Dirichlet series
| L(s) = 1 | + 2-s − 15·3-s + 10·4-s + 25·5-s − 15·6-s + 56·7-s − 31·8-s + 90·9-s + 25·10-s + 33·11-s − 150·12-s − 46·13-s + 56·14-s − 375·15-s + 23·16-s + 136·17-s + 90·18-s + 39·19-s + 250·20-s − 840·21-s + 33·22-s + 133·23-s + 465·24-s + 250·25-s − 46·26-s − 135·27-s + 560·28-s + ⋯ |
| L(s) = 1 | + 0.353·2-s − 2.88·3-s + 5/4·4-s + 2.23·5-s − 1.02·6-s + 3.02·7-s − 1.37·8-s + 10/3·9-s + 0.790·10-s + 0.904·11-s − 3.60·12-s − 0.981·13-s + 1.06·14-s − 6.45·15-s + 0.359·16-s + 1.94·17-s + 1.17·18-s + 0.470·19-s + 2.79·20-s − 8.72·21-s + 0.319·22-s + 1.20·23-s + 3.95·24-s + 2·25-s − 0.346·26-s − 0.962·27-s + 3.77·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{10} \cdot 5^{10} \cdot 7^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{10} \cdot 5^{10} \cdot 7^{10}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(3^{10} \cdot 5^{10} \cdot 7^{10}\) |
| Sign: | $1$ |
| Analytic conductor: | \(8.32824\times 10^{7}\) |
| Root analytic conductor: | \(2.48901\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((20,\ 3^{10} \cdot 5^{10} \cdot 7^{10} ,\ ( \ : [3/2]^{10} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(19.66236177\) |
| \(L(\frac12)\) | \(\approx\) | \(19.66236177\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( ( 1 + p T + p^{2} T^{2} )^{5} \) |
| 5 | \( ( 1 - p T + p^{2} T^{2} )^{5} \) | |
| 7 | \( 1 - 8 p T + 1761 T^{2} - 6700 p T^{3} + 147635 p T^{4} - 397668 p^{2} T^{5} + 147635 p^{4} T^{6} - 6700 p^{7} T^{7} + 1761 p^{9} T^{8} - 8 p^{13} T^{9} + p^{15} T^{10} \) | |
| good | 2 | \( 1 - T - 9 T^{2} + 25 p T^{3} - 7 p T^{4} - 25 p^{4} T^{5} + 71 p^{4} T^{6} + 31 p^{3} T^{7} - 1123 p^{3} T^{8} + 105 p^{4} T^{9} + 1457 p^{4} T^{10} + 105 p^{7} T^{11} - 1123 p^{9} T^{12} + 31 p^{12} T^{13} + 71 p^{16} T^{14} - 25 p^{19} T^{15} - 7 p^{19} T^{16} + 25 p^{22} T^{17} - 9 p^{24} T^{18} - p^{27} T^{19} + p^{30} T^{20} \) |
| 11 | \( 1 - 3 p T - 670 T^{2} + 29721 T^{3} - 422065 T^{4} + 60807414 T^{5} + 3843117816 T^{6} - 264648812490 T^{7} + 484736115169 T^{8} + 178222331833149 T^{9} - 2105070196908054 T^{10} + 178222331833149 p^{3} T^{11} + 484736115169 p^{6} T^{12} - 264648812490 p^{9} T^{13} + 3843117816 p^{12} T^{14} + 60807414 p^{15} T^{15} - 422065 p^{18} T^{16} + 29721 p^{21} T^{17} - 670 p^{24} T^{18} - 3 p^{28} T^{19} + p^{30} T^{20} \) | |
| 13 | \( ( 1 + 23 T + 5115 T^{2} + 67458 T^{3} + 14193793 T^{4} + 48898957 T^{5} + 14193793 p^{3} T^{6} + 67458 p^{6} T^{7} + 5115 p^{9} T^{8} + 23 p^{12} T^{9} + p^{15} T^{10} )^{2} \) | |
| 17 | \( 1 - 8 p T + 17627 T^{2} - 1261192 T^{3} + 51283667 T^{4} + 1474736384 T^{5} - 302802583826 T^{6} + 22322448865408 T^{7} + 436909427829105 T^{8} - 170853787341953928 T^{9} + 16229413764409058493 T^{10} - 170853787341953928 p^{3} T^{11} + 436909427829105 p^{6} T^{12} + 22322448865408 p^{9} T^{13} - 302802583826 p^{12} T^{14} + 1474736384 p^{15} T^{15} + 51283667 p^{18} T^{16} - 1261192 p^{21} T^{17} + 17627 p^{24} T^{18} - 8 p^{28} T^{19} + p^{30} T^{20} \) | |
