Dirichlet series
| L(s) = 1 | + 4-s − 8·7-s − 4·8-s + 12·9-s + 16-s − 8·17-s + 28·25-s − 8·28-s + 16·31-s − 8·32-s + 12·36-s + 16·41-s + 24·47-s − 12·49-s + 32·56-s − 96·63-s + 11·64-s − 8·68-s + 48·71-s − 48·72-s − 48·79-s + 50·81-s − 16·89-s + 32·97-s + 28·100-s − 8·112-s + 64·119-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 3.02·7-s − 1.41·8-s + 4·9-s + 1/4·16-s − 1.94·17-s + 28/5·25-s − 1.51·28-s + 2.87·31-s − 1.41·32-s + 2·36-s + 2.49·41-s + 3.50·47-s − 1.71·49-s + 4.27·56-s − 12.0·63-s + 11/8·64-s − 0.970·68-s + 5.69·71-s − 5.65·72-s − 5.40·79-s + 50/9·81-s − 1.69·89-s + 3.24·97-s + 14/5·100-s − 0.755·112-s + 5.86·119-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 19^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 19^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{48} \cdot 19^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(22.1784\) |
| Root analytic conductor: | \(1.10169\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{48} \cdot 19^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(2.244788745\) |
| \(L(\frac12)\) | \(\approx\) | \(2.244788745\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 - T^{2} + p^{2} T^{3} + 3 p T^{6} + p^{3} T^{8} + 3 p^{3} T^{10} + p^{7} T^{13} - p^{6} T^{14} + p^{8} T^{16} \) |
| 19 | \( ( 1 + T^{2} )^{8} \) | |
| good | 3 | \( 1 - 4 p T^{2} + 94 T^{4} - 544 T^{6} + 2617 T^{8} - 10984 T^{10} + 13882 p T^{12} - 143372 T^{14} + 150556 p T^{16} - 143372 p^{2} T^{18} + 13882 p^{5} T^{20} - 10984 p^{6} T^{22} + 2617 p^{8} T^{24} - 544 p^{10} T^{26} + 94 p^{12} T^{28} - 4 p^{15} T^{30} + p^{16} T^{32} \) |
| 5 | \( 1 - 28 T^{2} + 438 T^{4} - 4848 T^{6} + 42529 T^{8} - 62616 p T^{10} + 2009318 T^{12} - 11529948 T^{14} + 60231524 T^{16} - 11529948 p^{2} T^{18} + 2009318 p^{4} T^{20} - 62616 p^{7} T^{22} + 42529 p^{8} T^{24} - 4848 p^{10} T^{26} + 438 p^{12} T^{28} - 28 p^{14} T^{30} + p^{16} T^{32} \) | |
| 7 | \( ( 1 + 4 T + 30 T^{2} + 68 T^{3} + 360 T^{4} + 584 T^{5} + 3284 T^{6} + 4880 T^{7} + 26161 T^{8} + 4880 p T^{9} + 3284 p^{2} T^{10} + 584 p^{3} T^{11} + 360 p^{4} T^{12} + 68 p^{5} T^{13} + 30 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 11 | \( 1 - 92 T^{2} + 4326 T^{4} - 138936 T^{6} + 3407825 T^{8} - 67421256 T^{10} + 1109052726 T^{12} - 15437378244 T^{14} + 183500018436 T^{16} - 15437378244 p^{2} T^{18} + 1109052726 p^{4} T^{20} - 67421256 p^{6} T^{22} + 3407825 p^{8} T^{24} - 138936 p^{10} T^{26} + 4326 p^{12} T^{28} - 92 p^{14} T^{30} + p^{16} T^{32} \) | |
| 13 | \( 1 - 96 T^{2} + 4590 T^{4} - 145832 T^{6} + 3460001 T^{8} - 65542128 T^{10} + 1045890990 T^{12} - 14852004936 T^{14} + 197390117924 T^{16} - 14852004936 p^{2} T^{18} + 1045890990 p^{4} T^{20} - 65542128 p^{6} T^{22} + 3460001 p^{8} T^{24} - 145832 p^{10} T^{26} + 4590 p^{12} T^{28} - 96 p^{14} T^{30} + p^{16} T^{32} \) | |
| 17 | \( ( 1 + 4 T + 60 T^{2} + 216 T^{3} + 1706 T^{4} + 4476 T^{5} + 30416 T^{6} + 51164 T^{7} + 480003 T^{8} + 51164 p T^{9} + 30416 p^{2} T^{10} + 4476 p^{3} T^{11} + 1706 p^{4} T^{12} + 216 p^{5} T^{13} + 60 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 23 | \( ( 1 + 60 T^{2} + 76 T^{3} + 109 p T^{4} + 4156 T^{5} + 75872 T^{6} + 152912 T^{7} + 1893888 T^{8} + 152912 p T^{9} + 75872 p^{2} T^{10} + 4156 p^{3} T^{11} + 109 p^{5} T^{12} + 76 p^{5} T^{13} + 60 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 29 | \( 1 - 216 T^{2} + 21422 T^{4} - 1343568 T^{6} + 63937425 T^{8} - 2636294320 T^{10} + 99364061438 T^{12} - 3391411721192 T^{14} + 103804339922532 T^{16} - 3391411721192 p^{2} T^{18} + 99364061438 p^{4} T^{20} - 2636294320 p^{6} T^{22} + 63937425 p^{8} T^{24} - 1343568 p^{10} T^{26} + 21422 p^{12} T^{28} - 216 p^{14} T^{30} + p^{16} T^{32} \) | |
