Dirichlet series
| L(s) = 1 | − 4·7-s − 16·17-s − 8·23-s + 24·25-s + 4·31-s + 36·41-s + 24·47-s − 24·49-s − 32·73-s + 60·89-s + 40·97-s + 56·103-s − 72·113-s + 64·119-s + 48·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 32·161-s + 163-s + 167-s − 8·169-s + 173-s + ⋯ |
| L(s) = 1 | − 1.51·7-s − 3.88·17-s − 1.66·23-s + 24/5·25-s + 0.718·31-s + 5.62·41-s + 3.50·47-s − 3.42·49-s − 3.74·73-s + 6.35·89-s + 4.06·97-s + 5.51·103-s − 6.77·113-s + 5.86·119-s + 4.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 2.52·161-s + 0.0783·163-s + 0.0773·167-s − 0.615·169-s + 0.0760·173-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{80} \cdot 3^{32} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{80} \cdot 3^{32} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{80} \cdot 3^{32} \cdot 13^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.07199\times 10^{23}\) |
| Root analytic conductor: | \(5.46772\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{80} \cdot 3^{32} \cdot 13^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(6.030919913\) |
| \(L(\frac12)\) | \(\approx\) | \(6.030919913\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) | |
| 13 | \( ( 1 + T^{2} )^{8} \) | |
| good | 5 | \( 1 - 24 T^{2} + 12 p^{2} T^{4} - 2648 T^{6} + 18052 T^{8} - 97192 T^{10} + 423508 T^{12} - 1647592 T^{14} + 7179574 T^{16} - 1647592 p^{2} T^{18} + 423508 p^{4} T^{20} - 97192 p^{6} T^{22} + 18052 p^{8} T^{24} - 2648 p^{10} T^{26} + 12 p^{14} T^{28} - 24 p^{14} T^{30} + p^{16} T^{32} \) |
| 7 | \( ( 1 + 2 T + 18 T^{2} + 38 T^{3} + 220 T^{4} + 374 T^{5} + 2014 T^{6} + 2962 T^{7} + 14566 T^{8} + 2962 p T^{9} + 2014 p^{2} T^{10} + 374 p^{3} T^{11} + 220 p^{4} T^{12} + 38 p^{5} T^{13} + 18 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 11 | \( 1 - 48 T^{2} + 1368 T^{4} - 2736 p T^{6} + 545916 T^{8} - 8663216 T^{10} + 122742888 T^{12} - 1567517840 T^{14} + 18148303430 T^{16} - 1567517840 p^{2} T^{18} + 122742888 p^{4} T^{20} - 8663216 p^{6} T^{22} + 545916 p^{8} T^{24} - 2736 p^{11} T^{26} + 1368 p^{12} T^{28} - 48 p^{14} T^{30} + p^{16} T^{32} \) | |
| 17 | \( ( 1 + 8 T + 4 p T^{2} + 232 T^{3} + 4 p^{2} T^{4} + 2072 T^{5} + 17404 T^{6} + 40696 T^{7} + 360438 T^{8} + 40696 p T^{9} + 17404 p^{2} T^{10} + 2072 p^{3} T^{11} + 4 p^{6} T^{12} + 232 p^{5} T^{13} + 4 p^{7} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 19 | \( 1 - 80 T^{2} + 3940 T^{4} - 145216 T^{6} + 4404036 T^{8} - 6115232 p T^{10} + 2727772700 T^{12} - 58301165328 T^{14} + 1149493565238 T^{16} - 58301165328 p^{2} T^{18} + 2727772700 p^{4} T^{20} - 6115232 p^{7} T^{22} + 4404036 p^{8} T^{24} - 145216 p^{10} T^{26} + 3940 p^{12} T^{28} - 80 p^{14} T^{30} + p^{16} T^{32} \) | |
| 23 | \( ( 1 + 4 T + 64 T^{2} + 20 T^{3} + 1788 T^{4} - 2876 T^{5} + 58048 T^{6} - 64268 T^{7} + 1609286 T^{8} - 64268 p T^{9} + 58048 p^{2} T^{10} - 2876 p^{3} T^{11} + 1788 p^{4} T^{12} + 20 p^{5} T^{13} + 64 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 29 | \( 1 - 216 T^{2} + 24184 T^{4} - 1875144 T^{6} + 112984540 T^{8} - 5590750936 T^{10} + 233769519048 T^{12} - 8393471730696 T^{14} + 261111468932614 T^{16} - 8393471730696 p^{2} T^{18} + 233769519048 p^{4} T^{20} - 5590750936 p^{6} T^{22} + 112984540 p^{8} T^{24} - 1875144 p^{10} T^{26} + 24184 p^{12} T^{28} - 216 p^{14} T^{30} + p^{16} T^{32} \) | |
