Dirichlet series
| L(s) = 1 | + 3·2-s + 6·3-s + 2·4-s − 2·5-s + 18·6-s + 12·7-s − 3·8-s + 21·9-s − 6·10-s + 12·12-s + 7·13-s + 36·14-s − 12·15-s − 4·16-s − 5·17-s + 63·18-s + 6·19-s − 4·20-s + 72·21-s − 11·23-s − 18·24-s − 9·25-s + 21·26-s + 56·27-s + 24·28-s + 7·29-s − 36·30-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 3.46·3-s + 4-s − 0.894·5-s + 7.34·6-s + 4.53·7-s − 1.06·8-s + 7·9-s − 1.89·10-s + 3.46·12-s + 1.94·13-s + 9.62·14-s − 3.09·15-s − 16-s − 1.21·17-s + 14.8·18-s + 1.37·19-s − 0.894·20-s + 15.7·21-s − 2.29·23-s − 3.67·24-s − 9/5·25-s + 4.11·26-s + 10.7·27-s + 4.53·28-s + 1.29·29-s − 6.57·30-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 11^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 11^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(12\) |
| Conductor: | \(3^{6} \cdot 11^{12} \cdot 19^{6}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.79012\times 10^{10}\) |
| Root analytic conductor: | \(7.42110\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((12,\ 3^{6} \cdot 11^{12} \cdot 19^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(462.8412206\) |
| \(L(\frac12)\) | \(\approx\) | \(462.8412206\) |
| \(L(\frac{3}{2})\) | 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 | \( ( 1 - T )^{6} \) | |
| 11 | \( 1 \) | ||
| 19 | \( ( 1 - T )^{6} \) | ||
| good | 2 | \( 1 - 3 T + 7 T^{2} - 3 p^{2} T^{3} + 17 T^{4} - 5 p^{2} T^{5} + 29 T^{6} - 5 p^{3} T^{7} + 17 p^{2} T^{8} - 3 p^{5} T^{9} + 7 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \) | 6.2.ad_h_am_r_au_bd |
| 5 | \( 1 + 2 T + 13 T^{2} + 4 p T^{3} + 94 T^{4} + 7 p^{2} T^{5} + 581 T^{6} + 7 p^{3} T^{7} + 94 p^{2} T^{8} + 4 p^{4} T^{9} + 13 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \) | 6.5.c_n_u_dq_gt_wj | |
| 7 | \( 1 - 12 T + 12 p T^{2} - 414 T^{3} + 1619 T^{4} - 5265 T^{5} + 14891 T^{6} - 5265 p T^{7} + 1619 p^{2} T^{8} - 414 p^{3} T^{9} + 12 p^{5} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \) | 6.7.am_dg_apy_ckh_ahun_wat | |
| 13 | \( 1 - 7 T + 33 T^{2} - 97 T^{3} + 168 T^{4} + 36 p T^{5} - 3629 T^{6} + 36 p^{2} T^{7} + 168 p^{2} T^{8} - 97 p^{3} T^{9} + 33 p^{4} T^{10} - 7 p^{5} T^{11} + p^{6} T^{12} \) | 6.13.ah_bh_adt_gm_sa_afjp | |
| 17 | \( 1 + 5 T + 94 T^{2} + 367 T^{3} + 3775 T^{4} + 11596 T^{5} + 83899 T^{6} + 11596 p T^{7} + 3775 p^{2} T^{8} + 367 p^{3} T^{9} + 94 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} \) | 6.17.f_dq_od_fpf_rea_eucx | |
| 23 | \( 1 + 11 T + 155 T^{2} + 1211 T^{3} + 9484 T^{4} + 54480 T^{5} + 296609 T^{6} + 54480 p T^{7} + 9484 p^{2} T^{8} + 1211 p^{3} T^{9} + 155 p^{4} T^{10} + 11 p^{5} T^{11} + p^{6} T^{12} \) | 6.23.l_fz_bup_oau_dcpk_qwub | |
