Dirichlet series
| L(s) = 1 | + 8·2-s − 6·3-s + 28·4-s − 48·6-s + 48·8-s + 11·9-s − 168·12-s + 6·16-s + 88·18-s + 8·23-s − 288·24-s + 21·25-s + 6·27-s − 16·29-s + 12·31-s − 168·32-s + 308·36-s + 64·46-s − 6·47-s − 36·48-s − 9·49-s + 168·50-s + 48·54-s − 128·58-s + 36·59-s + 96·62-s − 384·64-s + ⋯ |
| L(s) = 1 | + 5.65·2-s − 3.46·3-s + 14·4-s − 19.5·6-s + 16.9·8-s + 11/3·9-s − 48.4·12-s + 3/2·16-s + 20.7·18-s + 1.66·23-s − 58.7·24-s + 21/5·25-s + 1.15·27-s − 2.97·29-s + 2.15·31-s − 29.6·32-s + 51.3·36-s + 9.43·46-s − 0.875·47-s − 5.19·48-s − 9/7·49-s + 23.7·50-s + 6.53·54-s − 16.8·58-s + 4.68·59-s + 12.1·62-s − 48·64-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{16} \cdot 23^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{16} \cdot 23^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{16} \cdot 7^{16} \cdot 23^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.64861\times 10^{6}\) |
| Root analytic conductor: | \(1.60349\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{16} \cdot 7^{16} \cdot 23^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.1531516119\) |
| \(L(\frac12)\) | \(\approx\) | \(0.1531516119\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - T + T^{2} )^{8} \) |
| 7 | \( 1 + 9 T^{2} - 4 p T^{4} - 279 T^{6} + 435 T^{8} - 279 p^{2} T^{10} - 4 p^{5} T^{12} + 9 p^{6} T^{14} + p^{8} T^{16} \) | |
| 23 | \( 1 - 8 T + 76 T^{2} - 192 T^{3} + 1418 T^{4} - 16 p T^{5} + 33072 T^{6} + 15368 T^{7} + 739315 T^{8} + 15368 p T^{9} + 33072 p^{2} T^{10} - 16 p^{4} T^{11} + 1418 p^{4} T^{12} - 192 p^{5} T^{13} + 76 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| good | 3 | \( ( 1 + p T + 8 T^{2} + 5 p T^{3} + 8 p T^{4} + 8 p T^{5} - 13 T^{6} - 20 p T^{7} - 155 T^{8} - 20 p^{2} T^{9} - 13 p^{2} T^{10} + 8 p^{4} T^{11} + 8 p^{5} T^{12} + 5 p^{6} T^{13} + 8 p^{6} T^{14} + p^{8} T^{15} + p^{8} T^{16} )^{2} \) |
| 5 | \( 1 - 21 T^{2} + 44 p T^{4} - 1557 T^{6} + 8008 T^{8} - 4878 p T^{10} - 23207 T^{12} + 778908 T^{14} - 5159699 T^{16} + 778908 p^{2} T^{18} - 23207 p^{4} T^{20} - 4878 p^{7} T^{22} + 8008 p^{8} T^{24} - 1557 p^{10} T^{26} + 44 p^{13} T^{28} - 21 p^{14} T^{30} + p^{16} T^{32} \) | |
| 11 | \( 1 + 24 T^{2} + 244 T^{4} + 822 T^{6} - 17045 T^{8} - 374247 T^{10} - 1671683 T^{12} + 20448081 T^{14} + 391625287 T^{16} + 20448081 p^{2} T^{18} - 1671683 p^{4} T^{20} - 374247 p^{6} T^{22} - 17045 p^{8} T^{24} + 822 p^{10} T^{26} + 244 p^{12} T^{28} + 24 p^{14} T^{30} + p^{16} T^{32} \) | |
| 13 | \( ( 1 - 49 T^{2} + 1004 T^{4} - 11475 T^{6} + 117191 T^{8} - 11475 p^{2} T^{10} + 1004 p^{4} T^{12} - 49 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 17 | \( 1 - 87 T^{2} + 4111 T^{4} - 127380 T^{6} + 2773534 T^{8} - 39896685 T^{10} + 253725484 T^{12} + 4019936976 T^{14} - 129792990803 T^{16} + 4019936976 p^{2} T^{18} + 253725484 p^{4} T^{20} - 39896685 p^{6} T^{22} + 2773534 p^{8} T^{24} - 127380 p^{10} T^{26} + 4111 p^{12} T^{28} - 87 p^{14} T^{30} + p^{16} T^{32} \) | |