| 19 | \( 1 - 39 T - 520 p T^{2} + 624565 T^{3} + 34233842 T^{4} - 285070799 p T^{5} + 257964611298 T^{6} + 32510372903289 T^{7} - 248672962530545 p T^{8} - 95041416218862726 T^{9} + 40077725082249793964 T^{10} - 95041416218862726 p^{3} T^{11} - 248672962530545 p^{7} T^{12} + 32510372903289 p^{9} T^{13} + 257964611298 p^{12} T^{14} - 285070799 p^{16} T^{15} + 34233842 p^{18} T^{16} + 624565 p^{21} T^{17} - 520 p^{25} T^{18} - 39 p^{27} T^{19} + p^{30} T^{20} \) | |
| 23 | \( 1 - 133 T - 7186 T^{2} - 4075423 T^{3} + 803226335 T^{4} + 26486527262 T^{5} + 7194420605464 T^{6} - 2387572288321298 T^{7} - 21820426257674127 T^{8} - 4470686607756087351 T^{9} + \)\(40\!\cdots\!02\)\( T^{10} - 4470686607756087351 p^{3} T^{11} - 21820426257674127 p^{6} T^{12} - 2387572288321298 p^{9} T^{13} + 7194420605464 p^{12} T^{14} + 26486527262 p^{15} T^{15} + 803226335 p^{18} T^{16} - 4075423 p^{21} T^{17} - 7186 p^{24} T^{18} - 133 p^{27} T^{19} + p^{30} T^{20} \) | |
| 29 | \( ( 1 - 272 T + 89713 T^{2} - 18586464 T^{3} + 4032454334 T^{4} - 595833637376 T^{5} + 4032454334 p^{3} T^{6} - 18586464 p^{6} T^{7} + 89713 p^{9} T^{8} - 272 p^{12} T^{9} + p^{15} T^{10} )^{2} \) | |
| 31 | \( 1 - 430 T + 15907 T^{2} + 21798754 T^{3} - 2419402776 T^{4} - 860625141498 T^{5} + 180970103191009 T^{6} + 11431707811370410 T^{7} - 5289476269692339797 T^{8} - \)\(20\!\cdots\!76\)\( T^{9} + \)\(18\!\cdots\!32\)\( T^{10} - \)\(20\!\cdots\!76\)\( p^{3} T^{11} - 5289476269692339797 p^{6} T^{12} + 11431707811370410 p^{9} T^{13} + 180970103191009 p^{12} T^{14} - 860625141498 p^{15} T^{15} - 2419402776 p^{18} T^{16} + 21798754 p^{21} T^{17} + 15907 p^{24} T^{18} - 430 p^{27} T^{19} + p^{30} T^{20} \) | |
| 37 | \( 1 + 3 T - 197874 T^{2} - 9191603 T^{3} + 21726300338 T^{4} + 1331144162729 T^{5} - 1659517612754472 T^{6} - 84598271217222955 T^{7} + \)\(10\!\cdots\!81\)\( T^{8} + \)\(20\!\cdots\!78\)\( T^{9} - \)\(53\!\cdots\!56\)\( T^{10} + \)\(20\!\cdots\!78\)\( p^{3} T^{11} + \)\(10\!\cdots\!81\)\( p^{6} T^{12} - 84598271217222955 p^{9} T^{13} - 1659517612754472 p^{12} T^{14} + 1331144162729 p^{15} T^{15} + 21726300338 p^{18} T^{16} - 9191603 p^{21} T^{17} - 197874 p^{24} T^{18} + 3 p^{27} T^{19} + p^{30} T^{20} \) | |
| 41 | \( ( 1 + 627 T + 298461 T^{2} + 100221404 T^{3} + 28792732854 T^{4} + 7903095350774 T^{5} + 28792732854 p^{3} T^{6} + 100221404 p^{6} T^{7} + 298461 p^{9} T^{8} + 627 p^{12} T^{9} + p^{15} T^{10} )^{2} \) | |
| 43 | \( ( 1 - 108 T + 178357 T^{2} - 39070996 T^{3} + 22552051325 T^{4} - 3630162418200 T^{5} + 22552051325 p^{3} T^{6} - 39070996 p^{6} T^{7} + 178357 p^{9} T^{8} - 108 p^{12} T^{9} + p^{15} T^{10} )^{2} \) | |
| 47 | \( 1 - 553 T - 156682 T^{2} + 109233557 T^{3} + 28024627643 T^{4} - 16290620541022 T^{5} - 2638343597881880 T^{6} + 1559174107875594658 T^{7} + 79078601331925068813 T^{8} - \)\(43\!\cdots\!55\)\( T^{9} - \)\(90\!\cdots\!10\)\( T^{10} - \)\(43\!\cdots\!55\)\( p^{3} T^{11} + 79078601331925068813 p^{6} T^{12} + 1559174107875594658 p^{9} T^{13} - 2638343597881880 p^{12} T^{14} - 16290620541022 p^{15} T^{15} + 28024627643 p^{18} T^{16} + 109233557 p^{21} T^{17} - 156682 p^{24} T^{18} - 553 p^{27} T^{19} + p^{30} T^{20} \) | |
