Dirichlet series
| L(s) = 1 | − 10·2-s − 5·3-s + 40·4-s − 5-s + 50·6-s − 7-s − 40·8-s + 61·9-s + 10·10-s − 80·11-s − 200·12-s + 73·13-s + 10·14-s + 5·15-s − 320·16-s + 69·17-s − 610·18-s + 33·19-s − 40·20-s + 5·21-s + 800·22-s − 524·23-s + 200·24-s + 379·25-s − 730·26-s − 54·27-s − 40·28-s + ⋯ |
| L(s) = 1 | − 3.53·2-s − 0.962·3-s + 5·4-s − 0.0894·5-s + 3.40·6-s − 0.0539·7-s − 1.76·8-s + 2.25·9-s + 0.316·10-s − 2.19·11-s − 4.81·12-s + 1.55·13-s + 0.190·14-s + 0.0860·15-s − 5·16-s + 0.984·17-s − 7.98·18-s + 0.398·19-s − 0.447·20-s + 0.0519·21-s + 7.75·22-s − 4.75·23-s + 1.70·24-s + 3.03·25-s − 5.50·26-s − 0.384·27-s − 0.269·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 37^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 37^{10}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(2^{10} \cdot 37^{10}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.51754\times 10^{6}\) |
| Root analytic conductor: | \(2.08953\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((20,\ 2^{10} \cdot 37^{10} ,\ ( \ : [3/2]^{10} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(0.01269955807\) |
| \(L(\frac12)\) | \(\approx\) | \(0.01269955807\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + p T + p^{2} T^{2} )^{5} \) |
| 37 | \( 1 - 24 T + 68881 T^{2} - 24254 p T^{3} + 3594001 p^{2} T^{4} - 6046334 p^{3} T^{5} + 3594001 p^{5} T^{6} - 24254 p^{7} T^{7} + 68881 p^{9} T^{8} - 24 p^{12} T^{9} + p^{15} T^{10} \) | |
| good | 3 | \( 1 + 5 T - 4 p^{2} T^{2} - 431 T^{3} - 55 T^{4} + 5690 p T^{5} + 51962 T^{6} - 463724 T^{7} - 1034605 p T^{8} + 5650457 T^{9} + 114419590 T^{10} + 5650457 p^{3} T^{11} - 1034605 p^{7} T^{12} - 463724 p^{9} T^{13} + 51962 p^{12} T^{14} + 5690 p^{16} T^{15} - 55 p^{18} T^{16} - 431 p^{21} T^{17} - 4 p^{26} T^{18} + 5 p^{27} T^{19} + p^{30} T^{20} \) |
| 5 | \( 1 + T - 378 T^{2} - 4609 T^{3} + 82254 T^{4} + 1422567 T^{5} - 1972288 T^{6} - 257896749 T^{7} - 1468214039 T^{8} + 13350138286 T^{9} + 353775840004 T^{10} + 13350138286 p^{3} T^{11} - 1468214039 p^{6} T^{12} - 257896749 p^{9} T^{13} - 1972288 p^{12} T^{14} + 1422567 p^{15} T^{15} + 82254 p^{18} T^{16} - 4609 p^{21} T^{17} - 378 p^{24} T^{18} + p^{27} T^{19} + p^{30} T^{20} \) | |
| 7 | \( 1 + T - 932 T^{2} + 3037 T^{3} + 52039 p T^{4} - 2253782 T^{5} - 123930762 T^{6} - 2495580 T^{7} + 8116692135 p T^{8} + 129835895501 T^{9} - 22831481227126 T^{10} + 129835895501 p^{3} T^{11} + 8116692135 p^{7} T^{12} - 2495580 p^{9} T^{13} - 123930762 p^{12} T^{14} - 2253782 p^{15} T^{15} + 52039 p^{19} T^{16} + 3037 p^{21} T^{17} - 932 p^{24} T^{18} + p^{27} T^{19} + p^{30} T^{20} \) | |
| 11 | \( ( 1 + 40 T + 1927 T^{2} + 56688 T^{3} + 3876914 T^{4} + 157068880 T^{5} + 3876914 p^{3} T^{6} + 56688 p^{6} T^{7} + 1927 p^{9} T^{8} + 40 p^{12} T^{9} + p^{15} T^{10} )^{2} \) | |
| 13 | \( 1 - 73 T + 812 T^{2} - 69477 T^{3} - 1091929 T^{4} + 639165186 T^{5} - 22821117728 T^{6} + 624286096074 T^{7} + 32145954713 p T^{8} - 2363658162612351 T^{9} + 134496138438112708 T^{10} - 2363658162612351 p^{3} T^{11} + 32145954713 p^{7} T^{12} + 624286096074 p^{9} T^{13} - 22821117728 p^{12} T^{14} + 639165186 p^{15} T^{15} - 1091929 p^{18} T^{16} - 69477 p^{21} T^{17} + 812 p^{24} T^{18} - 73 p^{27} T^{19} + p^{30} T^{20} \) | |