| 31 | \( ( 1 - 8 T + 196 T^{2} - 1432 T^{3} + 17900 T^{4} - 117272 T^{5} + 997916 T^{6} - 5686664 T^{7} + 37305574 T^{8} - 5686664 p T^{9} + 997916 p^{2} T^{10} - 117272 p^{3} T^{11} + 17900 p^{4} T^{12} - 1432 p^{5} T^{13} + 196 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 37 | \( 1 - 448 T^{2} + 97096 T^{4} - 13538240 T^{6} + 1361799228 T^{8} - 104985163840 T^{10} + 6427935625976 T^{12} - 319300247763520 T^{14} + 13017076317741510 T^{16} - 319300247763520 p^{2} T^{18} + 6427935625976 p^{4} T^{20} - 104985163840 p^{6} T^{22} + 1361799228 p^{8} T^{24} - 13538240 p^{10} T^{26} + 97096 p^{12} T^{28} - 448 p^{14} T^{30} + p^{16} T^{32} \) | |
| 41 | \( ( 1 - 8 T + 200 T^{2} - 1464 T^{3} + 20020 T^{4} - 127208 T^{5} + 1294840 T^{6} - 7234008 T^{7} + 60783446 T^{8} - 7234008 p T^{9} + 1294840 p^{2} T^{10} - 127208 p^{3} T^{11} + 20020 p^{4} T^{12} - 1464 p^{5} T^{13} + 200 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 43 | \( 1 - 484 T^{2} + 114486 T^{4} - 17626832 T^{6} + 1981003681 T^{8} - 172489189384 T^{10} + 12041902947430 T^{12} - 687681116820964 T^{14} + 32467415307528932 T^{16} - 687681116820964 p^{2} T^{18} + 12041902947430 p^{4} T^{20} - 172489189384 p^{6} T^{22} + 1981003681 p^{8} T^{24} - 17626832 p^{10} T^{26} + 114486 p^{12} T^{28} - 484 p^{14} T^{30} + p^{16} T^{32} \) | |
| 47 | \( ( 1 - 12 T + 230 T^{2} - 1740 T^{3} + 17497 T^{4} - 72208 T^{5} + 426938 T^{6} + 505824 T^{7} + 864076 T^{8} + 505824 p T^{9} + 426938 p^{2} T^{10} - 72208 p^{3} T^{11} + 17497 p^{4} T^{12} - 1740 p^{5} T^{13} + 230 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 53 | \( 1 - 568 T^{2} + 160894 T^{4} - 30026272 T^{6} + 4120038209 T^{8} - 439876082224 T^{10} + 37748747628942 T^{12} - 2653751888034360 T^{14} + 154373495002173956 T^{16} - 2653751888034360 p^{2} T^{18} + 37748747628942 p^{4} T^{20} - 439876082224 p^{6} T^{22} + 4120038209 p^{8} T^{24} - 30026272 p^{10} T^{26} + 160894 p^{12} T^{28} - 568 p^{14} T^{30} + p^{16} T^{32} \) | |
| 59 | \( 1 - 548 T^{2} + 149614 T^{4} - 27223768 T^{6} + 3714016473 T^{8} - 403655596264 T^{10} + 36158344547598 T^{12} - 2721990080866300 T^{14} + 173995217146714804 T^{16} - 2721990080866300 p^{2} T^{18} + 36158344547598 p^{4} T^{20} - 403655596264 p^{6} T^{22} + 3714016473 p^{8} T^{24} - 27223768 p^{10} T^{26} + 149614 p^{12} T^{28} - 548 p^{14} T^{30} + p^{16} T^{32} \) | |
| 61 | \( 1 - 4 p T^{2} + 29622 T^{4} - 1899400 T^{6} + 41603905 T^{8} + 2312073928 T^{10} - 36116466618 T^{12} - 26518643483884 T^{14} + 2619416083527844 T^{16} - 26518643483884 p^{2} T^{18} - 36116466618 p^{4} T^{20} + 2312073928 p^{6} T^{22} + 41603905 p^{8} T^{24} - 1899400 p^{10} T^{26} + 29622 p^{12} T^{28} - 4 p^{15} T^{30} + p^{16} T^{32} \) | |
| 67 | \( 1 - 588 T^{2} + 169518 T^{4} - 32357584 T^{6} + 4649137449 T^{8} - 538935344360 T^{10} + 52394328139710 T^{12} - 4360644811515548 T^{14} + 313763277516961716 T^{16} - 4360644811515548 p^{2} T^{18} + 52394328139710 p^{4} T^{20} - 538935344360 p^{6} T^{22} + 4649137449 p^{8} T^{24} - 32357584 p^{10} T^{26} + 169518 p^{12} T^{28} - 588 p^{14} T^{30} + p^{16} T^{32} \) | |
