Dirichlet series
| L(s) = 1 | + 3·3-s − 3·4-s + 9-s − 9·12-s + 7·16-s − 6·27-s + 42·31-s − 3·36-s + 24·37-s + 36·43-s + 21·48-s − 2·49-s − 90·61-s − 20·64-s + 20·67-s − 48·73-s + 46·79-s + 81-s + 126·93-s − 42·103-s + 18·108-s + 24·109-s + 72·111-s − 60·121-s − 126·124-s + 127-s + 108·129-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 3/2·4-s + 1/3·9-s − 2.59·12-s + 7/4·16-s − 1.15·27-s + 7.54·31-s − 1/2·36-s + 3.94·37-s + 5.48·43-s + 3.03·48-s − 2/7·49-s − 11.5·61-s − 5/2·64-s + 2.44·67-s − 5.61·73-s + 5.17·79-s + 1/9·81-s + 13.0·93-s − 4.13·103-s + 1.73·108-s + 2.29·109-s + 6.83·111-s − 5.45·121-s − 11.3·124-s + 0.0887·127-s + 9.50·129-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{20} \cdot 5^{40} \cdot 7^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{20} \cdot 5^{40} \cdot 7^{20}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(40\) |
| Conductor: | \(3^{20} \cdot 5^{40} \cdot 7^{20}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.81014\times 10^{12}\) |
| Root analytic conductor: | \(2.04747\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((40,\ 3^{20} \cdot 5^{40} \cdot 7^{20} ,\ ( \ : [1/2]^{20} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(1.197735641\) |
| \(L(\frac12)\) | \(\approx\) | \(1.197735641\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 - p T + 8 T^{2} - 5 p T^{3} + 2 p^{2} T^{4} - 5 p^{2} T^{5} + 59 T^{6} - 49 p T^{7} + 271 T^{8} - 110 p T^{9} + 293 p T^{10} - 110 p^{2} T^{11} + 271 p^{2} T^{12} - 49 p^{4} T^{13} + 59 p^{4} T^{14} - 5 p^{7} T^{15} + 2 p^{8} T^{16} - 5 p^{8} T^{17} + 8 p^{8} T^{18} - p^{10} T^{19} + p^{10} T^{20} \) |
| 5 | \( 1 \) | |
| 7 | \( ( 1 + T^{2} - 13 T^{3} + p T^{4} - 19 p T^{5} + p^{2} T^{6} - 13 p^{2} T^{7} + p^{3} T^{8} + p^{5} T^{10} )^{2} \) | |
| good | 2 | \( 1 + 3 T^{2} + p T^{4} + 5 T^{6} + 9 T^{8} - 29 T^{10} + 47 T^{12} + 147 p T^{14} + 5 p^{3} T^{16} - 9 p^{4} T^{18} + 1017 T^{20} - 9 p^{6} T^{22} + 5 p^{7} T^{24} + 147 p^{7} T^{26} + 47 p^{8} T^{28} - 29 p^{10} T^{30} + 9 p^{12} T^{32} + 5 p^{14} T^{34} + p^{17} T^{36} + 3 p^{18} T^{38} + p^{20} T^{40} \) |
| 11 | \( 1 + 60 T^{2} + 2114 T^{4} + 53114 T^{6} + 1038747 T^{8} + 16310488 T^{10} + 209475293 T^{12} + 2203457790 T^{14} + 19212402091 T^{16} + 148735009794 T^{18} + 1341299566641 T^{20} + 148735009794 p^{2} T^{22} + 19212402091 p^{4} T^{24} + 2203457790 p^{6} T^{26} + 209475293 p^{8} T^{28} + 16310488 p^{10} T^{30} + 1038747 p^{12} T^{32} + 53114 p^{14} T^{34} + 2114 p^{16} T^{36} + 60 p^{18} T^{38} + p^{20} T^{40} \) | |
| 13 | \( ( 1 - 58 T^{2} + 159 p T^{4} - 50385 T^{6} + 943419 T^{8} - 13724229 T^{10} + 943419 p^{2} T^{12} - 50385 p^{4} T^{14} + 159 p^{7} T^{16} - 58 p^{8} T^{18} + p^{10} T^{20} )^{2} \) | |