| 17 | \( 1 - 69 T - 9574 T^{2} + 36849 p T^{3} + 41453598 T^{4} - 1218204051 T^{5} - 152164327284 T^{6} - 12258784614267 T^{7} + 81728679968409 p T^{8} + 46711986819223218 T^{9} - 9257282506851127260 T^{10} + 46711986819223218 p^{3} T^{11} + 81728679968409 p^{7} T^{12} - 12258784614267 p^{9} T^{13} - 152164327284 p^{12} T^{14} - 1218204051 p^{15} T^{15} + 41453598 p^{18} T^{16} + 36849 p^{22} T^{17} - 9574 p^{24} T^{18} - 69 p^{27} T^{19} + p^{30} T^{20} \) | |
| 19 | \( 1 - 33 T - 11242 T^{2} + 241805 T^{3} + 14814063 T^{4} - 340937870 T^{5} + 125862657332 T^{6} + 30783131332614 T^{7} + 815073135445429 T^{8} - 204139467108051255 T^{9} - 12210610861688253486 T^{10} - 204139467108051255 p^{3} T^{11} + 815073135445429 p^{6} T^{12} + 30783131332614 p^{9} T^{13} + 125862657332 p^{12} T^{14} - 340937870 p^{15} T^{15} + 14814063 p^{18} T^{16} + 241805 p^{21} T^{17} - 11242 p^{24} T^{18} - 33 p^{27} T^{19} + p^{30} T^{20} \) | |
| 23 | \( ( 1 + 262 T + 63127 T^{2} + 8417840 T^{3} + 1184179366 T^{4} + 118529632212 T^{5} + 1184179366 p^{3} T^{6} + 8417840 p^{6} T^{7} + 63127 p^{9} T^{8} + 262 p^{12} T^{9} + p^{15} T^{10} )^{2} \) | |
| 29 | \( ( 1 - 148 T + 59629 T^{2} - 3673658 T^{3} + 1859178337 T^{4} - 106413657846 T^{5} + 1859178337 p^{3} T^{6} - 3673658 p^{6} T^{7} + 59629 p^{9} T^{8} - 148 p^{12} T^{9} + p^{15} T^{10} )^{2} \) | |
| 31 | \( ( 1 + 392 T + 134499 T^{2} + 25790224 T^{3} + 4897992770 T^{4} + 738585946704 T^{5} + 4897992770 p^{3} T^{6} + 25790224 p^{6} T^{7} + 134499 p^{9} T^{8} + 392 p^{12} T^{9} + p^{15} T^{10} )^{2} \) | |
| 41 | \( 1 - 345 T - 180502 T^{2} + 41908989 T^{3} + 27258607470 T^{4} - 2823067433199 T^{5} - 3146507235921636 T^{6} + 169996634266594545 T^{7} + \)\(26\!\cdots\!61\)\( T^{8} - \)\(44\!\cdots\!38\)\( T^{9} - \)\(19\!\cdots\!08\)\( T^{10} - \)\(44\!\cdots\!38\)\( p^{3} T^{11} + \)\(26\!\cdots\!61\)\( p^{6} T^{12} + 169996634266594545 p^{9} T^{13} - 3146507235921636 p^{12} T^{14} - 2823067433199 p^{15} T^{15} + 27258607470 p^{18} T^{16} + 41908989 p^{21} T^{17} - 180502 p^{24} T^{18} - 345 p^{27} T^{19} + p^{30} T^{20} \) | |
| 43 | \( ( 1 + 246 T + 177343 T^{2} + 40719104 T^{3} + 21039074338 T^{4} + 4203468930260 T^{5} + 21039074338 p^{3} T^{6} + 40719104 p^{6} T^{7} + 177343 p^{9} T^{8} + 246 p^{12} T^{9} + p^{15} T^{10} )^{2} \) | |
| 47 | \( ( 1 + 16 T + 394867 T^{2} - 15044496 T^{3} + 67799398370 T^{4} - 3494689887872 T^{5} + 67799398370 p^{3} T^{6} - 15044496 p^{6} T^{7} + 394867 p^{9} T^{8} + 16 p^{12} T^{9} + p^{15} T^{10} )^{2} \) | |
| 53 | \( 1 + 19 T - 198900 T^{2} + 128643471 T^{3} - 13152475289 T^{4} - 16891976066502 T^{5} + 9265838075165824 T^{6} - 3648758786212370478 T^{7} + \)\(45\!\cdots\!29\)\( T^{8} + \)\(11\!\cdots\!25\)\( p T^{9} - \)\(32\!\cdots\!92\)\( T^{10} + \)\(11\!\cdots\!25\)\( p^{4} T^{11} + \)\(45\!\cdots\!29\)\( p^{6} T^{12} - 3648758786212370478 p^{9} T^{13} + 9265838075165824 p^{12} T^{14} - 16891976066502 p^{15} T^{15} - 13152475289 p^{18} T^{16} + 128643471 p^{21} T^{17} - 198900 p^{24} T^{18} + 19 p^{27} T^{19} + p^{30} T^{20} \) | |
