Dirichlet series
| L(s) = 1 | − 4·3-s + 4·5-s + 8·9-s + 8·11-s − 12·13-s − 16·15-s − 8·17-s − 4·19-s + 8·25-s + 4·27-s + 8·31-s − 32·33-s + 8·37-s + 48·39-s + 24·43-s + 32·45-s + 40·47-s − 8·49-s + 32·51-s + 32·53-s + 32·55-s + 16·57-s + 4·59-s + 20·61-s − 48·65-s − 32·67-s − 32·75-s + ⋯ |
| L(s) = 1 | − 2.30·3-s + 1.78·5-s + 8/3·9-s + 2.41·11-s − 3.32·13-s − 4.13·15-s − 1.94·17-s − 0.917·19-s + 8/5·25-s + 0.769·27-s + 1.43·31-s − 5.57·33-s + 1.31·37-s + 7.68·39-s + 3.65·43-s + 4.77·45-s + 5.83·47-s − 8/7·49-s + 4.48·51-s + 4.39·53-s + 4.31·55-s + 2.11·57-s + 0.520·59-s + 2.56·61-s − 5.95·65-s − 3.90·67-s − 3.69·75-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{128} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{128} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{128} \cdot 7^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.08916\times 10^{18}\) |
| Root analytic conductor: | \(3.78274\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{128} \cdot 7^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.4572157187\) |
| \(L(\frac12)\) | \(\approx\) | \(0.4572157187\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 7 | \( ( 1 + T^{2} )^{8} \) | |
| good | 3 | \( 1 + 4 T + 8 T^{2} - 4 T^{3} - 56 T^{4} - 124 T^{5} - 40 T^{6} + 428 T^{7} + 944 T^{8} + 284 T^{9} - 2680 T^{10} - 4652 T^{11} + 728 p T^{12} + 6404 p T^{13} + 22040 T^{14} - 30988 T^{15} - 115970 T^{16} - 30988 p T^{17} + 22040 p^{2} T^{18} + 6404 p^{4} T^{19} + 728 p^{5} T^{20} - 4652 p^{5} T^{21} - 2680 p^{6} T^{22} + 284 p^{7} T^{23} + 944 p^{8} T^{24} + 428 p^{9} T^{25} - 40 p^{10} T^{26} - 124 p^{11} T^{27} - 56 p^{12} T^{28} - 4 p^{13} T^{29} + 8 p^{14} T^{30} + 4 p^{15} T^{31} + p^{16} T^{32} \) |
| 5 | \( 1 - 4 T + 8 T^{2} + 4 p T^{3} - 136 T^{4} + 196 T^{5} + 504 T^{6} - 3236 T^{7} + 912 p T^{8} + 9524 T^{9} - 51544 T^{10} + 69356 T^{11} + 159256 T^{12} - 745668 T^{13} + 890072 T^{14} + 1893684 T^{15} - 8957506 T^{16} + 1893684 p T^{17} + 890072 p^{2} T^{18} - 745668 p^{3} T^{19} + 159256 p^{4} T^{20} + 69356 p^{5} T^{21} - 51544 p^{6} T^{22} + 9524 p^{7} T^{23} + 912 p^{9} T^{24} - 3236 p^{9} T^{25} + 504 p^{10} T^{26} + 196 p^{11} T^{27} - 136 p^{12} T^{28} + 4 p^{14} T^{29} + 8 p^{14} T^{30} - 4 p^{15} T^{31} + p^{16} T^{32} \) | |
| 11 | \( 1 - 8 T + 32 T^{2} - 216 T^{3} + 1472 T^{4} - 552 p T^{5} + 24800 T^{6} - 133864 T^{7} + 563580 T^{8} - 1921256 T^{9} + 8179104 T^{10} - 34333432 T^{11} + 111359808 T^{12} - 387202648 T^{13} + 1559355232 T^{14} - 5104139592 T^{15} + 15281881542 T^{16} - 5104139592 p T^{17} + 1559355232 p^{2} T^{18} - 387202648 p^{3} T^{19} + 111359808 p^{4} T^{20} - 34333432 p^{5} T^{21} + 8179104 p^{6} T^{22} - 1921256 p^{7} T^{23} + 563580 p^{8} T^{24} - 133864 p^{9} T^{25} + 24800 p^{10} T^{26} - 552 p^{12} T^{27} + 1472 p^{12} T^{28} - 216 p^{13} T^{29} + 32 p^{14} T^{30} - 8 p^{15} T^{31} + p^{16} T^{32} \) | |