| 29 | \( 1 - 7 T + 118 T^{2} - 882 T^{3} + 7088 T^{4} - 46543 T^{5} + 260691 T^{6} - 46543 p T^{7} + 7088 p^{2} T^{8} - 882 p^{3} T^{9} + 118 p^{4} T^{10} - 7 p^{5} T^{11} + p^{6} T^{12} \) | 6.29.ah_eo_abhy_kmq_acqwd_ovqp | |
| 31 | \( 1 + 2 T + 138 T^{2} + 233 T^{3} + 8895 T^{4} + 12153 T^{5} + 345457 T^{6} + 12153 p T^{7} + 8895 p^{2} T^{8} + 233 p^{3} T^{9} + 138 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \) | 6.31.c_fi_iz_ned_rzl_trav | |
| 37 | \( 1 - 4 T + 92 T^{2} - 758 T^{3} + 6385 T^{4} - 37895 T^{5} + 339817 T^{6} - 37895 p T^{7} + 6385 p^{2} T^{8} - 758 p^{3} T^{9} + 92 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \) | 6.37.ae_do_abde_jlp_acebn_tirx | |
| 41 | \( 1 + 22 T + 394 T^{2} + 4593 T^{3} + 46769 T^{4} + 370111 T^{5} + 2637945 T^{6} + 370111 p T^{7} + 46769 p^{2} T^{8} + 4593 p^{3} T^{9} + 394 p^{4} T^{10} + 22 p^{5} T^{11} + p^{6} T^{12} \) | 6.41.w_pe_gur_crev_vbnb_fuchl | |
| 43 | \( 1 - 6 T + 169 T^{2} - 741 T^{3} + 13081 T^{4} - 47144 T^{5} + 666465 T^{6} - 47144 p T^{7} + 13081 p^{2} T^{8} - 741 p^{3} T^{9} + 169 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \) | 6.43.ag_gn_abcn_tjd_acrtg_blxxh | |
| 47 | \( 1 + 21 T + 457 T^{2} + 5673 T^{3} + 67688 T^{4} + 569804 T^{5} + 4552109 T^{6} + 569804 p T^{7} + 67688 p^{2} T^{8} + 5673 p^{3} T^{9} + 457 p^{4} T^{10} + 21 p^{5} T^{11} + p^{6} T^{12} \) | 6.47.v_rp_ikf_dwdk_bgkxo_jyzxd | |
| 53 | \( 1 - 5 T + 4 T^{2} - 74 T^{3} + 5602 T^{4} - 19633 T^{5} + 41051 T^{6} - 19633 p T^{7} + 5602 p^{2} T^{8} - 74 p^{3} T^{9} + 4 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} \) | 6.53.af_e_acw_ihm_abdbd_cisx | |
| 59 | \( 1 - 6 T + 136 T^{2} - 235 T^{3} + 9285 T^{4} + 3121 T^{5} + 548151 T^{6} + 3121 p T^{7} + 9285 p^{2} T^{8} - 235 p^{3} T^{9} + 136 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \) | 6.59.ag_fg_ajb_ntd_eqb_bfewt | |
| 61 | \( 1 - 17 T + 337 T^{2} - 3999 T^{3} + 46464 T^{4} - 419686 T^{5} + 3621187 T^{6} - 419686 p T^{7} + 46464 p^{2} T^{8} - 3999 p^{3} T^{9} + 337 p^{4} T^{10} - 17 p^{5} T^{11} + p^{6} T^{12} \) | 6.61.ar_mz_afxv_cqtc_axwvu_hyaul | |
| 67 | \( 1 - 12 T + 375 T^{2} - 3524 T^{3} + 59918 T^{4} - 442732 T^{5} + 5264507 T^{6} - 442732 p T^{7} + 59918 p^{2} T^{8} - 3524 p^{3} T^{9} + 375 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \) | 6.67.am_ol_affo_dkqo_azeye_lnntb | |
| 71 | \( 1 - 2 T + 97 T^{2} + 824 T^{3} + 7015 T^{4} + 42864 T^{5} + 1032469 T^{6} + 42864 p T^{7} + 7015 p^{2} T^{8} + 824 p^{3} T^{9} + 97 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \) | 6.71.ac_dt_bfs_kjv_clkq_cgtij | |
| 73 | \( 1 - 16 T + 479 T^{2} - 5619 T^{3} + 91293 T^{4} - 804404 T^{5} + 9002459 T^{6} - 804404 p T^{7} + 91293 p^{2} T^{8} - 5619 p^{3} T^{9} + 479 p^{4} T^{10} - 16 p^{5} T^{11} + p^{6} T^{12} \) | 6.73.aq_sl_aiid_ffbh_abttyq_tsfgl | |