| 17 | \( 1 - 4 p T^{2} + 1606 T^{4} - 19422 T^{6} + 550335 T^{8} - 18001236 T^{10} + 282479529 T^{12} - 4249504470 T^{14} + 113206321539 T^{16} - 2028696191474 T^{18} + 28292432369185 T^{20} - 2028696191474 p^{2} T^{22} + 113206321539 p^{4} T^{24} - 4249504470 p^{6} T^{26} + 282479529 p^{8} T^{28} - 18001236 p^{10} T^{30} + 550335 p^{12} T^{32} - 19422 p^{14} T^{34} + 1606 p^{16} T^{36} - 4 p^{19} T^{38} + p^{20} T^{40} \) | |
| 19 | \( ( 1 + 59 T^{2} + 1650 T^{4} + 38478 T^{6} + 13230 T^{7} + 896922 T^{8} + 780570 T^{9} + 18506640 T^{10} + 780570 p T^{11} + 896922 p^{2} T^{12} + 13230 p^{3} T^{13} + 38478 p^{4} T^{14} + 1650 p^{6} T^{16} + 59 p^{8} T^{18} + p^{10} T^{20} )^{2} \) | |
| 23 | \( 1 + 99 T^{2} + 4250 T^{4} + 102299 T^{6} + 1576890 T^{8} + 17142031 T^{10} + 181616261 T^{12} + 10191303513 T^{14} + 591206119363 T^{16} + 862389579114 p T^{18} + 930279860055 p^{2} T^{20} + 862389579114 p^{3} T^{22} + 591206119363 p^{4} T^{24} + 10191303513 p^{6} T^{26} + 181616261 p^{8} T^{28} + 17142031 p^{10} T^{30} + 1576890 p^{12} T^{32} + 102299 p^{14} T^{34} + 4250 p^{16} T^{36} + 99 p^{18} T^{38} + p^{20} T^{40} \) | |
| 29 | \( ( 1 - 132 T^{2} + 8782 T^{4} - 413831 T^{6} + 16000213 T^{8} - 513486626 T^{10} + 16000213 p^{2} T^{12} - 413831 p^{4} T^{14} + 8782 p^{6} T^{16} - 132 p^{8} T^{18} + p^{10} T^{20} )^{2} \) | |
| 31 | \( ( 1 - 21 T + 326 T^{2} - 3759 T^{3} + 37146 T^{4} - 10221 p T^{5} + 2454105 T^{6} - 17262699 T^{7} + 113223339 T^{8} - 689960868 T^{9} + 3971445087 T^{10} - 689960868 p T^{11} + 113223339 p^{2} T^{12} - 17262699 p^{3} T^{13} + 2454105 p^{4} T^{14} - 10221 p^{6} T^{15} + 37146 p^{6} T^{16} - 3759 p^{7} T^{17} + 326 p^{8} T^{18} - 21 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 37 | \( ( 1 - 12 T - 49 T^{2} + 490 T^{3} + 7278 T^{4} - 25228 T^{5} - 430924 T^{6} + 718902 T^{7} + 17278462 T^{8} + 143460 p T^{9} - 787415472 T^{10} + 143460 p^{2} T^{11} + 17278462 p^{2} T^{12} + 718902 p^{3} T^{13} - 430924 p^{4} T^{14} - 25228 p^{5} T^{15} + 7278 p^{6} T^{16} + 490 p^{7} T^{17} - 49 p^{8} T^{18} - 12 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 41 | \( ( 1 + 149 T^{2} + 12123 T^{4} + 648375 T^{6} + 26231484 T^{8} + 1031235516 T^{10} + 26231484 p^{2} T^{12} + 648375 p^{4} T^{14} + 12123 p^{6} T^{16} + 149 p^{8} T^{18} + p^{10} T^{20} )^{2} \) | |
| 43 | \( ( 1 - 9 T + 103 T^{2} - 589 T^{3} + 6250 T^{4} - 34300 T^{5} + 6250 p T^{6} - 589 p^{2} T^{7} + 103 p^{3} T^{8} - 9 p^{4} T^{9} + p^{5} T^{10} )^{4} \) | |
| 47 | \( 1 - 320 T^{2} + 55258 T^{4} - 6481134 T^{6} + 564305583 T^{8} - 37299364464 T^{10} + 1825870594845 T^{12} - 57025827049134 T^{14} + 130434887196171 T^{16} + 112418780517368770 T^{18} - 7605195418763792351 T^{20} + 112418780517368770 p^{2} T^{22} + 130434887196171 p^{4} T^{24} - 57025827049134 p^{6} T^{26} + 1825870594845 p^{8} T^{28} - 37299364464 p^{10} T^{30} + 564305583 p^{12} T^{32} - 6481134 p^{14} T^{34} + 55258 p^{16} T^{36} - 320 p^{18} T^{38} + p^{20} T^{40} \) | |