| 71 | \( ( 1 - 24 T + 580 T^{2} - 8776 T^{3} + 132836 T^{4} - 1549960 T^{5} + 17835132 T^{6} - 167522904 T^{7} + 1548536566 T^{8} - 167522904 p T^{9} + 17835132 p^{2} T^{10} - 1549960 p^{3} T^{11} + 132836 p^{4} T^{12} - 8776 p^{5} T^{13} + 580 p^{6} T^{14} - 24 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 73 | \( ( 1 + 268 T^{2} - 560 T^{3} + 37634 T^{4} - 139728 T^{5} + 3640384 T^{6} - 16876272 T^{7} + 288747435 T^{8} - 16876272 p T^{9} + 3640384 p^{2} T^{10} - 139728 p^{3} T^{11} + 37634 p^{4} T^{12} - 560 p^{5} T^{13} + 268 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 79 | \( ( 1 + 24 T + 696 T^{2} + 11288 T^{3} + 192468 T^{4} + 2366392 T^{5} + 29698376 T^{6} + 291409976 T^{7} + 2898297046 T^{8} + 291409976 p T^{9} + 29698376 p^{2} T^{10} + 2366392 p^{3} T^{11} + 192468 p^{4} T^{12} + 11288 p^{5} T^{13} + 696 p^{6} T^{14} + 24 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 83 | \( 1 - 888 T^{2} + 388120 T^{4} - 111209896 T^{6} + 23425602588 T^{8} - 3849513927800 T^{10} + 510595309926440 T^{12} - 55757709197515496 T^{14} + 5065547522744531590 T^{16} - 55757709197515496 p^{2} T^{18} + 510595309926440 p^{4} T^{20} - 3849513927800 p^{6} T^{22} + 23425602588 p^{8} T^{24} - 111209896 p^{10} T^{26} + 388120 p^{12} T^{28} - 888 p^{14} T^{30} + p^{16} T^{32} \) | |
| 89 | \( ( 1 + 8 T + 360 T^{2} + 1944 T^{3} + 59412 T^{4} + 285448 T^{5} + 7491096 T^{6} + 38771288 T^{7} + 773903254 T^{8} + 38771288 p T^{9} + 7491096 p^{2} T^{10} + 285448 p^{3} T^{11} + 59412 p^{4} T^{12} + 1944 p^{5} T^{13} + 360 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 97 | \( ( 1 - 16 T + 484 T^{2} - 7360 T^{3} + 115940 T^{4} - 1500384 T^{5} + 18527132 T^{6} - 192518480 T^{7} + 2118921782 T^{8} - 192518480 p T^{9} + 18527132 p^{2} T^{10} - 1500384 p^{3} T^{11} + 115940 p^{4} T^{12} - 7360 p^{5} T^{13} + 484 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
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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
−4.02133115179029488749776409348, −3.79601122281903407543512973027, −3.60142182447235088395660189524, −3.50628925825918335350294183087, −3.47366888083275812214273162905, −3.45840191616352357356818660026, −3.32584049335855546582369680780, −3.05861702195791530640353808453, −3.02581813848704798331632236622, −2.96289491803091372956333676219, −2.89171658228689230911127777542, −2.77437871939406430210849309083, −2.75629307518976522902126641723, −2.74923909236795286168599604780, −2.30519157334385807502225078658, −2.26304990662688588552318336379, −2.25104613418163899567233956431, −2.07194012362618841419490790404, −1.77424552184790539447964079363, −1.66859414021996525581259853281, −1.60403985319325714110510460330, −1.16657140684239661570957978392, −1.01220196122543408428068787677, −0.891674383068029056808757982441, −0.71644535900500256772558095458, 0.71644535900500256772558095458, 0.891674383068029056808757982441, 1.01220196122543408428068787677, 1.16657140684239661570957978392, 1.60403985319325714110510460330, 1.66859414021996525581259853281, 1.77424552184790539447964079363, 2.07194012362618841419490790404, 2.25104613418163899567233956431, 2.26304990662688588552318336379, 2.30519157334385807502225078658, 2.74923909236795286168599604780, 2.75629307518976522902126641723, 2.77437871939406430210849309083, 2.89171658228689230911127777542, 2.96289491803091372956333676219, 3.02581813848704798331632236622, 3.05861702195791530640353808453, 3.32584049335855546582369680780, 3.45840191616352357356818660026, 3.47366888083275812214273162905, 3.50628925825918335350294183087, 3.60142182447235088395660189524, 3.79601122281903407543512973027, 4.02133115179029488749776409348
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.