| 19 | \( 1 - 69 T^{2} + 2476 T^{4} - 62181 T^{6} + 1051420 T^{8} - 5491566 T^{10} - 315101939 T^{12} + 13166863692 T^{14} - 305587110539 T^{16} + 13166863692 p^{2} T^{18} - 315101939 p^{4} T^{20} - 5491566 p^{6} T^{22} + 1051420 p^{8} T^{24} - 62181 p^{10} T^{26} + 2476 p^{12} T^{28} - 69 p^{14} T^{30} + p^{16} T^{32} \) | |
| 29 | \( ( 1 + 4 T + 105 T^{2} + 300 T^{3} + 4393 T^{4} + 300 p T^{5} + 105 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{4} \) | |
| 31 | \( ( 1 - 6 T + 128 T^{2} - 696 T^{3} + 9203 T^{4} - 42585 T^{5} + 439233 T^{6} - 1785153 T^{7} + 15638981 T^{8} - 1785153 p T^{9} + 439233 p^{2} T^{10} - 42585 p^{3} T^{11} + 9203 p^{4} T^{12} - 696 p^{5} T^{13} + 128 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 37 | \( 1 + 234 T^{2} + 790 p T^{4} + 2563506 T^{6} + 176434561 T^{8} + 10178643303 T^{10} + 510233136685 T^{12} + 22588433362719 T^{14} + 887288864997523 T^{16} + 22588433362719 p^{2} T^{18} + 510233136685 p^{4} T^{20} + 10178643303 p^{6} T^{22} + 176434561 p^{8} T^{24} + 2563506 p^{10} T^{26} + 790 p^{13} T^{28} + 234 p^{14} T^{30} + p^{16} T^{32} \) | |
| 41 | \( ( 1 - 197 T^{2} + 15772 T^{4} - 707111 T^{6} + 26461951 T^{8} - 707111 p^{2} T^{10} + 15772 p^{4} T^{12} - 197 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 43 | \( ( 1 - 264 T^{2} + 32888 T^{4} - 2527551 T^{6} + 130892385 T^{8} - 2527551 p^{2} T^{10} + 32888 p^{4} T^{12} - 264 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 47 | \( ( 1 + 3 T + 76 T^{2} + 219 T^{3} + 2608 T^{4} - 10452 T^{5} - 80465 T^{6} - 1727478 T^{7} - 9364823 T^{8} - 1727478 p T^{9} - 80465 p^{2} T^{10} - 10452 p^{3} T^{11} + 2608 p^{4} T^{12} + 219 p^{5} T^{13} + 76 p^{6} T^{14} + 3 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 53 | \( 1 + 109 T^{2} + 3746 T^{4} - 70297 T^{6} - 7672384 T^{8} + 507596474 T^{10} + 34360415895 T^{12} - 2075915368374 T^{14} - 230333268032387 T^{16} - 2075915368374 p^{2} T^{18} + 34360415895 p^{4} T^{20} + 507596474 p^{6} T^{22} - 7672384 p^{8} T^{24} - 70297 p^{10} T^{26} + 3746 p^{12} T^{28} + 109 p^{14} T^{30} + p^{16} T^{32} \) | |
| 59 | \( ( 1 - 18 T + 280 T^{2} - 3096 T^{3} + 33529 T^{4} - 330525 T^{5} + 2959021 T^{6} - 25320627 T^{7} + 193895521 T^{8} - 25320627 p T^{9} + 2959021 p^{2} T^{10} - 330525 p^{3} T^{11} + 33529 p^{4} T^{12} - 3096 p^{5} T^{13} + 280 p^{6} T^{14} - 18 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 61 | \( 1 - 143 T^{2} + 9131 T^{4} - 89656 T^{6} - 30747310 T^{8} + 2670761891 T^{10} - 69633958116 T^{12} - 3255675829176 T^{14} + 421113566711821 T^{16} - 3255675829176 p^{2} T^{18} - 69633958116 p^{4} T^{20} + 2670761891 p^{6} T^{22} - 30747310 p^{8} T^{24} - 89656 p^{10} T^{26} + 9131 p^{12} T^{28} - 143 p^{14} T^{30} + p^{16} T^{32} \) | |
| 67 | \( 1 + 131 T^{2} + 12393 T^{4} + 409522 T^{6} - 2975920 T^{8} - 2049336531 T^{10} + 85073116420 T^{12} + 21082817976008 T^{14} + 2323677203911233 T^{16} + 21082817976008 p^{2} T^{18} + 85073116420 p^{4} T^{20} - 2049336531 p^{6} T^{22} - 2975920 p^{8} T^{24} + 409522 p^{10} T^{26} + 12393 p^{12} T^{28} + 131 p^{14} T^{30} + p^{16} T^{32} \) | |
| 71 | \( ( 1 + 13 T + 330 T^{2} + 2805 T^{3} + 36763 T^{4} + 2805 p T^{5} + 330 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} )^{4} \) | |
