Properties

Label 18-95e18-1.1-c1e9-0-1
Degree $18$
Conductor $3.972\times 10^{35}$
Sign $1$
Analytic cond. $5.24226\times 10^{16}$
Root an. cond. $8.48910$
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  + 6·2-s + 9·3-s + 12·4-s + 54·6-s − 2·8-s + 30·9-s + 108·12-s + 9·13-s − 48·16-s + 9·17-s + 180·18-s + 12·23-s − 18·24-s + 54·26-s + 26·27-s − 9·29-s − 18·31-s − 69·32-s + 54·34-s + 360·36-s + 18·37-s + 81·39-s − 6·41-s + 12·43-s + 72·46-s − 15·47-s − 432·48-s + ⋯
L(s)  = 1  + 4.24·2-s + 5.19·3-s + 6·4-s + 22.0·6-s − 0.707·8-s + 10·9-s + 31.1·12-s + 2.49·13-s − 12·16-s + 2.18·17-s + 42.4·18-s + 2.50·23-s − 3.67·24-s + 10.5·26-s + 5.00·27-s − 1.67·29-s − 3.23·31-s − 12.1·32-s + 9.26·34-s + 60·36-s + 2.95·37-s + 12.9·39-s − 0.937·41-s + 1.82·43-s + 10.6·46-s − 2.18·47-s − 62.3·48-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{18} \cdot 19^{18}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{18} \cdot 19^{18}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(18\)
Conductor: \(5^{18} \cdot 19^{18}\)
Sign: $1$
Analytic conductor: \(5.24226\times 10^{16}\)
Root analytic conductor: \(8.48910\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((18,\ 5^{18} \cdot 19^{18} ,\ ( \ : [1/2]^{9} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(769.1008153\)
\(L(\frac12)\) \(\approx\) \(769.1008153\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
19 \( 1 \)
good2 \( 1 - 3 p T + 3 p^{3} T^{2} - 35 p T^{3} + 21 p^{3} T^{4} - 339 T^{5} + 303 p T^{6} - 975 T^{7} + 1473 T^{8} - 2115 T^{9} + 1473 p T^{10} - 975 p^{2} T^{11} + 303 p^{4} T^{12} - 339 p^{4} T^{13} + 21 p^{8} T^{14} - 35 p^{7} T^{15} + 3 p^{10} T^{16} - 3 p^{9} T^{17} + p^{9} T^{18} \)
3 \( 1 - p^{2} T + 17 p T^{2} - 215 T^{3} + 247 p T^{4} - 241 p^{2} T^{5} + 5543 T^{6} - 4183 p T^{7} + 2827 p^{2} T^{8} - 46403 T^{9} + 2827 p^{3} T^{10} - 4183 p^{3} T^{11} + 5543 p^{3} T^{12} - 241 p^{6} T^{13} + 247 p^{6} T^{14} - 215 p^{6} T^{15} + 17 p^{8} T^{16} - p^{10} T^{17} + p^{9} T^{18} \)
7 \( 1 + 36 T^{2} + 24 T^{3} + 654 T^{4} + 666 T^{5} + 8207 T^{6} + 1245 p T^{7} + 76791 T^{8} + 73187 T^{9} + 76791 p T^{10} + 1245 p^{3} T^{11} + 8207 p^{3} T^{12} + 666 p^{4} T^{13} + 654 p^{5} T^{14} + 24 p^{6} T^{15} + 36 p^{7} T^{16} + p^{9} T^{18} \)
11 \( 1 + 39 T^{2} - 52 T^{3} + 954 T^{4} - 1395 T^{5} + 18353 T^{6} - 24852 T^{7} + 252720 T^{8} - 346805 T^{9} + 252720 p T^{10} - 24852 p^{2} T^{11} + 18353 p^{3} T^{12} - 1395 p^{4} T^{13} + 954 p^{5} T^{14} - 52 p^{6} T^{15} + 39 p^{7} T^{16} + p^{9} T^{18} \)