| 53 | \( 1 + 348 T^{2} + 65006 T^{4} + 8168402 T^{6} + 754331535 T^{8} + 52520483476 T^{10} + 2690516083409 T^{12} + 86297805646650 T^{14} - 153773100809741 T^{16} - 235981740156615306 T^{18} - 17360136557008258575 T^{20} - 235981740156615306 p^{2} T^{22} - 153773100809741 p^{4} T^{24} + 86297805646650 p^{6} T^{26} + 2690516083409 p^{8} T^{28} + 52520483476 p^{10} T^{30} + 754331535 p^{12} T^{32} + 8168402 p^{14} T^{34} + 65006 p^{16} T^{36} + 348 p^{18} T^{38} + p^{20} T^{40} \) | |
| 59 | \( 1 - 281 T^{2} + 43822 T^{4} - 4385565 T^{6} + 4988346 p T^{8} - 11205257169 T^{10} - 28140925407 T^{12} + 33950080736241 T^{14} - 2072476947516237 T^{16} + 52044476827656370 T^{18} - 492615579799524713 T^{20} + 52044476827656370 p^{2} T^{22} - 2072476947516237 p^{4} T^{24} + 33950080736241 p^{6} T^{26} - 28140925407 p^{8} T^{28} - 11205257169 p^{10} T^{30} + 4988346 p^{13} T^{32} - 4385565 p^{14} T^{34} + 43822 p^{16} T^{36} - 281 p^{18} T^{38} + p^{20} T^{40} \) | |
| 61 | \( ( 1 + 45 T + 1163 T^{2} + 360 p T^{3} + 334221 T^{4} + 4323609 T^{5} + 49194321 T^{6} + 504078669 T^{7} + 4728952065 T^{8} + 41065652565 T^{9} + 332029517802 T^{10} + 41065652565 p T^{11} + 4728952065 p^{2} T^{12} + 504078669 p^{3} T^{13} + 49194321 p^{4} T^{14} + 4323609 p^{5} T^{15} + 334221 p^{6} T^{16} + 360 p^{8} T^{17} + 1163 p^{8} T^{18} + 45 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 67 | \( ( 1 - 10 T - 194 T^{2} + 1284 T^{3} + 29439 T^{4} - 97824 T^{5} - 3224136 T^{6} + 5037786 T^{7} + 270996189 T^{8} - 90868948 T^{9} - 19829421434 T^{10} - 90868948 p T^{11} + 270996189 p^{2} T^{12} + 5037786 p^{3} T^{13} - 3224136 p^{4} T^{14} - 97824 p^{5} T^{15} + 29439 p^{6} T^{16} + 1284 p^{7} T^{17} - 194 p^{8} T^{18} - 10 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 71 | \( ( 1 - 417 T^{2} + 83080 T^{4} - 10852019 T^{6} + 1067389843 T^{8} - 84042150656 T^{10} + 1067389843 p^{2} T^{12} - 10852019 p^{4} T^{14} + 83080 p^{6} T^{16} - 417 p^{8} T^{18} + p^{10} T^{20} )^{2} \) | |
| 73 | \( ( 1 + 24 T + 572 T^{2} + 9120 T^{3} + 139305 T^{4} + 1702236 T^{5} + 20269650 T^{6} + 207027012 T^{7} + 2089522809 T^{8} + 18747463404 T^{9} + 169031552934 T^{10} + 18747463404 p T^{11} + 2089522809 p^{2} T^{12} + 207027012 p^{3} T^{13} + 20269650 p^{4} T^{14} + 1702236 p^{5} T^{15} + 139305 p^{6} T^{16} + 9120 p^{7} T^{17} + 572 p^{8} T^{18} + 24 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 79 | \( ( 1 - 23 T + 19 T^{2} + 1650 T^{3} + 22269 T^{4} - 308997 T^{5} - 2262141 T^{6} + 18480543 T^{7} + 3241509 p T^{8} - 288694607 T^{9} - 26891047898 T^{10} - 288694607 p T^{11} + 3241509 p^{3} T^{12} + 18480543 p^{3} T^{13} - 2262141 p^{4} T^{14} - 308997 p^{5} T^{15} + 22269 p^{6} T^{16} + 1650 p^{7} T^{17} + 19 p^{8} T^{18} - 23 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 83 | \( ( 1 + 200 T^{2} + 27258 T^{4} + 1670937 T^{6} + 63680961 T^{8} - 932587014 T^{10} + 63680961 p^{2} T^{12} + 1670937 p^{4} T^{14} + 27258 p^{6} T^{16} + 200 p^{8} T^{18} + p^{10} T^{20} )^{2} \) | |