| 73 | \( ( 1 - 12 T + 212 T^{2} - 1968 T^{3} + 18029 T^{4} - 26481 T^{5} - 159573 T^{6} + 12236655 T^{7} - 106391125 T^{8} + 12236655 p T^{9} - 159573 p^{2} T^{10} - 26481 p^{3} T^{11} + 18029 p^{4} T^{12} - 1968 p^{5} T^{13} + 212 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 79 | \( 1 + 309 T^{2} + 38899 T^{4} + 3629832 T^{6} + 425685682 T^{8} + 538002081 p T^{10} + 3080793746956 T^{12} + 265239883313640 T^{14} + 24463983360059053 T^{16} + 265239883313640 p^{2} T^{18} + 3080793746956 p^{4} T^{20} + 538002081 p^{7} T^{22} + 425685682 p^{8} T^{24} + 3629832 p^{10} T^{26} + 38899 p^{12} T^{28} + 309 p^{14} T^{30} + p^{16} T^{32} \) | |
| 83 | \( ( 1 + 468 T^{2} + 106997 T^{4} + 15388152 T^{6} + 1522400685 T^{8} + 15388152 p^{2} T^{10} + 106997 p^{4} T^{12} + 468 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 89 | \( 1 - 361 T^{2} + 61380 T^{4} - 6965969 T^{6} + 620585852 T^{8} - 40825114326 T^{10} + 1456858643149 T^{12} + 4300740561392 T^{14} - 3308905175667507 T^{16} + 4300740561392 p^{2} T^{18} + 1456858643149 p^{4} T^{20} - 40825114326 p^{6} T^{22} + 620585852 p^{8} T^{24} - 6965969 p^{10} T^{26} + 61380 p^{12} T^{28} - 361 p^{14} T^{30} + p^{16} T^{32} \) | |
| 97 | \( ( 1 - 124 T^{2} + 41849 T^{4} - 3529440 T^{6} + 610492013 T^{8} - 3529440 p^{2} T^{10} + 41849 p^{4} T^{12} - 124 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.40315553335043166198899207820, −3.33233974131551787492997329280, −3.21414019968349163069021178907, −3.02407497183484540445419981071, −2.89398810949506200080876307207, −2.86789035443006757047311345705, −2.75062683859895025633003967121, −2.73784565340108360657949106252, −2.72852990613516760878946861298, −2.60628709238562299731206157607, −2.57772952109510380461745327915, −2.57287581791995300234680747862, −2.39238038569114773970692172017, −2.35974589523856266995407892842, −1.83683731662926946361019555194, −1.70911562895900039234653316030, −1.69259765303818018665724547660, −1.52581527430685771937455537118, −1.47184863354212302951381440263, −1.24105577694972906786792514004, −1.19115452752006809458059855517, −0.72771580023753207505863273070, −0.62982557020075671258811033686, −0.54614618050650743619397588456, −0.03748518596492980902858070773, 0.03748518596492980902858070773, 0.54614618050650743619397588456, 0.62982557020075671258811033686, 0.72771580023753207505863273070, 1.19115452752006809458059855517, 1.24105577694972906786792514004, 1.47184863354212302951381440263, 1.52581527430685771937455537118, 1.69259765303818018665724547660, 1.70911562895900039234653316030, 1.83683731662926946361019555194, 2.35974589523856266995407892842, 2.39238038569114773970692172017, 2.57287581791995300234680747862, 2.57772952109510380461745327915, 2.60628709238562299731206157607, 2.72852990613516760878946861298, 2.73784565340108360657949106252, 2.75062683859895025633003967121, 2.86789035443006757047311345705, 2.89398810949506200080876307207, 3.02407497183484540445419981071, 3.21414019968349163069021178907, 3.33233974131551787492997329280, 3.40315553335043166198899207820
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.