| 13 | \( 1 + 12 T + 72 T^{2} + 268 T^{3} + 40 p T^{4} - 20 T^{5} - 136 p T^{6} + 7836 T^{7} + 115728 T^{8} + 538676 T^{9} + 1151304 T^{10} - 2236444 T^{11} - 34939128 T^{12} - 164829596 T^{13} - 435847976 T^{14} - 431042140 T^{15} + 741453246 T^{16} - 431042140 p T^{17} - 435847976 p^{2} T^{18} - 164829596 p^{3} T^{19} - 34939128 p^{4} T^{20} - 2236444 p^{5} T^{21} + 1151304 p^{6} T^{22} + 538676 p^{7} T^{23} + 115728 p^{8} T^{24} + 7836 p^{9} T^{25} - 136 p^{11} T^{26} - 20 p^{11} T^{27} + 40 p^{13} T^{28} + 268 p^{13} T^{29} + 72 p^{14} T^{30} + 12 p^{15} T^{31} + p^{16} T^{32} \) | |
| 17 | \( ( 1 + 4 T + 36 T^{2} + 12 p T^{3} + 1420 T^{4} + 6148 T^{5} + 35260 T^{6} + 140460 T^{7} + 681382 T^{8} + 140460 p T^{9} + 35260 p^{2} T^{10} + 6148 p^{3} T^{11} + 1420 p^{4} T^{12} + 12 p^{6} T^{13} + 36 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 19 | \( 1 + 4 T + 8 T^{2} + 36 T^{3} + 440 T^{4} + 2284 T^{5} + 6264 T^{6} + 43996 T^{7} + 7984 T^{8} - 270868 T^{9} + 288680 T^{10} + 9508092 T^{11} - 48857064 T^{12} - 534506956 T^{13} - 1232749480 T^{14} - 6811311532 T^{15} - 29688733186 T^{16} - 6811311532 p T^{17} - 1232749480 p^{2} T^{18} - 534506956 p^{3} T^{19} - 48857064 p^{4} T^{20} + 9508092 p^{5} T^{21} + 288680 p^{6} T^{22} - 270868 p^{7} T^{23} + 7984 p^{8} T^{24} + 43996 p^{9} T^{25} + 6264 p^{10} T^{26} + 2284 p^{11} T^{27} + 440 p^{12} T^{28} + 36 p^{13} T^{29} + 8 p^{14} T^{30} + 4 p^{15} T^{31} + p^{16} T^{32} \) | |
| 23 | \( 1 - 136 T^{2} + 9696 T^{4} - 480664 T^{6} + 18591548 T^{8} - 595598600 T^{10} + 16552944672 T^{12} - 416833926104 T^{14} + 9840485267974 T^{16} - 416833926104 p^{2} T^{18} + 16552944672 p^{4} T^{20} - 595598600 p^{6} T^{22} + 18591548 p^{8} T^{24} - 480664 p^{10} T^{26} + 9696 p^{12} T^{28} - 136 p^{14} T^{30} + p^{16} T^{32} \) | |
| 29 | \( 1 - 64 T^{3} - 1768 T^{4} - 64 T^{5} + 2048 T^{6} + 42624 T^{7} + 1265052 T^{8} - 282496 T^{9} + 894976 T^{10} - 30326976 T^{11} + 184626984 T^{12} + 598875456 T^{13} + 1858899968 T^{14} - 16846719232 T^{15} - 789123496314 T^{16} - 16846719232 p T^{17} + 1858899968 p^{2} T^{18} + 598875456 p^{3} T^{19} + 184626984 p^{4} T^{20} - 30326976 p^{5} T^{21} + 894976 p^{6} T^{22} - 282496 p^{7} T^{23} + 1265052 p^{8} T^{24} + 42624 p^{9} T^{25} + 2048 p^{10} T^{26} - 64 p^{11} T^{27} - 1768 p^{12} T^{28} - 64 p^{13} T^{29} + p^{16} T^{32} \) | |
| 31 | \( ( 1 - 4 T + 116 T^{2} - 292 T^{3} + 7660 T^{4} - 556 p T^{5} + 363628 T^{6} - 667188 T^{7} + 12635238 T^{8} - 667188 p T^{9} + 363628 p^{2} T^{10} - 556 p^{4} T^{11} + 7660 p^{4} T^{12} - 292 p^{5} T^{13} + 116 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 37 | \( 1 - 8 T + 32 T^{2} - 24 T^{3} - 3336 T^{4} + 20664 T^{5} - 58272 T^{6} - 204376 T^{7} + 4401628 T^{8} - 3547688 T^{9} - 88460512 T^{10} + 1017297032 T^{11} - 4513434040 T^{12} - 15095202536 T^{13} + 149739129440 T^{14} - 912347147320 T^{15} + 6228314829446 T^{16} - 912347147320 p T^{17} + 149739129440 p^{2} T^{18} - 15095202536 p^{3} T^{19} - 4513434040 p^{4} T^{20} + 1017297032 p^{5} T^{21} - 88460512 p^{6} T^{22} - 3547688 p^{7} T^{23} + 4401628 p^{8} T^{24} - 204376 p^{9} T^{25} - 58272 p^{10} T^{26} + 20664 p^{11} T^{27} - 3336 p^{12} T^{28} - 24 p^{13} T^{29} + 32 p^{14} T^{30} - 8 p^{15} T^{31} + p^{16} T^{32} \) | |
