# 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 of factors

## 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 $$1 + 2234 T + 7747865 T^{2} + 13656982720 T^{3} + 26987814014020 T^{4} + 38339711774685656 T^{5} + 55519290330743069780 T^{6} +$$$$64\!\cdots\!44$$$$T^{7} +$$$$74\!\cdots\!54$$$$T^{8} +$$$$71\!\cdots\!56$$$$T^{9} +$$$$74\!\cdots\!54$$$$p^{3} T^{10} +$$$$64\!\cdots\!44$$$$p^{6} T^{11} + 55519290330743069780 p^{9} T^{12} + 38339711774685656 p^{12} T^{13} + 26987814014020 p^{15} T^{14} + 13656982720 p^{18} T^{15} + 7747865 p^{21} T^{16} + 2234 p^{24} T^{17} + p^{27} T^{18}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{18} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$