| 89 | \( 1 - 548 T^{2} + 149638 T^{4} - 27617742 T^{6} + 3982290303 T^{8} - 496827970068 T^{10} + 645389915649 p T^{12} - 6298039134882534 T^{14} + 650541332844392067 T^{16} - 62975586452546418242 T^{18} + \)\(57\!\cdots\!53\)\( T^{20} - 62975586452546418242 p^{2} T^{22} + 650541332844392067 p^{4} T^{24} - 6298039134882534 p^{6} T^{26} + 645389915649 p^{9} T^{28} - 496827970068 p^{10} T^{30} + 3982290303 p^{12} T^{32} - 27617742 p^{14} T^{34} + 149638 p^{16} T^{36} - 548 p^{18} T^{38} + p^{20} T^{40} \) | |
| 97 | \( ( 1 - 673 T^{2} + 225150 T^{4} - 48497952 T^{6} + 7395278745 T^{8} - 830846466435 T^{10} + 7395278745 p^{2} T^{12} - 48497952 p^{4} T^{14} + 225150 p^{6} T^{16} - 673 p^{8} T^{18} + p^{10} T^{20} )^{2} \) | |
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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
−2.49626533802679722462184312015, −2.47476187274558089172529908777, −2.46706921142808290657280690553, −2.43566567677352751772516644496, −2.32722805188296696772730703469, −2.32568711360185095610722266091, −2.27017253411333420177266358105, −1.97860821401724933882185055064, −1.95538505079440529126489927259, −1.84014065756468585223434312137, −1.74950580766077526432704933288, −1.66436517452125308140821011554, −1.61860029609932339897986388949, −1.54900779627635093111738692562, −1.27608703098156438346946464861, −1.23179282213492080517932925955, −1.10906271609944392084578945038, −1.09915576848393248846034261154, −1.03819961434114609252795123804, −1.00966579435215258755029984030, −0.846362969491703340207734719995, −0.72609227626148214806139097716, −0.64241320062656623951696365498, −0.15657843726763388044727192067, −0.10981055316661673948656525705, 0.10981055316661673948656525705, 0.15657843726763388044727192067, 0.64241320062656623951696365498, 0.72609227626148214806139097716, 0.846362969491703340207734719995, 1.00966579435215258755029984030, 1.03819961434114609252795123804, 1.09915576848393248846034261154, 1.10906271609944392084578945038, 1.23179282213492080517932925955, 1.27608703098156438346946464861, 1.54900779627635093111738692562, 1.61860029609932339897986388949, 1.66436517452125308140821011554, 1.74950580766077526432704933288, 1.84014065756468585223434312137, 1.95538505079440529126489927259, 1.97860821401724933882185055064, 2.27017253411333420177266358105, 2.32568711360185095610722266091, 2.32722805188296696772730703469, 2.43566567677352751772516644496, 2.46706921142808290657280690553, 2.47476187274558089172529908777, 2.49626533802679722462184312015
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.