Dirichlet series
| L(s) = 1 | − 4-s + 6·7-s − 8-s − 3·11-s + 6·13-s − 13·17-s + 11·19-s − 13·23-s − 6·28-s + 3·29-s − 11·31-s − 32-s + 21·37-s + 41-s + 2·43-s + 3·44-s − 14·47-s − 10·49-s − 6·52-s − 23·53-s − 6·56-s − 9·59-s + 11·61-s − 9·64-s + 8·67-s + 13·68-s + 8·71-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 2.26·7-s − 0.353·8-s − 0.904·11-s + 1.66·13-s − 3.15·17-s + 2.52·19-s − 2.71·23-s − 1.13·28-s + 0.557·29-s − 1.97·31-s − 0.176·32-s + 3.45·37-s + 0.156·41-s + 0.304·43-s + 0.452·44-s − 2.04·47-s − 1.42·49-s − 0.832·52-s − 3.15·53-s − 0.801·56-s − 1.17·59-s + 1.40·61-s − 9/8·64-s + 0.977·67-s + 1.57·68-s + 0.949·71-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(12\) |
| Conductor: | \(3^{12} \cdot 5^{24}\) |
| Sign: | $1$ |
| Analytic conductor: | \(8.21103\times 10^{9}\) |
| Root analytic conductor: | \(6.70192\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((12,\ 3^{12} \cdot 5^{24} ,\ ( \ : [1/2]^{6} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(5.870268462\) |
| \(L(\frac12)\) | \(\approx\) | \(5.870268462\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ | |
|---|---|---|---|
| bad | 3 | \( 1 \) | |
| 5 | \( 1 \) | ||
| good | 2 | \( 1 + T^{2} + T^{3} + T^{4} + 3 T^{5} + 11 T^{6} + 3 p T^{7} + p^{2} T^{8} + p^{3} T^{9} + p^{4} T^{10} + p^{6} T^{12} \) | 6.2.a_b_b_b_d_l |
| 7 | \( 1 - 6 T + 46 T^{2} - 185 T^{3} + 822 T^{4} - 2435 T^{5} + 7706 T^{6} - 2435 p T^{7} + 822 p^{2} T^{8} - 185 p^{3} T^{9} + 46 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \) | 6.7.ag_bu_ahd_bfq_adpr_lkk | |
| 11 | \( 1 + 3 T + 42 T^{2} + 9 p T^{3} + 870 T^{4} + 1842 T^{5} + 11882 T^{6} + 1842 p T^{7} + 870 p^{2} T^{8} + 9 p^{4} T^{9} + 42 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \) | 6.11.d_bq_dv_bhm_csw_rpa | |
| 13 | \( 1 - 6 T + 4 p T^{2} - 217 T^{3} + 1284 T^{4} - 4378 T^{5} + 20303 T^{6} - 4378 p T^{7} + 1284 p^{2} T^{8} - 217 p^{3} T^{9} + 4 p^{5} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \) | 6.13.ag_ca_aij_bxk_agmk_beax | |
| 17 | \( 1 + 13 T + 117 T^{2} + 40 p T^{3} + 3545 T^{4} + 15273 T^{5} + 67369 T^{6} + 15273 p T^{7} + 3545 p^{2} T^{8} + 40 p^{4} T^{9} + 117 p^{4} T^{10} + 13 p^{5} T^{11} + p^{6} T^{12} \) | 6.17.n_en_bae_fgj_wpl_dvrd | |
| 19 | \( 1 - 11 T + 125 T^{2} - 876 T^{3} + 6052 T^{4} - 30827 T^{5} + 424 p^{2} T^{6} - 30827 p T^{7} + 6052 p^{2} T^{8} - 876 p^{3} T^{9} + 125 p^{4} T^{10} - 11 p^{5} T^{11} + p^{6} T^{12} \) | 6.19.al_ev_abhs_iyu_abtpr_islc | |
| 23 | \( 1 + 13 T + 144 T^{2} + 1185 T^{3} + 364 p T^{4} + 49500 T^{5} + 255074 T^{6} + 49500 p T^{7} + 364 p^{3} T^{8} + 1185 p^{3} T^{9} + 144 p^{4} T^{10} + 13 p^{5} T^{11} + p^{6} T^{12} \) | 6.23.n_fo_btp_mka_cvfw_onio | |
