Dirichlet series
| L(s) = 1 | + 10·9-s − 14·19-s + 16·29-s + 8·31-s + 26·41-s − 48·49-s − 4·59-s + 2·61-s − 2·71-s + 16·79-s + 64·81-s + 40·89-s − 2·101-s + 34·109-s − 112·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 52·169-s − 140·171-s + 173-s + ⋯ |
| L(s) = 1 | + 10/3·9-s − 3.21·19-s + 2.97·29-s + 1.43·31-s + 4.06·41-s − 6.85·49-s − 0.520·59-s + 0.256·61-s − 0.237·71-s + 1.80·79-s + 64/9·81-s + 4.23·89-s − 0.199·101-s + 3.25·109-s − 10.1·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 4·169-s − 10.7·171-s + 0.0760·173-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 5^{40} \cdot 19^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 5^{40} \cdot 19^{20}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(40\) |
| Conductor: | \(2^{40} \cdot 5^{40} \cdot 19^{20}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.17459\times 10^{23}\) |
| Root analytic conductor: | \(3.89507\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((40,\ 2^{40} \cdot 5^{40} \cdot 19^{20} ,\ ( \ : [1/2]^{20} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(13.57564844\) |
| \(L(\frac12)\) | \(\approx\) | \(13.57564844\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) | |
| 19 | \( ( 1 + 7 T + 48 T^{2} + 14 T^{3} - 454 T^{4} - 5658 T^{5} - 454 p T^{6} + 14 p^{2} T^{7} + 48 p^{3} T^{8} + 7 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| good | 3 | \( 1 - 10 T^{2} + 4 p^{2} T^{4} - 22 T^{6} - 200 T^{8} + 412 T^{10} + 685 T^{12} - 2927 T^{14} - 428 p T^{16} + 3176 p^{2} T^{18} - 106442 T^{20} + 3176 p^{4} T^{22} - 428 p^{5} T^{24} - 2927 p^{6} T^{26} + 685 p^{8} T^{28} + 412 p^{10} T^{30} - 200 p^{12} T^{32} - 22 p^{14} T^{34} + 4 p^{18} T^{36} - 10 p^{18} T^{38} + p^{20} T^{40} \) |
| 7 | \( ( 1 + 24 T^{2} + 40 p T^{4} + 2105 T^{6} + 247 p^{2} T^{8} + 71998 T^{10} + 247 p^{4} T^{12} + 2105 p^{4} T^{14} + 40 p^{7} T^{16} + 24 p^{8} T^{18} + p^{10} T^{20} )^{2} \) | |
| 11 | \( ( 1 + 28 T^{2} - 29 T^{3} + 426 T^{4} - 490 T^{5} + 426 p T^{6} - 29 p^{2} T^{7} + 28 p^{3} T^{8} + p^{5} T^{10} )^{4} \) | |
| 13 | \( 1 - 4 p T^{2} + 115 p T^{4} - 38544 T^{6} + 851648 T^{8} - 1177688 p T^{10} + 256990861 T^{12} - 4043461036 T^{14} + 56316013675 T^{16} - 59168223320 p T^{18} + 61685695744 p^{2} T^{20} - 59168223320 p^{3} T^{22} + 56316013675 p^{4} T^{24} - 4043461036 p^{6} T^{26} + 256990861 p^{8} T^{28} - 1177688 p^{11} T^{30} + 851648 p^{12} T^{32} - 38544 p^{14} T^{34} + 115 p^{17} T^{36} - 4 p^{19} T^{38} + p^{20} T^{40} \) | |
| 17 | \( 1 - 72 T^{2} + 2972 T^{4} - 97226 T^{6} + 2642130 T^{8} - 61215192 T^{10} + 1285507661 T^{12} - 25113283229 T^{14} + 460403465844 T^{16} - 8172408097376 T^{18} + 141458143418418 T^{20} - 8172408097376 p^{2} T^{22} + 460403465844 p^{4} T^{24} - 25113283229 p^{6} T^{26} + 1285507661 p^{8} T^{28} - 61215192 p^{10} T^{30} + 2642130 p^{12} T^{32} - 97226 p^{14} T^{34} + 2972 p^{16} T^{36} - 72 p^{18} T^{38} + p^{20} T^{40} \) | |