| 41 | \( 1 - 360 T^{2} + 62888 T^{4} - 7220152 T^{6} + 621792572 T^{8} - 43221811688 T^{10} + 2528834046232 T^{12} - 127405324980216 T^{14} + 5588806117303942 T^{16} - 127405324980216 p^{2} T^{18} + 2528834046232 p^{4} T^{20} - 43221811688 p^{6} T^{22} + 621792572 p^{8} T^{24} - 7220152 p^{10} T^{26} + 62888 p^{12} T^{28} - 360 p^{14} T^{30} + p^{16} T^{32} \) | |
| 43 | \( 1 - 24 T + 288 T^{2} - 2600 T^{3} + 21760 T^{4} - 185480 T^{5} + 1564640 T^{6} - 12850680 T^{7} + 104125692 T^{8} - 801969016 T^{9} + 5825673888 T^{10} - 40932747976 T^{11} + 281054739456 T^{12} - 1923512990504 T^{13} + 13279006331488 T^{14} - 92014074112600 T^{15} + 620497328996550 T^{16} - 92014074112600 p T^{17} + 13279006331488 p^{2} T^{18} - 1923512990504 p^{3} T^{19} + 281054739456 p^{4} T^{20} - 40932747976 p^{5} T^{21} + 5825673888 p^{6} T^{22} - 801969016 p^{7} T^{23} + 104125692 p^{8} T^{24} - 12850680 p^{9} T^{25} + 1564640 p^{10} T^{26} - 185480 p^{11} T^{27} + 21760 p^{12} T^{28} - 2600 p^{13} T^{29} + 288 p^{14} T^{30} - 24 p^{15} T^{31} + p^{16} T^{32} \) | |
| 47 | \( ( 1 - 20 T + 444 T^{2} - 5796 T^{3} + 76348 T^{4} - 746420 T^{5} + 7220260 T^{6} - 55738788 T^{7} + 423816262 T^{8} - 55738788 p T^{9} + 7220260 p^{2} T^{10} - 746420 p^{3} T^{11} + 76348 p^{4} T^{12} - 5796 p^{5} T^{13} + 444 p^{6} T^{14} - 20 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 53 | \( 1 - 32 T + 512 T^{2} - 5792 T^{3} + 51448 T^{4} - 347232 T^{5} + 1543680 T^{6} - 648160 T^{7} - 77911332 T^{8} + 1083882720 T^{9} - 10173720064 T^{10} + 75210276448 T^{11} - 416179543480 T^{12} + 1155415431584 T^{13} + 2859279872 p^{2} T^{14} - 3039863062112 p T^{15} + 1480255202813318 T^{16} - 3039863062112 p^{2} T^{17} + 2859279872 p^{4} T^{18} + 1155415431584 p^{3} T^{19} - 416179543480 p^{4} T^{20} + 75210276448 p^{5} T^{21} - 10173720064 p^{6} T^{22} + 1083882720 p^{7} T^{23} - 77911332 p^{8} T^{24} - 648160 p^{9} T^{25} + 1543680 p^{10} T^{26} - 347232 p^{11} T^{27} + 51448 p^{12} T^{28} - 5792 p^{13} T^{29} + 512 p^{14} T^{30} - 32 p^{15} T^{31} + p^{16} T^{32} \) | |
| 59 | \( 1 - 4 T + 8 T^{2} - 436 T^{3} - 8 T^{4} + 13572 T^{5} + 40824 T^{6} - 624764 T^{7} - 5768016 T^{8} + 61514388 T^{9} + 164909864 T^{10} + 3670383764 T^{11} - 16903696936 T^{12} - 262495882532 T^{13} + 851756850392 T^{14} - 9139950487220 T^{15} + 107683331209598 T^{16} - 9139950487220 p T^{17} + 851756850392 p^{2} T^{18} - 262495882532 p^{3} T^{19} - 16903696936 p^{4} T^{20} + 3670383764 p^{5} T^{21} + 164909864 p^{6} T^{22} + 61514388 p^{7} T^{23} - 5768016 p^{8} T^{24} - 624764 p^{9} T^{25} + 40824 p^{10} T^{26} + 13572 p^{11} T^{27} - 8 p^{12} T^{28} - 436 p^{13} T^{29} + 8 p^{14} T^{30} - 4 p^{15} T^{31} + p^{16} T^{32} \) | |