| 29 | \( 1 - 3 T + 133 T^{2} - 262 T^{3} + 8085 T^{4} - 12009 T^{5} + 296107 T^{6} - 12009 p T^{7} + 8085 p^{2} T^{8} - 262 p^{3} T^{9} + 133 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \) | 6.29.ad_fd_akc_lyz_artx_qwat | |
| 31 | \( 1 + 11 T + 140 T^{2} + 1025 T^{3} + 8226 T^{4} + 46520 T^{5} + 297614 T^{6} + 46520 p T^{7} + 8226 p^{2} T^{8} + 1025 p^{3} T^{9} + 140 p^{4} T^{10} + 11 p^{5} T^{11} + p^{6} T^{12} \) | 6.31.l_fk_bnl_mek_cqvg_qygs | |
| 37 | \( 1 - 21 T + 361 T^{2} - 4000 T^{3} + 38967 T^{4} - 7915 p T^{5} + 1990421 T^{6} - 7915 p^{2} T^{7} + 38967 p^{2} T^{8} - 4000 p^{3} T^{9} + 361 p^{4} T^{10} - 21 p^{5} T^{11} + p^{6} T^{12} \) | 6.37.av_nx_afxw_cfqt_aqrfr_ejgkr | |
| 41 | \( 1 - T + 65 T^{2} - 315 T^{3} + 4526 T^{4} - 16885 T^{5} + 207789 T^{6} - 16885 p T^{7} + 4526 p^{2} T^{8} - 315 p^{3} T^{9} + 65 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \) | 6.41.ab_cn_amd_gsc_ayzl_lvjx | |
| 43 | \( 1 - 2 T + 167 T^{2} - 256 T^{3} + 13452 T^{4} - 15956 T^{5} + 692036 T^{6} - 15956 p T^{7} + 13452 p^{2} T^{8} - 256 p^{3} T^{9} + 167 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \) | 6.43.ac_gl_ajw_txk_axps_bnjsu | |
| 47 | \( 1 + 14 T + 278 T^{2} + 2603 T^{3} + 30072 T^{4} + 211487 T^{5} + 1808494 T^{6} + 211487 p T^{7} + 30072 p^{2} T^{8} + 2603 p^{3} T^{9} + 278 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} \) | 6.47.o_ks_dwd_bsmq_mawd_dyxhm | |
| 53 | \( 1 + 23 T + 379 T^{2} + 4310 T^{3} + 40637 T^{4} + 323685 T^{5} + 2441559 T^{6} + 323685 p T^{7} + 40637 p^{2} T^{8} + 4310 p^{3} T^{9} + 379 p^{4} T^{10} + 23 p^{5} T^{11} + p^{6} T^{12} \) | 6.53.x_op_gju_cicz_skvl_fixud | |
| 59 | \( 1 + 9 T + 265 T^{2} + 2264 T^{3} + 32442 T^{4} + 249123 T^{5} + 2397904 T^{6} + 249123 p T^{7} + 32442 p^{2} T^{8} + 2264 p^{3} T^{9} + 265 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \) | 6.59.j_kf_djc_bvzu_oenr_fglfc | |
| 61 | \( 1 - 11 T + 227 T^{2} - 1507 T^{3} + 23288 T^{4} - 138147 T^{5} + 1774033 T^{6} - 138147 p T^{7} + 23288 p^{2} T^{8} - 1507 p^{3} T^{9} + 227 p^{4} T^{10} - 11 p^{5} T^{11} + p^{6} T^{12} \) | 6.61.al_it_acfz_bils_ahwjj_dwyib | |
| 67 | \( 1 - 8 T + 247 T^{2} - 1290 T^{3} + 26550 T^{4} - 92768 T^{5} + 1956344 T^{6} - 92768 p T^{7} + 26550 p^{2} T^{8} - 1290 p^{3} T^{9} + 247 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \) | 6.67.ai_jn_abxq_bnhe_afhga_ehiaa | |
| 71 | \( 1 - 8 T + 276 T^{2} - 1095 T^{3} + 27860 T^{4} - 28473 T^{5} + 1889114 T^{6} - 28473 p T^{7} + 27860 p^{2} T^{8} - 1095 p^{3} T^{9} + 276 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \) | 6.71.ai_kq_abqd_bpfo_abqdd_edmog | |
| 73 | \( 1 - 13 T + 364 T^{2} - 3570 T^{3} + 58057 T^{4} - 450270 T^{5} + 5397049 T^{6} - 450270 p T^{7} + 58057 p^{2} T^{8} - 3570 p^{3} T^{9} + 364 p^{4} T^{10} - 13 p^{5} T^{11} + p^{6} T^{12} \) | 6.73.an_oa_afhi_dhwz_azqcc_lvbuv | |