13 \( 1 - 9 T + 102 T^{2} - 578 T^{3} + 3603 T^{4} - 13758 T^{5} + 59376 T^{6} - 153369 T^{7} + 582609 T^{8} - 1362971 T^{9} + 582609 p T^{10} - 153369 p^{2} T^{11} + 59376 p^{3} T^{12} - 13758 p^{4} T^{13} + 3603 p^{5} T^{14} - 578 p^{6} T^{15} + 102 p^{7} T^{16} - 9 p^{8} T^{17} + p^{9} T^{18} \)
17 \( 1 - 9 T + 135 T^{2} - 820 T^{3} + 7125 T^{4} - 32703 T^{5} + 216633 T^{6} - 814356 T^{7} + 4653000 T^{8} - 15341403 T^{9} + 4653000 p T^{10} - 814356 p^{2} T^{11} + 216633 p^{3} T^{12} - 32703 p^{4} T^{13} + 7125 p^{5} T^{14} - 820 p^{6} T^{15} + 135 p^{7} T^{16} - 9 p^{8} T^{17} + p^{9} T^{18} \)
23 \( 1 - 12 T + 171 T^{2} - 1240 T^{3} + 10428 T^{4} - 53955 T^{5} + 346252 T^{6} - 1395789 T^{7} + 8202441 T^{8} - 30534213 T^{9} + 8202441 p T^{10} - 1395789 p^{2} T^{11} + 346252 p^{3} T^{12} - 53955 p^{4} T^{13} + 10428 p^{5} T^{14} - 1240 p^{6} T^{15} + 171 p^{7} T^{16} - 12 p^{8} T^{17} + p^{9} T^{18} \)
29 \( 1 + 9 T + 195 T^{2} + 1415 T^{3} + 18156 T^{4} + 111693 T^{5} + 1066401 T^{6} + 5608599 T^{7} + 43244664 T^{8} + 193931331 T^{9} + 43244664 p T^{10} + 5608599 p^{2} T^{11} + 1066401 p^{3} T^{12} + 111693 p^{4} T^{13} + 18156 p^{5} T^{14} + 1415 p^{6} T^{15} + 195 p^{7} T^{16} + 9 p^{8} T^{17} + p^{9} T^{18} \)
31 \( 1 + 18 T + 300 T^{2} + 3280 T^{3} + 34206 T^{4} + 284967 T^{5} + 2280228 T^{6} + 15470571 T^{7} + 101189040 T^{8} + 573434041 T^{9} + 101189040 p T^{10} + 15470571 p^{2} T^{11} + 2280228 p^{3} T^{12} + 284967 p^{4} T^{13} + 34206 p^{5} T^{14} + 3280 p^{6} T^{15} + 300 p^{7} T^{16} + 18 p^{8} T^{17} + p^{9} T^{18} \)
37 \( 1 - 18 T + 318 T^{2} - 3704 T^{3} + 39882 T^{4} - 353355 T^{5} + 2925055 T^{6} - 21419565 T^{7} + 3990912 p T^{8} - 925340573 T^{9} + 3990912 p^{2} T^{10} - 21419565 p^{2} T^{11} + 2925055 p^{3} T^{12} - 353355 p^{4} T^{13} + 39882 p^{5} T^{14} - 3704 p^{6} T^{15} + 318 p^{7} T^{16} - 18 p^{8} T^{17} + p^{9} T^{18} \)
41 \( 1 + 6 T + 291 T^{2} + 1591 T^{3} + 972 p T^{4} + 196821 T^{5} + 3380643 T^{6} + 14786358 T^{7} + 195808266 T^{8} + 736166283 T^{9} + 195808266 p T^{10} + 14786358 p^{2} T^{11} + 3380643 p^{3} T^{12} + 196821 p^{4} T^{13} + 972 p^{6} T^{14} + 1591 p^{6} T^{15} + 291 p^{7} T^{16} + 6 p^{8} T^{17} + p^{9} T^{18} \)
43 \( 1 - 12 T + 234 T^{2} - 2182 T^{3} + 25842 T^{4} - 212256 T^{5} + 1911582 T^{6} - 14168982 T^{7} + 105505830 T^{8} - 700118107 T^{9} + 105505830 p T^{10} - 14168982 p^{2} T^{11} + 1911582 p^{3} T^{12} - 212256 p^{4} T^{13} + 25842 p^{5} T^{14} - 2182 p^{6} T^{15} + 234 p^{7} T^{16} - 12 p^{8} T^{17} + p^{9} T^{18} \)
47 \( 1 + 15 T + 315 T^{2} + 3069 T^{3} + 38685 T^{4} + 293229 T^{5} + 2956529 T^{6} + 19533285 T^{7} + 173649807 T^{8} + 1030092269 T^{9} + 173649807 p T^{10} + 19533285 p^{2} T^{11} + 2956529 p^{3} T^{12} + 293229 p^{4} T^{13} + 38685 p^{5} T^{14} + 3069 p^{6} T^{15} + 315 p^{7} T^{16} + 15 p^{8} T^{17} + p^{9} T^{18} \)