| 59 | \( 1 + 105 T - 954514 T^{2} - 49843453 T^{3} + 540728578479 T^{4} + 14647622180974 T^{5} - 209426054124223196 T^{6} - 2396400623217226422 T^{7} + \)\(61\!\cdots\!29\)\( T^{8} + \)\(18\!\cdots\!23\)\( T^{9} - \)\(14\!\cdots\!74\)\( T^{10} + \)\(18\!\cdots\!23\)\( p^{3} T^{11} + \)\(61\!\cdots\!29\)\( p^{6} T^{12} - 2396400623217226422 p^{9} T^{13} - 209426054124223196 p^{12} T^{14} + 14647622180974 p^{15} T^{15} + 540728578479 p^{18} T^{16} - 49843453 p^{21} T^{17} - 954514 p^{24} T^{18} + 105 p^{27} T^{19} + p^{30} T^{20} \) | |
| 61 | \( 1 - 219 T - 605746 T^{2} - 112881989 T^{3} + 243240749142 T^{4} + 83887887678227 T^{5} - 32681506955069440 T^{6} - 31597762144134754857 T^{7} - \)\(12\!\cdots\!55\)\( T^{8} + \)\(28\!\cdots\!58\)\( T^{9} + \)\(20\!\cdots\!12\)\( T^{10} + \)\(28\!\cdots\!58\)\( p^{3} T^{11} - \)\(12\!\cdots\!55\)\( p^{6} T^{12} - 31597762144134754857 p^{9} T^{13} - 32681506955069440 p^{12} T^{14} + 83887887678227 p^{15} T^{15} + 243240749142 p^{18} T^{16} - 112881989 p^{21} T^{17} - 605746 p^{24} T^{18} - 219 p^{27} T^{19} + p^{30} T^{20} \) | |
| 67 | \( 1 - 773 T - 167894 T^{2} + 246924445 T^{3} - 29325206837 T^{4} + 5573205707530 T^{5} + 32641381413614232 T^{6} - 38598067449736461654 T^{7} + \)\(37\!\cdots\!89\)\( T^{8} + \)\(49\!\cdots\!69\)\( T^{9} - \)\(16\!\cdots\!66\)\( T^{10} + \)\(49\!\cdots\!69\)\( p^{3} T^{11} + \)\(37\!\cdots\!89\)\( p^{6} T^{12} - 38598067449736461654 p^{9} T^{13} + 32641381413614232 p^{12} T^{14} + 5573205707530 p^{15} T^{15} - 29325206837 p^{18} T^{16} + 246924445 p^{21} T^{17} - 167894 p^{24} T^{18} - 773 p^{27} T^{19} + p^{30} T^{20} \) | |
| 71 | \( 1 - 555 T - 1354192 T^{2} + 555669717 T^{3} + 1204952895381 T^{4} - 342650776851654 T^{5} - 755452935670685094 T^{6} + \)\(11\!\cdots\!64\)\( T^{7} + \)\(37\!\cdots\!89\)\( T^{8} - \)\(19\!\cdots\!67\)\( T^{9} - \)\(14\!\cdots\!14\)\( T^{10} - \)\(19\!\cdots\!67\)\( p^{3} T^{11} + \)\(37\!\cdots\!89\)\( p^{6} T^{12} + \)\(11\!\cdots\!64\)\( p^{9} T^{13} - 755452935670685094 p^{12} T^{14} - 342650776851654 p^{15} T^{15} + 1204952895381 p^{18} T^{16} + 555669717 p^{21} T^{17} - 1354192 p^{24} T^{18} - 555 p^{27} T^{19} + p^{30} T^{20} \) | |
| 73 | \( ( 1 - 174 T + 1424197 T^{2} - 135214792 T^{3} + 907802198290 T^{4} - 53088974134228 T^{5} + 907802198290 p^{3} T^{6} - 135214792 p^{6} T^{7} + 1424197 p^{9} T^{8} - 174 p^{12} T^{9} + p^{15} T^{10} )^{2} \) | |
| 79 | \( 1 + 727 T - 1082258 T^{2} - 401785139 T^{3} + 799407470323 T^{4} - 36563813268302 T^{5} - 491697774548283840 T^{6} + 59503864255646522202 T^{7} + \)\(23\!\cdots\!25\)\( T^{8} - \)\(44\!\cdots\!35\)\( T^{9} - \)\(10\!\cdots\!14\)\( T^{10} - \)\(44\!\cdots\!35\)\( p^{3} T^{11} + \)\(23\!\cdots\!25\)\( p^{6} T^{12} + 59503864255646522202 p^{9} T^{13} - 491697774548283840 p^{12} T^{14} - 36563813268302 p^{15} T^{15} + 799407470323 p^{18} T^{16} - 401785139 p^{21} T^{17} - 1082258 p^{24} T^{18} + 727 p^{27} T^{19} + p^{30} T^{20} \) | |
