Dirichlet series
| L(s) = 1 | − 2·2-s − 8·4-s − 35·5-s + 22·7-s + 18·8-s + 70·10-s − 23·11-s + 96·13-s − 44·14-s + 41·16-s − 161·17-s + 279·19-s + 280·20-s + 46·22-s − 96·23-s + 700·25-s − 192·26-s − 176·28-s + 296·29-s + 244·31-s − 8·32-s + 322·34-s − 770·35-s + 404·37-s − 558·38-s − 630·40-s + 47·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 4-s − 3.13·5-s + 1.18·7-s + 0.795·8-s + 2.21·10-s − 0.630·11-s + 2.04·13-s − 0.839·14-s + 0.640·16-s − 2.29·17-s + 3.36·19-s + 3.13·20-s + 0.445·22-s − 0.870·23-s + 28/5·25-s − 1.44·26-s − 1.18·28-s + 1.89·29-s + 1.41·31-s − 0.0441·32-s + 1.62·34-s − 3.71·35-s + 1.79·37-s − 2.38·38-s − 2.49·40-s + 0.179·41-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{28} \cdot 5^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{28} \cdot 5^{7}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(14\) |
| Conductor: | \(3^{28} \cdot 5^{7}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.44884\times 10^{9}\) |
| Root analytic conductor: | \(4.88833\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((14,\ 3^{28} \cdot 5^{7} ,\ ( \ : [3/2]^{7} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(5.232068230\) |
| \(L(\frac12)\) | \(\approx\) | \(5.232068230\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 \) |
| 5 | \( ( 1 + p T )^{7} \) | |
| good | 2 | \( 1 + p T + 3 p^{2} T^{2} + 11 p T^{3} + 63 T^{4} + 3 p^{2} T^{5} + 7 p^{3} T^{6} - 27 p^{5} T^{7} + 7 p^{6} T^{8} + 3 p^{8} T^{9} + 63 p^{9} T^{10} + 11 p^{13} T^{11} + 3 p^{17} T^{12} + p^{19} T^{13} + p^{21} T^{14} \) |
| 7 | \( 1 - 22 T + 830 T^{2} - 19134 T^{3} + 426701 T^{4} - 7825194 T^{5} + 165254493 T^{6} - 347285588 p T^{7} + 165254493 p^{3} T^{8} - 7825194 p^{6} T^{9} + 426701 p^{9} T^{10} - 19134 p^{12} T^{11} + 830 p^{15} T^{12} - 22 p^{18} T^{13} + p^{21} T^{14} \) | |
| 11 | \( 1 + 23 T + 5151 T^{2} + 137494 T^{3} + 15125367 T^{4} + 32802843 p T^{5} + 29040094313 T^{6} + 601859871012 T^{7} + 29040094313 p^{3} T^{8} + 32802843 p^{7} T^{9} + 15125367 p^{9} T^{10} + 137494 p^{12} T^{11} + 5151 p^{15} T^{12} + 23 p^{18} T^{13} + p^{21} T^{14} \) | |
| 13 | \( 1 - 96 T + 12191 T^{2} - 828700 T^{3} + 67939809 T^{4} - 3670430048 T^{5} + 227859076023 T^{6} - 10004745881352 T^{7} + 227859076023 p^{3} T^{8} - 3670430048 p^{6} T^{9} + 67939809 p^{9} T^{10} - 828700 p^{12} T^{11} + 12191 p^{15} T^{12} - 96 p^{18} T^{13} + p^{21} T^{14} \) | |
| 17 | \( 1 + 161 T + 24747 T^{2} + 2620018 T^{3} + 242733069 T^{4} + 19012626063 T^{5} + 1428967389431 T^{6} + 5791825964988 p T^{7} + 1428967389431 p^{3} T^{8} + 19012626063 p^{6} T^{9} + 242733069 p^{9} T^{10} + 2620018 p^{12} T^{11} + 24747 p^{15} T^{12} + 161 p^{18} T^{13} + p^{21} T^{14} \) | |
| 19 | \( 1 - 279 T + 63455 T^{2} - 9627646 T^{3} + 1292727231 T^{4} - 139417672577 T^{5} + 13934197302465 T^{6} - 1188087181318116 T^{7} + 13934197302465 p^{3} T^{8} - 139417672577 p^{6} T^{9} + 1292727231 p^{9} T^{10} - 9627646 p^{12} T^{11} + 63455 p^{15} T^{12} - 279 p^{18} T^{13} + p^{21} T^{14} \) | |
| 23 | \( 1 + 96 T + 70730 T^{2} + 5128416 T^{3} + 2231256537 T^{4} + 129120969582 T^{5} + 1826857392719 p T^{6} + 1973532783601374 T^{7} + 1826857392719 p^{4} T^{8} + 129120969582 p^{6} T^{9} + 2231256537 p^{9} T^{10} + 5128416 p^{12} T^{11} + 70730 p^{15} T^{12} + 96 p^{18} T^{13} + p^{21} T^{14} \) | |
| 29 | \( 1 - 296 T + 154422 T^{2} - 29519614 T^{3} + 8749743807 T^{4} - 1202308386924 T^{5} + 280424950326131 T^{6} - 32104391095886106 T^{7} + 280424950326131 p^{3} T^{8} - 1202308386924 p^{6} T^{9} + 8749743807 p^{9} T^{10} - 29519614 p^{12} T^{11} + 154422 p^{15} T^{12} - 296 p^{18} T^{13} + p^{21} T^{14} \) | |