53 \( 1 - 15 T + 6 p T^{2} - 4282 T^{3} + 52845 T^{4} - 588384 T^{5} + 5698942 T^{6} - 51883791 T^{7} + 425056737 T^{8} - 3232620591 T^{9} + 425056737 p T^{10} - 51883791 p^{2} T^{11} + 5698942 p^{3} T^{12} - 588384 p^{4} T^{13} + 52845 p^{5} T^{14} - 4282 p^{6} T^{15} + 6 p^{8} T^{16} - 15 p^{8} T^{17} + p^{9} T^{18} \)
59 \( 1 + 21 T + 537 T^{2} + 7480 T^{3} + 114057 T^{4} + 1221237 T^{5} + 14025070 T^{6} + 123301956 T^{7} + 1159161036 T^{8} + 8610185721 T^{9} + 1159161036 p T^{10} + 123301956 p^{2} T^{11} + 14025070 p^{3} T^{12} + 1221237 p^{4} T^{13} + 114057 p^{5} T^{14} + 7480 p^{6} T^{15} + 537 p^{7} T^{16} + 21 p^{8} T^{17} + p^{9} T^{18} \)
61 \( 1 + 12 T + 426 T^{2} + 3506 T^{3} + 73122 T^{4} + 406104 T^{5} + 7002567 T^{6} + 25180341 T^{7} + 473206089 T^{8} + 1309127783 T^{9} + 473206089 p T^{10} + 25180341 p^{2} T^{11} + 7002567 p^{3} T^{12} + 406104 p^{4} T^{13} + 73122 p^{5} T^{14} + 3506 p^{6} T^{15} + 426 p^{7} T^{16} + 12 p^{8} T^{17} + p^{9} T^{18} \)
67 \( 1 - 60 T + 2070 T^{2} - 50489 T^{3} + 965169 T^{4} - 15142848 T^{5} + 201176990 T^{6} - 2301545778 T^{7} + 22946242653 T^{8} - 200413450191 T^{9} + 22946242653 p T^{10} - 2301545778 p^{2} T^{11} + 201176990 p^{3} T^{12} - 15142848 p^{4} T^{13} + 965169 p^{5} T^{14} - 50489 p^{6} T^{15} + 2070 p^{7} T^{16} - 60 p^{8} T^{17} + p^{9} T^{18} \)
71 \( 1 - 18 T + 414 T^{2} - 6040 T^{3} + 92103 T^{4} - 1069749 T^{5} + 12742747 T^{6} - 125860989 T^{7} + 1245066777 T^{8} - 10442555541 T^{9} + 1245066777 p T^{10} - 125860989 p^{2} T^{11} + 12742747 p^{3} T^{12} - 1069749 p^{4} T^{13} + 92103 p^{5} T^{14} - 6040 p^{6} T^{15} + 414 p^{7} T^{16} - 18 p^{8} T^{17} + p^{9} T^{18} \)
73 \( 1 + 27 T + 624 T^{2} + 9206 T^{3} + 123630 T^{4} + 1340889 T^{5} + 14287750 T^{6} + 137267877 T^{7} + 1315413105 T^{8} + 11403188273 T^{9} + 1315413105 p T^{10} + 137267877 p^{2} T^{11} + 14287750 p^{3} T^{12} + 1340889 p^{4} T^{13} + 123630 p^{5} T^{14} + 9206 p^{6} T^{15} + 624 p^{7} T^{16} + 27 p^{8} T^{17} + p^{9} T^{18} \)
79 \( 1 + 15 T + 570 T^{2} + 7467 T^{3} + 142881 T^{4} + 1678953 T^{5} + 21458316 T^{6} + 229461801 T^{7} + 2243660358 T^{8} + 21492811375 T^{9} + 2243660358 p T^{10} + 229461801 p^{2} T^{11} + 21458316 p^{3} T^{12} + 1678953 p^{4} T^{13} + 142881 p^{5} T^{14} + 7467 p^{6} T^{15} + 570 p^{7} T^{16} + 15 p^{8} T^{17} + p^{9} T^{18} \)
83 \( 1 + 240 T^{2} + 1147 T^{3} + 32373 T^{4} + 298044 T^{5} + 3460414 T^{6} + 41146956 T^{7} + 340549653 T^{8} + 3845806785 T^{9} + 340549653 p T^{10} + 41146956 p^{2} T^{11} + 3460414 p^{3} T^{12} + 298044 p^{4} T^{13} + 32373 p^{5} T^{14} + 1147 p^{6} T^{15} + 240 p^{7} T^{16} + p^{9} T^{18} \)