| 31 | \( ( 1 - 2 T + 138 T^{2} - 294 T^{3} + 10252 T^{4} - 21158 T^{5} + 508934 T^{6} - 970178 T^{7} + 18335302 T^{8} - 970178 p T^{9} + 508934 p^{2} T^{10} - 21158 p^{3} T^{11} + 10252 p^{4} T^{12} - 294 p^{5} T^{13} + 138 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 37 | \( 1 - 336 T^{2} + 56184 T^{4} - 6252912 T^{6} + 521916124 T^{8} - 34818111056 T^{10} + 1925502426568 T^{12} - 90081303233520 T^{14} + 3600631860689030 T^{16} - 90081303233520 p^{2} T^{18} + 1925502426568 p^{4} T^{20} - 34818111056 p^{6} T^{22} + 521916124 p^{8} T^{24} - 6252912 p^{10} T^{26} + 56184 p^{12} T^{28} - 336 p^{14} T^{30} + p^{16} T^{32} \) | |
| 41 | \( ( 1 - 18 T + 342 T^{2} - 4050 T^{3} + 46504 T^{4} - 422038 T^{5} + 3633186 T^{6} - 26569398 T^{7} + 182925454 T^{8} - 26569398 p T^{9} + 3633186 p^{2} T^{10} - 422038 p^{3} T^{11} + 46504 p^{4} T^{12} - 4050 p^{5} T^{13} + 342 p^{6} T^{14} - 18 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 43 | \( 1 - 360 T^{2} + 69624 T^{4} - 9265784 T^{6} + 936787036 T^{8} - 75641941096 T^{10} + 5020117969224 T^{12} - 278439638019128 T^{14} + 13021117594062982 T^{16} - 278439638019128 p^{2} T^{18} + 5020117969224 p^{4} T^{20} - 75641941096 p^{6} T^{22} + 936787036 p^{8} T^{24} - 9265784 p^{10} T^{26} + 69624 p^{12} T^{28} - 360 p^{14} T^{30} + p^{16} T^{32} \) | |
| 47 | \( ( 1 - 12 T + 8 p T^{2} - 3580 T^{3} + 61372 T^{4} - 473148 T^{5} + 5725864 T^{6} - 35892300 T^{7} + 335405302 T^{8} - 35892300 p T^{9} + 5725864 p^{2} T^{10} - 473148 p^{3} T^{11} + 61372 p^{4} T^{12} - 3580 p^{5} T^{13} + 8 p^{7} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 53 | \( 1 - 352 T^{2} + 64184 T^{4} - 8128288 T^{6} + 794839132 T^{8} - 63647867232 T^{10} + 4365698295944 T^{12} - 265511374218272 T^{14} + 14681503391564422 T^{16} - 265511374218272 p^{2} T^{18} + 4365698295944 p^{4} T^{20} - 63647867232 p^{6} T^{22} + 794839132 p^{8} T^{24} - 8128288 p^{10} T^{26} + 64184 p^{12} T^{28} - 352 p^{14} T^{30} + p^{16} T^{32} \) | |
| 59 | \( 1 - 400 T^{2} + 84184 T^{4} - 12223024 T^{6} + 1358469244 T^{8} - 122789490064 T^{10} + 9440933031656 T^{12} - 640782072309680 T^{14} + 39471908248607558 T^{16} - 640782072309680 p^{2} T^{18} + 9440933031656 p^{4} T^{20} - 122789490064 p^{6} T^{22} + 1358469244 p^{8} T^{24} - 12223024 p^{10} T^{26} + 84184 p^{12} T^{28} - 400 p^{14} T^{30} + p^{16} T^{32} \) | |
| 61 | \( 1 - 456 T^{2} + 97176 T^{4} - 12919448 T^{6} + 1230523164 T^{8} - 94494774344 T^{10} + 6581808027304 T^{12} - 442726183599320 T^{14} + 28146003812821382 T^{16} - 442726183599320 p^{2} T^{18} + 6581808027304 p^{4} T^{20} - 94494774344 p^{6} T^{22} + 1230523164 p^{8} T^{24} - 12919448 p^{10} T^{26} + 97176 p^{12} T^{28} - 456 p^{14} T^{30} + p^{16} T^{32} \) | |
| 67 | \( 1 - 544 T^{2} + 150916 T^{4} - 28617392 T^{6} + 4159667716 T^{8} - 490341594416 T^{10} + 48352853840124 T^{12} - 4060137639343744 T^{14} + 293009017988910774 T^{16} - 4060137639343744 p^{2} T^{18} + 48352853840124 p^{4} T^{20} - 490341594416 p^{6} T^{22} + 4159667716 p^{8} T^{24} - 28617392 p^{10} T^{26} + 150916 p^{12} T^{28} - 544 p^{14} T^{30} + p^{16} T^{32} \) | |
