582158888, −1.48598136447226439093564673186, −1.45900456278828047682619166542, −1.38079618037005683156235546907, −1.19418256649454417993238471225, −1.11108465961738298836708632840, −0.946099914143722747363122126533, −0.822590028748596357402277513252, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.822590028748596357402277513252, 0.946099914143722747363122126533, 1.11108465961738298836708632840, 1.19418256649454417993238471225, 1.38079618037005683156235546907, 1.45900456278828047682619166542, 1.48598136447226439093564673186, 1.48862076186139928477582158888, 1.76325034318848125178284070203, 2.14753571347409760857434373270, 2.15485781443355369864955877017, 2.27707437038743652981665384999, 2.29884281728275968152478526153, 2.37997804740801584917375860951, 2.60493701803437284423164880060, 2.64230120125822972450369506668, 2.72629218893552971598967717870, 2.93624279897876652459736183688, 3.06602712297177920667608180478, 3.10803653125865008567545667265, 3.11902011848613314061275842880, 3.34062641581840989356124829938, 3.50972779520580033447092189958, 3.51758822375033546467664669720, 3.58351772455205149355166883734

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

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