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
| \(L(2)\) | \(\approx\) | \(154.8007180\) |
| \(L(\frac12)\) | \(\approx\) | \(154.8007180\) |
| \(L(\frac{5}{2})\) | 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
−5.10888145568238799067577258423, −5.06044731752337269234862956210, −4.86030637029309612321018450784, −4.77479722836887099753179412606, −4.14844552906101273265005299630, −4.07298826479745293117054001121, −4.02819183758594387174035246374, −3.77669688988645145085316105998, −3.70795399960364442790559628130, −3.69799060440442134815543456491, −3.29910180872050006304963293809, −3.04068617154193777979897618394, −2.79306455443702338177852482022, −2.78674971994921559722705499755, −2.77856551342686772519634712802, −2.17885220313087044144120637342, −2.07140470923719495859051141070, −1.87431982926382245707977424244, −1.60773687503104671799500724727, −1.48979554034648124568603871584, −1.05306787406970586401934536607, −0.999433502649739041219744056893, −0.924802202868821414164549930099, −0.69071239505713741566432729649, −0.65008926217116654458751986923, 0.65008926217116654458751986923, 0.69071239505713741566432729649, 0.924802202868821414164549930099, 0.999433502649739041219744056893, 1.05306787406970586401934536607, 1.48979554034648124568603871584, 1.60773687503104671799500724727, 1.87431982926382245707977424244, 2.07140470923719495859051141070, 2.17885220313087044144120637342, 2.77856551342686772519634712802, 2.78674971994921559722705499755, 2.79306455443702338177852482022, 3.04068617154193777979897618394, 3.29910180872050006304963293809, 3.69799060440442134815543456491, 3.70795399960364442790559628130, 3.77669688988645145085316105998, 4.02819183758594387174035246374, 4.07298826479745293117054001121, 4.14844552906101273265005299630, 4.77479722836887099753179412606, 4.86030637029309612321018450784, 5.06044731752337269234862956210, 5.10888145568238799067577258423
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.