| 23 | \( 1 - 19 T^{2} - 55 T^{4} + 57832 T^{6} - 1248558 T^{8} + 8662708 T^{10} + 1440915788 T^{12} - 36791066226 T^{14} + 486459947676 T^{16} + 19357867822502 T^{18} - 593622492464142 T^{20} + 19357867822502 p^{2} T^{22} + 486459947676 p^{4} T^{24} - 36791066226 p^{6} T^{26} + 1440915788 p^{8} T^{28} + 8662708 p^{10} T^{30} - 1248558 p^{12} T^{32} + 57832 p^{14} T^{34} - 55 p^{16} T^{36} - 19 p^{18} T^{38} + p^{20} T^{40} \) | |
| 29 | \( ( 1 - 8 T - 67 T^{2} + 658 T^{3} + 3106 T^{4} - 32234 T^{5} - 89750 T^{6} + 917986 T^{7} + 2232290 T^{8} - 12140742 T^{9} - 47451544 T^{10} - 12140742 p T^{11} + 2232290 p^{2} T^{12} + 917986 p^{3} T^{13} - 89750 p^{4} T^{14} - 32234 p^{5} T^{15} + 3106 p^{6} T^{16} + 658 p^{7} T^{17} - 67 p^{8} T^{18} - 8 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 31 | \( ( 1 - 2 T + 2 p T^{2} + 3 T^{3} + 1870 T^{4} + 4418 T^{5} + 1870 p T^{6} + 3 p^{2} T^{7} + 2 p^{4} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} )^{4} \) | |
| 37 | \( ( 1 + 284 T^{2} + 38804 T^{4} + 3335085 T^{6} + 199025587 T^{8} + 8601154798 T^{10} + 199025587 p^{2} T^{12} + 3335085 p^{4} T^{14} + 38804 p^{6} T^{16} + 284 p^{8} T^{18} + p^{10} T^{20} )^{2} \) | |
| 41 | \( ( 1 - 13 T - 19 T^{2} + 372 T^{3} + 4322 T^{4} - 6956 T^{5} - 313604 T^{6} + 365434 T^{7} + 9626720 T^{8} + 1032286 T^{9} - 368366230 T^{10} + 1032286 p T^{11} + 9626720 p^{2} T^{12} + 365434 p^{3} T^{13} - 313604 p^{4} T^{14} - 6956 p^{5} T^{15} + 4322 p^{6} T^{16} + 372 p^{7} T^{17} - 19 p^{8} T^{18} - 13 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 43 | \( 1 - 164 T^{2} + 15262 T^{4} - 1141302 T^{6} + 72034784 T^{8} - 3945299798 T^{10} + 203667666497 T^{12} - 236506229051 p T^{14} + 487772236617298 T^{16} - 22674693786101858 T^{18} + 1007539293849702894 T^{20} - 22674693786101858 p^{2} T^{22} + 487772236617298 p^{4} T^{24} - 236506229051 p^{7} T^{26} + 203667666497 p^{8} T^{28} - 3945299798 p^{10} T^{30} + 72034784 p^{12} T^{32} - 1141302 p^{14} T^{34} + 15262 p^{16} T^{36} - 164 p^{18} T^{38} + p^{20} T^{40} \) | |
| 47 | \( 1 - 389 T^{2} + 80339 T^{4} - 11605110 T^{6} + 1307390252 T^{8} - 121625998724 T^{10} + 9669835557276 T^{12} - 671689529565208 T^{14} + 41357053308196426 T^{16} - 2277936799044487484 T^{18} + \)\(11\!\cdots\!78\)\( T^{20} - 2277936799044487484 p^{2} T^{22} + 41357053308196426 p^{4} T^{24} - 671689529565208 p^{6} T^{26} + 9669835557276 p^{8} T^{28} - 121625998724 p^{10} T^{30} + 1307390252 p^{12} T^{32} - 11605110 p^{14} T^{34} + 80339 p^{16} T^{36} - 389 p^{18} T^{38} + p^{20} T^{40} \) | |
| 53 | \( 1 - 4 p T^{2} + 20036 T^{4} - 864978 T^{6} - 4018654 T^{8} + 2258912620 T^{10} - 1176322215 p T^{12} - 4653562839781 T^{14} + 353906602119112 T^{16} - 1148372563747712 T^{18} - 609893422559628478 T^{20} - 1148372563747712 p^{2} T^{22} + 353906602119112 p^{4} T^{24} - 4653562839781 p^{6} T^{26} - 1176322215 p^{9} T^{28} + 2258912620 p^{10} T^{30} - 4018654 p^{12} T^{32} - 864978 p^{14} T^{34} + 20036 p^{16} T^{36} - 4 p^{19} T^{38} + p^{20} T^{40} \) | |