| 79 | \( 1 - 36 T + 933 T^{2} - 16576 T^{3} + 240332 T^{4} - 2780705 T^{5} + 27253337 T^{6} - 2780705 p T^{7} + 240332 p^{2} T^{8} - 16576 p^{3} T^{9} + 933 p^{4} T^{10} - 36 p^{5} T^{11} + p^{6} T^{12} \) | 6.79.abk_bjx_ayno_nrno_agcfmf_chqpph | |
| 83 | \( 1 - 11 T + 388 T^{2} - 3420 T^{3} + 64529 T^{4} - 474992 T^{5} + 6516945 T^{6} - 474992 p T^{7} + 64529 p^{2} T^{8} - 3420 p^{3} T^{9} + 388 p^{4} T^{10} - 11 p^{5} T^{11} + p^{6} T^{12} \) | 6.83.al_oy_afbo_drlx_abbaqy_ogult | |
| 89 | \( 1 + 3 T + 198 T^{2} + 540 T^{3} + 28790 T^{4} + 89001 T^{5} + 2874203 T^{6} + 89001 p T^{7} + 28790 p^{2} T^{8} + 540 p^{3} T^{9} + 198 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \) | 6.89.d_hq_uu_bqpi_fbrd_ghnuh | |
| 97 | \( 1 - 8 T + 142 T^{2} - 598 T^{3} + 13358 T^{4} - 3754 T^{5} + 420423 T^{6} - 3754 p T^{7} + 13358 p^{2} T^{8} - 598 p^{3} T^{9} + 142 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \) | 6.97.ai_fm_axa_ttu_afok_xxyd | |
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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
−4.00546787483904056247102120651, −3.92517470798340314600628638186, −3.77396391262038933505130638889, −3.72112710478033321076898693243, −3.60120842283819384136291294290, −3.47191145232977925369912898993, −3.35216846464537500078219039206, −3.23599643551008806587051203064, −3.13794660727291709011822543227, −2.94072969961214797213528649342, −2.71028386702752719436175843662, −2.54487301582684007915487332591, −2.25860002206310780172631720144, −2.15614106562475946923251180033, −2.06113688540447651192154437793, −1.91127705649358966812998929856, −1.85183034327710683827677104396, −1.78243043826287142247766089679, −1.71906520012946203938375537698, −1.36778696891081349324554014308, −1.23968474030013135982094864837, −1.09648922269214026256749235675, −0.64227143714677048537178060309, −0.57258764336475257151460717133, −0.49255142401501213920579512427, 0.49255142401501213920579512427, 0.57258764336475257151460717133, 0.64227143714677048537178060309, 1.09648922269214026256749235675, 1.23968474030013135982094864837, 1.36778696891081349324554014308, 1.71906520012946203938375537698, 1.78243043826287142247766089679, 1.85183034327710683827677104396, 1.91127705649358966812998929856, 2.06113688540447651192154437793, 2.15614106562475946923251180033, 2.25860002206310780172631720144, 2.54487301582684007915487332591, 2.71028386702752719436175843662, 2.94072969961214797213528649342, 3.13794660727291709011822543227, 3.23599643551008806587051203064, 3.35216846464537500078219039206, 3.47191145232977925369912898993, 3.60120842283819384136291294290, 3.72112710478033321076898693243, 3.77396391262038933505130638889, 3.92517470798340314600628638186, 4.00546787483904056247102120651
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.