| 31 | \( 1 - 244 T + 138953 T^{2} - 21581892 T^{3} + 6820038797 T^{4} - 643944456252 T^{5} + 180689501328357 T^{6} - 12932535760119704 T^{7} + 180689501328357 p^{3} T^{8} - 643944456252 p^{6} T^{9} + 6820038797 p^{9} T^{10} - 21581892 p^{12} T^{11} + 138953 p^{15} T^{12} - 244 p^{18} T^{13} + p^{21} T^{14} \) | |
| 37 | \( 1 - 404 T + 7243 p T^{2} - 88433668 T^{3} + 32537859593 T^{4} - 8785146905212 T^{5} + 2408172804310695 T^{6} - 541287630181037208 T^{7} + 2408172804310695 p^{3} T^{8} - 8785146905212 p^{6} T^{9} + 32537859593 p^{9} T^{10} - 88433668 p^{12} T^{11} + 7243 p^{16} T^{12} - 404 p^{18} T^{13} + p^{21} T^{14} \) | |
| 41 | \( 1 - 47 T + 321096 T^{2} + 12021155 T^{3} + 45171631701 T^{4} + 4630735558038 T^{5} + 4063682212222409 T^{6} + 491807687844214791 T^{7} + 4063682212222409 p^{3} T^{8} + 4630735558038 p^{6} T^{9} + 45171631701 p^{9} T^{10} + 12021155 p^{12} T^{11} + 321096 p^{15} T^{12} - 47 p^{18} T^{13} + p^{21} T^{14} \) | |
| 43 | \( 1 - 525 T + 461651 T^{2} - 177053890 T^{3} + 91552352643 T^{4} - 27963570378443 T^{5} + 10910759641065633 T^{6} - 2737206494978679516 T^{7} + 10910759641065633 p^{3} T^{8} - 27963570378443 p^{6} T^{9} + 91552352643 p^{9} T^{10} - 177053890 p^{12} T^{11} + 461651 p^{15} T^{12} - 525 p^{18} T^{13} + p^{21} T^{14} \) | |
| 47 | \( 1 + 164 T + 458970 T^{2} + 110392306 T^{3} + 108169816881 T^{4} + 25717707674850 T^{5} + 17081669577666257 T^{6} + 3320863886913274956 T^{7} + 17081669577666257 p^{3} T^{8} + 25717707674850 p^{6} T^{9} + 108169816881 p^{9} T^{10} + 110392306 p^{12} T^{11} + 458970 p^{15} T^{12} + 164 p^{18} T^{13} + p^{21} T^{14} \) | |
| 53 | \( 1 + 506 T + 541623 T^{2} + 172442740 T^{3} + 132333548265 T^{4} + 35763427746966 T^{5} + 25492045997195975 T^{6} + 6232790423469661848 T^{7} + 25492045997195975 p^{3} T^{8} + 35763427746966 p^{6} T^{9} + 132333548265 p^{9} T^{10} + 172442740 p^{12} T^{11} + 541623 p^{15} T^{12} + 506 p^{18} T^{13} + p^{21} T^{14} \) | |
| 59 | \( 1 - 85 T + 331341 T^{2} - 32993294 T^{3} + 125055414789 T^{4} - 9577822572435 T^{5} + 29633014670734625 T^{6} - 2027875987639118244 T^{7} + 29633014670734625 p^{3} T^{8} - 9577822572435 p^{6} T^{9} + 125055414789 p^{9} T^{10} - 32993294 p^{12} T^{11} + 331341 p^{15} T^{12} - 85 p^{18} T^{13} + p^{21} T^{14} \) | |
| 61 | \( 1 - 828 T + 1137794 T^{2} - 655662598 T^{3} + 472897824279 T^{4} - 204252976588544 T^{5} + 110585022303384087 T^{6} - 44527407364853870622 T^{7} + 110585022303384087 p^{3} T^{8} - 204252976588544 p^{6} T^{9} + 472897824279 p^{9} T^{10} - 655662598 p^{12} T^{11} + 1137794 p^{15} T^{12} - 828 p^{18} T^{13} + p^{21} T^{14} \) | |
| 67 | \( 1 - 1093 T + 2017106 T^{2} - 1699166505 T^{3} + 1757369253407 T^{4} - 1166070992255454 T^{5} + 864466170577418349 T^{6} - \)\(45\!\cdots\!53\)\( T^{7} + 864466170577418349 p^{3} T^{8} - 1166070992255454 p^{6} T^{9} + 1757369253407 p^{9} T^{10} - 1699166505 p^{12} T^{11} + 2017106 p^{15} T^{12} - 1093 p^{18} T^{13} + p^{21} T^{14} \) | |
| 71 | \( 1 + 328 T + 1276365 T^{2} + 268163756 T^{3} + 813660594393 T^{4} + 105690507940488 T^{5} + 374340014427629213 T^{6} + 36698946645864837192 T^{7} + 374340014427629213 p^{3} T^{8} + 105690507940488 p^{6} T^{9} + 813660594393 p^{9} T^{10} + 268163756 p^{12} T^{11} + 1276365 p^{15} T^{12} + 328 p^{18} T^{13} + p^{21} T^{14} \) | |