89 \( 1 - 39 T + 1239 T^{2} - 26191 T^{3} + 485151 T^{4} - 7199361 T^{5} + 97547514 T^{6} - 1133405652 T^{7} + 12381460932 T^{8} - 119728677873 T^{9} + 12381460932 p T^{10} - 1133405652 p^{2} T^{11} + 97547514 p^{3} T^{12} - 7199361 p^{4} T^{13} + 485151 p^{5} T^{14} - 26191 p^{6} T^{15} + 1239 p^{7} T^{16} - 39 p^{8} T^{17} + p^{9} T^{18} \)
97 \( 1 - 15 T + 630 T^{2} - 8450 T^{3} + 192372 T^{4} - 23709 p T^{5} + 37438131 T^{6} - 391237194 T^{7} + 5065045482 T^{8} - 45393097217 T^{9} + 5065045482 p T^{10} - 391237194 p^{2} T^{11} + 37438131 p^{3} T^{12} - 23709 p^{5} T^{13} + 192372 p^{5} T^{14} - 8450 p^{6} T^{15} + 630 p^{7} T^{16} - 15 p^{8} T^{17} + p^{9} T^{18} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{18} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−2.93143676117273412925634568013, −2.92706783235490357227377083626, −2.90841428474687272981550596901, −2.75626707721061680395564163971, −2.70841140700118639496247319693, −2.57996351544509528800610729511, −2.29155713407610786109714760554, −2.23566997070555733694348710518, −2.22213837184339648666915991964, −2.09264684037084794873596335628, −2.03025807931810821548104361509, −1.93779796184365924428759314056, −1.89038179641265735118164407683, −1.62238806367056848927661365934, −1.54880892148787914560204457997, −1.52043261714483481506179689343, −1.35827573453025065419281819295, −1.10791784860229896630846604771, −1.09261862133830114643268369867, −0.818733642924098121464851027166, −0.71956635515993537143925250575, −0.69202828714867567264399663876, −0.43607357739790409420710869370, −0.31909071972204186832953974839, −0.23070071225265816930031850902, 0.23070071225265816930031850902, 0.31909071972204186832953974839, 0.43607357739790409420710869370, 0.69202828714867567264399663876, 0.71956635515993537143925250575, 0.818733642924098121464851027166, 1.09261862133830114643268369867, 1.10791784860229896630846604771, 1.35827573453025065419281819295, 1.52043261714483481506179689343, 1.54880892148787914560204457997, 1.62238806367056848927661365934, 1.89038179641265735118164407683, 1.93779796184365924428759314056, 2.03025807931810821548104361509, 2.09264684037084794873596335628, 2.22213837184339648666915991964, 2.23566997070555733694348710518, 2.29155713407610786109714760554, 2.57996351544509528800610729511, 2.70841140700118639496247319693, 2.75626707721061680395564163971, 2.90841428474687272981550596901, 2.92706783235490357227377083626, 2.93143676117273412925634568013

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

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