| 79 | \( 1 + 5 T + 314 T^{2} + 1445 T^{3} + 45010 T^{4} + 190240 T^{5} + 4170310 T^{6} + 190240 p T^{7} + 45010 p^{2} T^{8} + 1445 p^{3} T^{9} + 314 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} \) | 6.79.f_mc_cdp_cope_kvky_jdhco | |
| 83 | \( 1 - 20 T + 524 T^{2} - 7048 T^{3} + 106936 T^{4} - 1073796 T^{5} + 11716314 T^{6} - 1073796 p T^{7} + 106936 p^{2} T^{8} - 7048 p^{3} T^{9} + 524 p^{4} T^{10} - 20 p^{5} T^{11} + p^{6} T^{12} \) | 6.83.au_ue_aklc_gcey_acjclw_zqpvm | |
| 89 | \( 1 - 4 T + 70 T^{2} - 1649 T^{3} + 9072 T^{4} - 84358 T^{5} + 1538759 T^{6} - 84358 p T^{7} + 9072 p^{2} T^{8} - 1649 p^{3} T^{9} + 70 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \) | 6.89.ae_cs_acll_nky_aeuuo_djohb | |
| 97 | \( 1 + 7 T + 367 T^{2} + 1905 T^{3} + 69210 T^{4} + 304027 T^{5} + 8348279 T^{6} + 304027 p T^{7} + 69210 p^{2} T^{8} + 1905 p^{3} T^{9} + 367 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} \) | 6.97.h_od_cvh_dyjy_rhtj_sgznr | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.39349382703147158969647520908, −4.05947882063679958955153258214, −3.91366331788131046016456354563, −3.78955206961991850990536937232, −3.50853527140452202814292572056, −3.49477456506453883719513429316, −3.26808051443242256142262360133, −3.22557522456088259888086503157, −3.03754414463849198759619151508, −2.99742551485765443730800414480, −2.89012647474747905461983074841, −2.40182987704959499103122413944, −2.39489810737147324306282910965, −2.14207255947012100854716914697, −2.10905834109457320171002966295, −1.81642737932227625191335261394, −1.75172172774445019279899813926, −1.72035845958284085184918126020, −1.68361734046066454933135768735, −1.27685585851809504853268690073, −1.11108895344512822448540520459, −0.75532988907201590714008960982, −0.65815820433299664980996561691, −0.33166971020873316769226741897, −0.29046410226323060869455092292, 0.29046410226323060869455092292, 0.33166971020873316769226741897, 0.65815820433299664980996561691, 0.75532988907201590714008960982, 1.11108895344512822448540520459, 1.27685585851809504853268690073, 1.68361734046066454933135768735, 1.72035845958284085184918126020, 1.75172172774445019279899813926, 1.81642737932227625191335261394, 2.10905834109457320171002966295, 2.14207255947012100854716914697, 2.39489810737147324306282910965, 2.40182987704959499103122413944, 2.89012647474747905461983074841, 2.99742551485765443730800414480, 3.03754414463849198759619151508, 3.22557522456088259888086503157, 3.26808051443242256142262360133, 3.49477456506453883719513429316, 3.50853527140452202814292572056, 3.78955206961991850990536937232, 3.91366331788131046016456354563, 4.05947882063679958955153258214, 4.39349382703147158969647520908
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.