| 71 | \( ( 1 + 160 T^{2} + 600 T^{3} + 17004 T^{4} + 100616 T^{5} + 1410080 T^{6} + 9416592 T^{7} + 113936278 T^{8} + 9416592 p T^{9} + 1410080 p^{2} T^{10} + 100616 p^{3} T^{11} + 17004 p^{4} T^{12} + 600 p^{5} T^{13} + 160 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 73 | \( ( 1 + 16 T + 264 T^{2} + 2864 T^{3} + 31292 T^{4} + 244624 T^{5} + 2081336 T^{6} + 13416496 T^{7} + 122295622 T^{8} + 13416496 p T^{9} + 2081336 p^{2} T^{10} + 244624 p^{3} T^{11} + 31292 p^{4} T^{12} + 2864 p^{5} T^{13} + 264 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 79 | \( ( 1 + 296 T^{2} - 320 T^{3} + 50844 T^{4} - 72640 T^{5} + 6227736 T^{6} - 8459904 T^{7} + 565492422 T^{8} - 8459904 p T^{9} + 6227736 p^{2} T^{10} - 72640 p^{3} T^{11} + 50844 p^{4} T^{12} - 320 p^{5} T^{13} + 296 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 83 | \( 1 - 768 T^{2} + 303384 T^{4} - 80932928 T^{6} + 16205155452 T^{8} - 2570471286080 T^{10} + 333073566768040 T^{12} - 35890410674183552 T^{14} + 3246507673622242758 T^{16} - 35890410674183552 p^{2} T^{18} + 333073566768040 p^{4} T^{20} - 2570471286080 p^{6} T^{22} + 16205155452 p^{8} T^{24} - 80932928 p^{10} T^{26} + 303384 p^{12} T^{28} - 768 p^{14} T^{30} + p^{16} T^{32} \) | |
| 89 | \( ( 1 - 30 T + 798 T^{2} - 12774 T^{3} + 197976 T^{4} - 2314498 T^{5} + 28341178 T^{6} - 280901994 T^{7} + 2945621870 T^{8} - 280901994 p T^{9} + 28341178 p^{2} T^{10} - 2314498 p^{3} T^{11} + 197976 p^{4} T^{12} - 12774 p^{5} T^{13} + 798 p^{6} T^{14} - 30 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 97 | \( ( 1 - 20 T + 648 T^{2} - 9308 T^{3} + 173372 T^{4} - 1971188 T^{5} + 27562552 T^{6} - 264044700 T^{7} + 3089748742 T^{8} - 264044700 p T^{9} + 27562552 p^{2} T^{10} - 1971188 p^{3} T^{11} + 173372 p^{4} T^{12} - 9308 p^{5} T^{13} + 648 p^{6} T^{14} - 20 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
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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
−1.98801883758191680564243366838, −1.94310955903543679788559722786, −1.91844373632553967972450255268, −1.89368313842384863162829794134, −1.87837356456569113079547302474, −1.79572695195964216741704380846, −1.78504280076139053518374969828, −1.75274869440338282291457910911, −1.40395973759119636274084227421, −1.35851378964354768249983645358, −1.31970152416985032412445877762, −1.13936743611562739427445281709, −1.07447764133946236133829774892, −1.03386395813026676588732984706, −1.01350646650106426927053562596, −0.941046688140224049672335337923, −0.923697689349561300239674471997, −0.70663591449722707678992669563, −0.67928572975716067257269326050, −0.64202001964230725779439721365, −0.57139754032398852302276957739, −0.40333742198462400791117613555, −0.20974032814779323406137569778, −0.19199743655628374169696922445, −0.12105249206826448109086267598, 0.12105249206826448109086267598, 0.19199743655628374169696922445, 0.20974032814779323406137569778, 0.40333742198462400791117613555, 0.57139754032398852302276957739, 0.64202001964230725779439721365, 0.67928572975716067257269326050, 0.70663591449722707678992669563, 0.923697689349561300239674471997, 0.941046688140224049672335337923, 1.01350646650106426927053562596, 1.03386395813026676588732984706, 1.07447764133946236133829774892, 1.13936743611562739427445281709, 1.31970152416985032412445877762, 1.35851378964354768249983645358, 1.40395973759119636274084227421, 1.75274869440338282291457910911, 1.78504280076139053518374969828, 1.79572695195964216741704380846, 1.87837356456569113079547302474, 1.89368313842384863162829794134, 1.91844373632553967972450255268, 1.94310955903543679788559722786, 1.98801883758191680564243366838
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.