| 59 | \( ( 1 + 2 T - 233 T^{2} - 272 T^{3} + 30854 T^{4} + 20216 T^{5} - 2844764 T^{6} - 901342 T^{7} + 205382320 T^{8} + 14529896 T^{9} - 12884877308 T^{10} + 14529896 p T^{11} + 205382320 p^{2} T^{12} - 901342 p^{3} T^{13} - 2844764 p^{4} T^{14} + 20216 p^{5} T^{15} + 30854 p^{6} T^{16} - 272 p^{7} T^{17} - 233 p^{8} T^{18} + 2 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 61 | \( ( 1 - T - 190 T^{2} - 267 T^{3} + 18650 T^{4} + 53321 T^{5} - 1221192 T^{6} - 4213739 T^{7} + 67080925 T^{8} + 113002070 T^{9} - 3673592740 T^{10} + 113002070 p T^{11} + 67080925 p^{2} T^{12} - 4213739 p^{3} T^{13} - 1221192 p^{4} T^{14} + 53321 p^{5} T^{15} + 18650 p^{6} T^{16} - 267 p^{7} T^{17} - 190 p^{8} T^{18} - p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 67 | \( 1 - 235 T^{2} + 22580 T^{4} - 521263 T^{6} - 91121829 T^{8} + 8945612102 T^{10} - 157942111692 T^{12} - 25295186066830 T^{14} + 1931740701942333 T^{16} - 28515711949759585 T^{18} - 2195444246537719120 T^{20} - 28515711949759585 p^{2} T^{22} + 1931740701942333 p^{4} T^{24} - 25295186066830 p^{6} T^{26} - 157942111692 p^{8} T^{28} + 8945612102 p^{10} T^{30} - 91121829 p^{12} T^{32} - 521263 p^{14} T^{34} + 22580 p^{16} T^{36} - 235 p^{18} T^{38} + p^{20} T^{40} \) | |
| 71 | \( ( 1 + T - 191 T^{2} + 584 T^{3} + 15245 T^{4} - 93875 T^{5} - 1039721 T^{6} + 1128481 T^{7} + 105345013 T^{8} + 160195123 T^{9} - 9264636086 T^{10} + 160195123 p T^{11} + 105345013 p^{2} T^{12} + 1128481 p^{3} T^{13} - 1039721 p^{4} T^{14} - 93875 p^{5} T^{15} + 15245 p^{6} T^{16} + 584 p^{7} T^{17} - 191 p^{8} T^{18} + p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 73 | \( 1 - 503 T^{2} + 133553 T^{4} - 24259792 T^{6} + 3335039790 T^{8} - 362534625048 T^{10} + 31777405459672 T^{12} - 2259723013851214 T^{14} + 131650094778511764 T^{16} - 6746769496613478970 T^{18} + \)\(40\!\cdots\!74\)\( T^{20} - 6746769496613478970 p^{2} T^{22} + 131650094778511764 p^{4} T^{24} - 2259723013851214 p^{6} T^{26} + 31777405459672 p^{8} T^{28} - 362534625048 p^{10} T^{30} + 3335039790 p^{12} T^{32} - 24259792 p^{14} T^{34} + 133553 p^{16} T^{36} - 503 p^{18} T^{38} + p^{20} T^{40} \) | |
| 79 | \( ( 1 - 8 T - 182 T^{2} + 1064 T^{3} + 19019 T^{4} - 58000 T^{5} - 1525960 T^{6} + 3698288 T^{7} + 86943629 T^{8} - 191150392 T^{9} - 4365214726 T^{10} - 191150392 p T^{11} + 86943629 p^{2} T^{12} + 3698288 p^{3} T^{13} - 1525960 p^{4} T^{14} - 58000 p^{5} T^{15} + 19019 p^{6} T^{16} + 1064 p^{7} T^{17} - 182 p^{8} T^{18} - 8 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 83 | \( ( 1 + 507 T^{2} + 122231 T^{4} + 18724815 T^{6} + 2102072492 T^{8} + 190434605588 T^{10} + 2102072492 p^{2} T^{12} + 18724815 p^{4} T^{14} + 122231 p^{6} T^{16} + 507 p^{8} T^{18} + p^{10} T^{20} )^{2} \) | |