| 61 | \( 1 - 20 T + 200 T^{2} - 1740 T^{3} + 19576 T^{4} - 220476 T^{5} + 32920 p T^{6} - 17012500 T^{7} + 145845072 T^{8} - 1289298812 T^{9} + 11212588008 T^{10} - 97956422516 T^{11} + 802408738328 T^{12} - 6008404699012 T^{13} + 46730515662744 T^{14} - 395910028026652 T^{15} + 3294587522023742 T^{16} - 395910028026652 p T^{17} + 46730515662744 p^{2} T^{18} - 6008404699012 p^{3} T^{19} + 802408738328 p^{4} T^{20} - 97956422516 p^{5} T^{21} + 11212588008 p^{6} T^{22} - 1289298812 p^{7} T^{23} + 145845072 p^{8} T^{24} - 17012500 p^{9} T^{25} + 32920 p^{11} T^{26} - 220476 p^{11} T^{27} + 19576 p^{12} T^{28} - 1740 p^{13} T^{29} + 200 p^{14} T^{30} - 20 p^{15} T^{31} + p^{16} T^{32} \) | |
| 67 | \( 1 + 32 T + 512 T^{2} + 96 p T^{3} + 52960 T^{4} + 97824 T^{5} - 3299840 T^{6} - 63355616 T^{7} - 629028420 T^{8} - 2383339744 T^{9} + 17836651008 T^{10} + 6989602400 p T^{11} + 5064819975200 T^{12} + 24652433524000 T^{13} - 38685704452608 T^{14} - 2319332270069984 T^{15} - 27642923121176122 T^{16} - 2319332270069984 p T^{17} - 38685704452608 p^{2} T^{18} + 24652433524000 p^{3} T^{19} + 5064819975200 p^{4} T^{20} + 6989602400 p^{6} T^{21} + 17836651008 p^{6} T^{22} - 2383339744 p^{7} T^{23} - 629028420 p^{8} T^{24} - 63355616 p^{9} T^{25} - 3299840 p^{10} T^{26} + 97824 p^{11} T^{27} + 52960 p^{12} T^{28} + 96 p^{14} T^{29} + 512 p^{14} T^{30} + 32 p^{15} T^{31} + p^{16} T^{32} \) | |
| 71 | \( 1 - 848 T^{2} + 348632 T^{4} - 92363376 T^{6} + 17679086172 T^{8} - 2597376864720 T^{10} + 303579608231656 T^{12} - 28842829381255792 T^{14} + 2253844595954270534 T^{16} - 28842829381255792 p^{2} T^{18} + 303579608231656 p^{4} T^{20} - 2597376864720 p^{6} T^{22} + 17679086172 p^{8} T^{24} - 92363376 p^{10} T^{26} + 348632 p^{12} T^{28} - 848 p^{14} T^{30} + p^{16} T^{32} \) | |
| 73 | \( 1 - 576 T^{2} + 176152 T^{4} - 37053504 T^{6} + 5953382364 T^{8} - 771520557504 T^{10} + 1141007667560 p T^{12} - 7636383437363136 T^{14} + 600792482529621830 T^{16} - 7636383437363136 p^{2} T^{18} + 1141007667560 p^{5} T^{20} - 771520557504 p^{6} T^{22} + 5953382364 p^{8} T^{24} - 37053504 p^{10} T^{26} + 176152 p^{12} T^{28} - 576 p^{14} T^{30} + p^{16} T^{32} \) | |
| 79 | \( ( 1 + 344 T^{2} - 704 T^{3} + 63740 T^{4} - 170560 T^{5} + 8074984 T^{6} - 22406016 T^{7} + 736732678 T^{8} - 22406016 p T^{9} + 8074984 p^{2} T^{10} - 170560 p^{3} T^{11} + 63740 p^{4} T^{12} - 704 p^{5} T^{13} + 344 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 83 | \( 1 + 36 T + 648 T^{2} + 6988 T^{3} + 37128 T^{4} - 132396 T^{5} - 4409128 T^{6} - 42843796 T^{7} - 215261392 T^{8} + 331971804 T^{9} + 24514375304 T^{10} + 412774717540 