| 53 | \( 1 - 1135 T + 344368 T^{2} + 107494169 T^{3} - 79422582717 T^{4} + 1522579460342 T^{5} + 9430009439322196 T^{6} - 3330261248690177614 T^{7} + \)\(61\!\cdots\!73\)\( T^{8} + \)\(99\!\cdots\!43\)\( T^{9} - \)\(11\!\cdots\!60\)\( T^{10} + \)\(99\!\cdots\!43\)\( p^{3} T^{11} + \)\(61\!\cdots\!73\)\( p^{6} T^{12} - 3330261248690177614 p^{9} T^{13} + 9430009439322196 p^{12} T^{14} + 1522579460342 p^{15} T^{15} - 79422582717 p^{18} T^{16} + 107494169 p^{21} T^{17} + 344368 p^{24} T^{18} - 1135 p^{27} T^{19} + p^{30} T^{20} \) | |
| 59 | \( 1 - 332 T - 320471 T^{2} + 243978636 T^{3} - 20389370525 T^{4} - 31533377981208 T^{5} + 20583073309028818 T^{6} - 8827744675821656792 T^{7} + \)\(10\!\cdots\!81\)\( T^{8} + \)\(16\!\cdots\!84\)\( T^{9} - \)\(12\!\cdots\!45\)\( T^{10} + \)\(16\!\cdots\!84\)\( p^{3} T^{11} + \)\(10\!\cdots\!81\)\( p^{6} T^{12} - 8827744675821656792 p^{9} T^{13} + 20583073309028818 p^{12} T^{14} - 31533377981208 p^{15} T^{15} - 20389370525 p^{18} T^{16} + 243978636 p^{21} T^{17} - 320471 p^{24} T^{18} - 332 p^{27} T^{19} + p^{30} T^{20} \) | |
| 61 | \( 1 - 584 T - 684009 T^{2} + 304553272 T^{3} + 343910273991 T^{4} - 91891274199664 T^{5} - 134703419439176310 T^{6} + 18304200035324266544 T^{7} + \)\(42\!\cdots\!49\)\( T^{8} - \)\(18\!\cdots\!52\)\( T^{9} - \)\(10\!\cdots\!07\)\( T^{10} - \)\(18\!\cdots\!52\)\( p^{3} T^{11} + \)\(42\!\cdots\!49\)\( p^{6} T^{12} + 18304200035324266544 p^{9} T^{13} - 134703419439176310 p^{12} T^{14} - 91891274199664 p^{15} T^{15} + 343910273991 p^{18} T^{16} + 304553272 p^{21} T^{17} - 684009 p^{24} T^{18} - 584 p^{27} T^{19} + p^{30} T^{20} \) | |
| 67 | \( 1 + 412 T - 633285 T^{2} - 520720460 T^{3} + 54577110124 T^{4} + 159720777575348 T^{5} + 31924035283508653 T^{6} + 10162758820279347212 T^{7} + \)\(13\!\cdots\!07\)\( T^{8} - \)\(68\!\cdots\!08\)\( T^{9} - \)\(11\!\cdots\!76\)\( T^{10} - \)\(68\!\cdots\!08\)\( p^{3} T^{11} + \)\(13\!\cdots\!07\)\( p^{6} T^{12} + 10162758820279347212 p^{9} T^{13} + 31924035283508653 p^{12} T^{14} + 159720777575348 p^{15} T^{15} + 54577110124 p^{18} T^{16} - 520720460 p^{21} T^{17} - 633285 p^{24} T^{18} + 412 p^{27} T^{19} + p^{30} T^{20} \) | |
| 71 | \( ( 1 + 2 p T + 1020275 T^{2} + 238977320 T^{3} + 602260486134 T^{4} + 107445186880572 T^{5} + 602260486134 p^{3} T^{6} + 238977320 p^{6} T^{7} + 1020275 p^{9} T^{8} + 2 p^{13} T^{9} + p^{15} T^{10} )^{2} \) | |
| 73 | \( 1 - 2074 T + 1006865 T^{2} - 22856718 T^{3} + 797613942736 T^{4} - 798346226345510 T^{5} - 93874831999907421 T^{6} + 62628506747320190882 T^{7} + \)\(25\!\cdots\!63\)\( T^{8} - \)\(66\!\cdots\!88\)\( T^{9} - \)\(65\!\cdots\!20\)\( T^{10} - \)\(66\!\cdots\!88\)\( p^{3} T^{11} + \)\(25\!\cdots\!63\)\( p^{6} T^{12} + 62628506747320190882 p^{9} T^{13} - 93874831999907421 p^{12} T^{14} - 798346226345510 p^{15} T^{15} + 797613942736 p^{18} T^{16} - 22856718 p^{21} T^{17} + 1006865 p^{24} T^{18} - 2074 p^{27} T^{19} + p^{30} T^{20} \) | |