| 83 | \( 1 - 2229 T + 1605932 T^{2} - 25986145 T^{3} - 763292891343 T^{4} + 1078413519169546 T^{5} - 1188882592537097078 T^{6} + \)\(60\!\cdots\!20\)\( T^{7} + \)\(20\!\cdots\!73\)\( T^{8} - \)\(50\!\cdots\!05\)\( T^{9} + \)\(42\!\cdots\!18\)\( T^{10} - \)\(50\!\cdots\!05\)\( p^{3} T^{11} + \)\(20\!\cdots\!73\)\( p^{6} T^{12} + \)\(60\!\cdots\!20\)\( p^{9} T^{13} - 1188882592537097078 p^{12} T^{14} + 1078413519169546 p^{15} T^{15} - 763292891343 p^{18} T^{16} - 25986145 p^{21} T^{17} + 1605932 p^{24} T^{18} - 2229 p^{27} T^{19} + p^{30} T^{20} \) | |
| 89 | \( 1 - 901 T - 1705902 T^{2} + 1622687993 T^{3} + 1439096595950 T^{4} - 1080004533793731 T^{5} - 1369411880778392620 T^{6} + \)\(23\!\cdots\!89\)\( T^{7} + \)\(15\!\cdots\!41\)\( T^{8} + \)\(19\!\cdots\!46\)\( T^{9} - \)\(13\!\cdots\!20\)\( T^{10} + \)\(19\!\cdots\!46\)\( p^{3} T^{11} + \)\(15\!\cdots\!41\)\( p^{6} T^{12} + \)\(23\!\cdots\!89\)\( p^{9} T^{13} - 1369411880778392620 p^{12} T^{14} - 1080004533793731 p^{15} T^{15} + 1439096595950 p^{18} T^{16} + 1622687993 p^{21} T^{17} - 1705902 p^{24} T^{18} - 901 p^{27} T^{19} + p^{30} T^{20} \) | |
| 97 | \( ( 1 + 2298 T + 3147103 T^{2} + 1017337428 T^{3} - 1711129472123 T^{4} - 3479748561617322 T^{5} - 1711129472123 p^{3} T^{6} + 1017337428 p^{6} T^{7} + 3147103 p^{9} T^{8} + 2298 p^{12} T^{9} + p^{15} T^{10} )^{2} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{20} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.57241185521436466269864120629, −5.41203087906894068017960168909, −4.97710166721376665152160135037, −4.95733764336215381822847515229, −4.94909333503195643492063748764, −4.71476732805425893519458043292, −4.67654268088849477023687186482, −4.38104399999551971787853362999, −4.02511660668479867841496286188, −3.97987066633208370631233787498, −3.70739109035617461649988587880, −3.68754271223862249323571001576, −3.58192854373335035608437158625, −3.29766180828592836270553868620, −2.78569588061363515472380408982, −2.59321280994575655754726392539, −2.40452090431143133372207346355, −2.01655542674722780162518115911, −1.99121214621547992790247548151, −1.57162865213331440461807386484, −1.23076582867424794785665752118, −1.12102056398389703499491159389, −0.804525395352835630687511448152, −0.54919644440612391123735505458, −0.05633372896207988072311280360, 0.05633372896207988072311280360, 0.54919644440612391123735505458, 0.804525395352835630687511448152, 1.12102056398389703499491159389, 1.23076582867424794785665752118, 1.57162865213331440461807386484, 1.99121214621547992790247548151, 2.01655542674722780162518115911, 2.40452090431143133372207346355, 2.59321280994575655754726392539, 2.78569588061363515472380408982, 3.29766180828592836270553868620, 3.58192854373335035608437158625, 3.68754271223862249323571001576, 3.70739109035617461649988587880, 3.97987066633208370631233787498, 4.02511660668479867841496286188, 4.38104399999551971787853362999, 4.67654268088849477023687186482, 4.71476732805425893519458043292, 4.94909333503195643492063748764, 4.95733764336215381822847515229, 4.97710166721376665152160135037, 5.41203087906894068017960168909, 5.57241185521436466269864120629
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.