| 73 | \( 1 - 2085 T + 4050563 T^{2} - 4984423486 T^{3} + 5631570806793 T^{4} - 4895773027898939 T^{5} + 3922468709989734339 T^{6} - \)\(25\!\cdots\!16\)\( T^{7} + 3922468709989734339 p^{3} T^{8} - 4895773027898939 p^{6} T^{9} + 5631570806793 p^{9} T^{10} - 4984423486 p^{12} T^{11} + 4050563 p^{15} T^{12} - 2085 p^{18} T^{13} + p^{21} T^{14} \) | |
| 79 | \( 1 - 2110 T + 3466397 T^{2} - 4074662340 T^{3} + 4200201567425 T^{4} - 3637140873668514 T^{5} + 2929970636702454981 T^{6} - \)\(20\!\cdots\!84\)\( T^{7} + 2929970636702454981 p^{3} T^{8} - 3637140873668514 p^{6} T^{9} + 4200201567425 p^{9} T^{10} - 4074662340 p^{12} T^{11} + 3466397 p^{15} T^{12} - 2110 p^{18} T^{13} + p^{21} T^{14} \) | |
| 83 | \( 1 + 1290 T + 3282374 T^{2} + 3191089158 T^{3} + 4600431414309 T^{4} + 3678070946162670 T^{5} + 3902415474964514617 T^{6} + \)\(26\!\cdots\!88\)\( T^{7} + 3902415474964514617 p^{3} T^{8} + 3678070946162670 p^{6} T^{9} + 4600431414309 p^{9} T^{10} + 3191089158 p^{12} T^{11} + 3282374 p^{15} T^{12} + 1290 p^{18} T^{13} + p^{21} T^{14} \) | |
| 89 | \( 1 - 3048 T + 5723102 T^{2} - 6323602086 T^{3} + 4534976524791 T^{4} - 780789931274556 T^{5} - 1899913405596725465 T^{6} + \)\(27\!\cdots\!50\)\( T^{7} - 1899913405596725465 p^{3} T^{8} - 780789931274556 p^{6} T^{9} + 4534976524791 p^{9} T^{10} - 6323602086 p^{12} T^{11} + 5723102 p^{15} T^{12} - 3048 p^{18} T^{13} + p^{21} T^{14} \) | |
| 97 | \( 1 - 1787 T + 4084243 T^{2} - 4789169974 T^{3} + 7296823674929 T^{4} - 7118143615887349 T^{5} + 8621884508580027771 T^{6} - \)\(72\!\cdots\!44\)\( T^{7} + 8621884508580027771 p^{3} T^{8} - 7118143615887349 p^{6} T^{9} + 7296823674929 p^{9} T^{10} - 4789169974 p^{12} T^{11} + 4084243 p^{15} T^{12} - 1787 p^{18} T^{13} + p^{21} T^{14} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{14} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.95335014449579350470096944370, −4.88855550172135036581938173551, −4.76177786178245292897715376416, −4.63407213640431552555192754310, −4.16463209098446068639553574114, −4.13546620660020685180015423952, −3.93456176029549216490959517391, −3.92329061184162276451883136560, −3.76939229706880367161895096057, −3.55269551529212876575052556696, −3.33075536230368044243124874591, −3.25031630265613507115998125630, −2.93701824830953775772850386143, −2.72549077903736379630300305387, −2.69859825181607269717792927407, −2.28402211704689360934952701559, −2.17658793496912945396131150029, −1.77040465006059863744687634367, −1.60077714316732582054481276997, −0.987617973996538241153943980658, −0.984246551985948356398853228608, −0.935079438675159345548818508351, −0.58877907005147055154530428951, −0.46470785885226256135455754199, −0.44145974167335774490302015681, 0.44145974167335774490302015681, 0.46470785885226256135455754199, 0.58877907005147055154530428951, 0.935079438675159345548818508351, 0.984246551985948356398853228608, 0.987617973996538241153943980658, 1.60077714316732582054481276997, 1.77040465006059863744687634367, 2.17658793496912945396131150029, 2.28402211704689360934952701559, 2.69859825181607269717792927407, 2.72549077903736379630300305387, 2.93701824830953775772850386143, 3.25031630265613507115998125630, 3.33075536230368044243124874591, 3.55269551529212876575052556696, 3.76939229706880367161895096057, 3.92329061184162276451883136560, 3.93456176029549216490959517391, 4.13546620660020685180015423952, 4.16463209098446068639553574114, 4.63407213640431552555192754310, 4.76177786178245292897715376416, 4.88855550172135036581938173551, 4.95335014449579350470096944370
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.