| 79 | \( 1 + 28 T - 1713233 T^{2} + 137836596 T^{3} + 1524513275000 T^{4} - 195374153482332 T^{5} - 1026627193743889947 T^{6} + 79301893796380256380 T^{7} + \)\(59\!\cdots\!51\)\( T^{8} - \)\(90\!\cdots\!64\)\( T^{9} - \)\(31\!\cdots\!36\)\( T^{10} - \)\(90\!\cdots\!64\)\( p^{3} T^{11} + \)\(59\!\cdots\!51\)\( p^{6} T^{12} + 79301893796380256380 p^{9} T^{13} - 1026627193743889947 p^{12} T^{14} - 195374153482332 p^{15} T^{15} + 1524513275000 p^{18} T^{16} + 137836596 p^{21} T^{17} - 1713233 p^{24} T^{18} + 28 p^{27} T^{19} + p^{30} T^{20} \) | |
| 83 | \( ( 1 + 840 T + 1394751 T^{2} + 1097470000 T^{3} + 1235101554942 T^{4} + 765253616219632 T^{5} + 1235101554942 p^{3} T^{6} + 1097470000 p^{6} T^{7} + 1394751 p^{9} T^{8} + 840 p^{12} T^{9} + p^{15} T^{10} )^{2} \) | |
| 89 | \( 1 - 2978 T + 2191615 T^{2} - 357555834 T^{3} + 3692071571515 T^{4} - 5614366883406660 T^{5} + 955110819277629838 T^{6} - \)\(88\!\cdots\!84\)\( T^{7} + \)\(51\!\cdots\!73\)\( T^{8} - \)\(27\!\cdots\!42\)\( T^{9} - \)\(42\!\cdots\!91\)\( T^{10} - \)\(27\!\cdots\!42\)\( p^{3} T^{11} + \)\(51\!\cdots\!73\)\( p^{6} T^{12} - \)\(88\!\cdots\!84\)\( p^{9} T^{13} + 955110819277629838 p^{12} T^{14} - 5614366883406660 p^{15} T^{15} + 3692071571515 p^{18} T^{16} - 357555834 p^{21} T^{17} + 2191615 p^{24} T^{18} - 2978 p^{27} T^{19} + p^{30} T^{20} \) | |
| 97 | \( ( 1 + 2168 T + 2916885 T^{2} + 25224480 p T^{3} + 2052444429082 T^{4} + 1555642076834128 T^{5} + 2052444429082 p^{3} T^{6} + 25224480 p^{7} T^{7} + 2916885 p^{9} T^{8} + 2168 p^{12} T^{9} + p^{15} T^{10} )^{2} \) | |
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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
−5.05554832144558460110712981789, −4.93636741087899934971038082892, −4.89540705714652550659377813731, −4.83438310701449878636744108371, −4.77515867893000584875773763433, −4.35946362263168456374938764499, −4.35543215036150837976215809969, −4.10062791612587876313215087943, −3.65104283835051045912610044816, −3.53795278747074475669565950385, −3.53760449866986078181581866039, −3.19426784280878819957129325815, −3.01996160889091424919029683794, −2.80481376191898164010301756676, −2.59504994274735853723777519592, −2.47634049253834188835748739228, −2.30499947698394323297720048032, −1.98833265554828714979898295007, −1.85317445675150874227886538373, −1.63546013867335008644626599272, −1.12925712937965342921742587316, −1.02838299717990802885501872602, −0.990804800995632984903878582809, −0.71357183840031704398453938089, −0.45653746705348151244030522429, 0.45653746705348151244030522429, 0.71357183840031704398453938089, 0.990804800995632984903878582809, 1.02838299717990802885501872602, 1.12925712937965342921742587316, 1.63546013867335008644626599272, 1.85317445675150874227886538373, 1.98833265554828714979898295007, 2.30499947698394323297720048032, 2.47634049253834188835748739228, 2.59504994274735853723777519592, 2.80481376191898164010301756676, 3.01996160889091424919029683794, 3.19426784280878819957129325815, 3.53760449866986078181581866039, 3.53795278747074475669565950385, 3.65104283835051045912610044816, 4.10062791612587876313215087943, 4.35543215036150837976215809969, 4.35946362263168456374938764499, 4.77515867893000584875773763433, 4.83438310701449878636744108371, 4.89540705714652550659377813731, 4.93636741087899934971038082892, 5.05554832144558460110712981789
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.