T^{11} + 4544512472904 T^{12} + 27679681680668 T^{13} - 49346883958888 T^{14} - 3450443463089516 T^{15} - 43564195590574850 T^{16} - 3450443463089516 p T^{17} - 49346883958888 p^{2} T^{18} + 27679681680668 p^{3} T^{19} + 4544512472904 p^{4} T^{20} + 412774717540 p^{5} T^{21} + 24514375304 p^{6} T^{22} + 331971804 p^{7} T^{23} - 215261392 p^{8} T^{24} - 42843796 p^{9} T^{25} - 4409128 p^{10} T^{26} - 132396 p^{11} T^{27} + 37128 p^{12} T^{28} + 6988 p^{13} T^{29} + 648 p^{14} T^{30} + 36 p^{15} T^{31} + p^{16} T^{32} \) | |
| 89 | \( ( 1 - 616 T^{2} + 171324 T^{4} - 28355288 T^{6} + 3076164038 T^{8} - 28355288 p^{2} T^{10} + 171324 p^{4} T^{12} - 616 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 97 | \( ( 1 + 36 T + 892 T^{2} + 16188 T^{3} + 246108 T^{4} + 3208884 T^{5} + 37592036 T^{6} + 406022892 T^{7} + 4110173318 T^{8} + 406022892 p T^{9} + 37592036 p^{2} T^{10} + 3208884 p^{3} T^{11} + 246108 p^{4} T^{12} + 16188 p^{5} T^{13} + 892 p^{6} T^{14} + 36 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
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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
−2.34483717812596202309186349115, −2.28960239255906098971651544694, −2.26745259399562635357916769005, −2.22913172792604079312245005668, −2.11144665083620146468923570586, −2.02293970671906016740116430959, −1.84716244783603199409001204043, −1.77580613197485032754792970789, −1.66470528622489210226352309317, −1.53478778776016476036274996983, −1.51976908008422964091190428045, −1.47118416131107862470698144663, −1.29244609665885917626610049086, −1.23294813043114005955042735017, −1.19737153443083969053616418228, −1.01635149282384684290815503194, −0.999687433244955444260777036309, −0.996573646598100863311341822327, −0.78273735830871347313984388476, −0.75297152309582751237777080785, −0.74092097793693176557812698129, −0.50270154654825922798790171436, −0.31082486553119927047935841161, −0.22088274805061217196728202003, −0.04328435743199586819985405425, 0.04328435743199586819985405425, 0.22088274805061217196728202003, 0.31082486553119927047935841161, 0.50270154654825922798790171436, 0.74092097793693176557812698129, 0.75297152309582751237777080785, 0.78273735830871347313984388476, 0.996573646598100863311341822327, 0.999687433244955444260777036309, 1.01635149282384684290815503194, 1.19737153443083969053616418228, 1.23294813043114005955042735017, 1.29244609665885917626610049086, 1.47118416131107862470698144663, 1.51976908008422964091190428045, 1.53478778776016476036274996983, 1.66470528622489210226352309317, 1.77580613197485032754792970789, 1.84716244783603199409001204043, 2.02293970671906016740116430959, 2.11144665083620146468923570586, 2.22913172792604079312245005668, 2.26745259399562635357916769005, 2.28960239255906098971651544694, 2.34483717812596202309186349115
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.