| 89 | \( ( 1 - 20 T + 34 T^{2} + 1556 T^{3} - 12469 T^{4} + 136264 T^{5} - 2065736 T^{6} + 5978128 T^{7} + 171717517 T^{8} - 1450257668 T^{9} + 4377047314 T^{10} - 1450257668 p T^{11} + 171717517 p^{2} T^{12} + 5978128 p^{3} T^{13} - 2065736 p^{4} T^{14} + 136264 p^{5} T^{15} - 12469 p^{6} T^{16} + 1556 p^{7} T^{17} + 34 p^{8} T^{18} - 20 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 97 | \( 1 - 642 T^{2} + 205412 T^{4} - 46027862 T^{6} + 8409057324 T^{8} - 1338659596956 T^{10} + 190448104827077 T^{12} - 24537963901843259 T^{14} + 2896011234112033380 T^{16} - \)\(31\!\cdots\!08\)\( T^{18} + \)\(31\!\cdots\!78\)\( T^{20} - \)\(31\!\cdots\!08\)\( p^{2} T^{22} + 2896011234112033380 p^{4} T^{24} - 24537963901843259 p^{6} T^{26} + 190448104827077 p^{8} T^{28} - 1338659596956 p^{10} T^{30} + 8409057324 p^{12} T^{32} - 46027862 p^{14} T^{34} + 205412 p^{16} T^{36} - 642 p^{18} T^{38} + p^{20} T^{40} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{40} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−1.84959163279818238054372253874, −1.83808933556959734894152391541, −1.78472206673415593598038184121, −1.77576403229660320455935181679, −1.73843291023829082473626473155, −1.66400604201829655175037242680, −1.56771897758567467692143218410, −1.42456140929434165814599837784, −1.39483464351349719680126395963, −1.37991742072317234844212957327, −1.26152020353580139009204990898, −1.20509539312166740212599407145, −1.18469787726902202159307723812, −1.14128873852603020942789396092, −1.02637183534941131803959385383, −0.915402674481901085167559164739, −0.902198720099450803939195994870, −0.854166870613623228559787421149, −0.75085873550615587350148426439, −0.52500707971126980228598242998, −0.50960587103059906189934030604, −0.44523806756299251153532478942, −0.22577860125079295931426759853, −0.21844597315166422374873537165, −0.11158201883957409702420438940, 0.11158201883957409702420438940, 0.21844597315166422374873537165, 0.22577860125079295931426759853, 0.44523806756299251153532478942, 0.50960587103059906189934030604, 0.52500707971126980228598242998, 0.75085873550615587350148426439, 0.854166870613623228559787421149, 0.902198720099450803939195994870, 0.915402674481901085167559164739, 1.02637183534941131803959385383, 1.14128873852603020942789396092, 1.18469787726902202159307723812, 1.20509539312166740212599407145, 1.26152020353580139009204990898, 1.37991742072317234844212957327, 1.39483464351349719680126395963, 1.42456140929434165814599837784, 1.56771897758567467692143218410, 1.66400604201829655175037242680, 1.73843291023829082473626473155, 1.77576403229660320455935181679, 1.78472206673415593598038184121, 1.83808933556959734894152391541, 1.84959163279818238054372253874
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.